Someone asked me about a confusing critique of the math in On the Historicity of Jesus by a YouTuber who goes by the moniker Fishers of Evidence. I don’t know his alignment in the debate. But he has posted a short eight minute video entitled The Error of Richard Carrier. And this is a good opportunity to teach some simple math you can use in your own probability reasoning.
The central gist of FoE’s argument isn’t well explained, and someone who doesn’t know math won’t know what he’s talking about. Even someone who knows math will be confused, because he confuses a variety of mathematical concepts, such as conflating odds with decimal probabilities. But to steelman his argument as best I can, he means to say this:
When you are going to multiply the probabilities of individual events, in order to determine the probability of the conjunction of all of those events, you can’t just multiply their individual probabilities if those events are in any way dependent on each other.
For example, the independent probability that I am rich may be 1 in 100. The independent probability that I am the winner of a multimillion dollar lottery may be 1 in 1,000,000. But the probability I am both is not 1/100 x 1/1,000,000. Because once I’ve won a multimillion dollar lottery, the probability that I am rich is no longer that independent base rate of 1 in 100. It’s now, let’s say, 2 in 3 (e.g., if only 1 in 3 such persons squander their millions and become poor again, all the rest will be rich, which means 2 in 3 will be, not 1 in 100 of them). And if it’s already known and confirmed that I won such a lottery, then the probability that I am rich is not 2/3 x 1,000,000 either, but in fact essentially just 2 in 3, given that it has to be 2/3 x ~1, the probability I really did win, given overwhelming evidence of it.
Notice this is still all multiplying. That’s important. Because not knowing how to do this correctly with odds ratios, is the Error of FoE. He thinks dependent probabilities can’t be multiplied. He must be high. Because that’s the only way they teach it in math class. What changes is not multiplying, but what probability you put into the multiplication.
Relevance to OHJ
The best way to fix up FoE’s argument is to say this:
When I count up all the evidence bearing on the historicity of Jesus, I multiply a string of probabilities in OHJ, in the form of likelihood ratios, one for each sub-division of evidence, to get a final probability (a final odds ratio) of all that evidence being that way. Which is therefore a probability of that conjunction of evidence. This is to make it easier for critics to examine and challenge or consider each step of reasoning that goes into the conclusion. But FoE claims that in doing this, I am making the same mistake as calculating the probability that I’m both rich and a lottery winner by multiplying 1/100 by 1/1,000,000, when really it should be 2/3 x 1/1 and thus in fact just 2/3. He doesn’t use that example. But it gives us a clearer form of his argument than he presents.
Importantly, he never shows this is the case. Ever. At no point in his video does he ever mention any ratio I assign to any item of evidence, nor does he ever explain how any particular one of those ratios should be changed to account for any dependence effect. This is lazy criticism. As I’ll explain below, there is actually a really good way to test and critique OHJ here, if only a critic actually did it, rather than assume, as if by magic, that any of my ratios is incorrectly assigned, without examining a single one of them. But before I get you there, let’s first circle back and tour his video a bit to see what’s going wrong here.
Did He Just Say That?
FoE correctly describes Bayes’ Theorem, but only in the short form, which is an incomplete formula to a lay observer, who won’t know what’s hidden in it (see Proving History, pp. 69 and 283-84). He says (and agrees) that the probability of historicity given the evidence equals the probability of that evidence given historicity, times the [prior/general] probability of historicity, all divided by the probability of the evidence “whether or not” Jesus existed. He makes an erroneous statement at this point, at timestamp 3:30, when he says, “which of course equals one because we have the evidence that we have.” That’s not how that works. The P(e) that completes the denominator of the short form is not simply “1” (100%) because e “is what we have.” That’s the Fallacy of Foregone Conclusions.
And he seems to know this, so either he misspoke or he is badly confused when applying his own examples to the question of historicity. In his own example of how this works, regarding meningitis, he gets it right. When M is “I have meningitis” and e, the evidence, is a positive test result, then P(M|E) = P(E|M) x P(M), divided by P(E| “whether M or not”). Just P(M) means the base rate, the probability before I get tested, of people like me having meningitis. But P(E| “whether M or not”) is not 1. If it were, that would mean everyone who gets tested for meningitis always gets a positive result, whether they had meningitis or not! That would render such a test completely useless.
And accordingly, FoE, despite saying P(e) is always 1 because “that’s the evidence we have,” correctly shows on screen that this is not true for the meningitis testing. He posits for his example that the probability of e, “a positive test result,” is 0.5% when you don’t have meningitis (aka ~M) and 99.5% if you do have meningitis (aka M); and he posits a base rate of having M of 1 in 1000, which means a prior probability of 0.001, or 0.1%, one tenth of one percent. And that means P(e), the probability of a positive test result “whether or not” you have M, is not 100% but in fact [P(“positive test result for M”|M) x P(base rate of M)] + [P(“positive test result for M”|~M) x P(base rate of ~M)], or, as he correctly shows on screen, (0.995 x 0.001) + (0.005 x 0.999) = (0.000995) + (0.004995) = 0.00599 (or about 0.6%). Which is nowhere near 1 (aka 100%).
One therefore would never say the probability of a positive test result “whether or not you have meningitis” is 100%. Because, in his own example, it’s 0.6%! And FoE seems to know this, as that’s what he shows on the screen. But he doesn’t connect the two examples, so he never notices his mistake in saying P(e) equals 1. The really weird thing here is that if he really thinks P(e), the whole denominator of every Bayesian equation, is always 1 because we always “have the evidence we have,” then you don’t need the denominator at all. The probability of anything is then just P(e|h) x P(h). That should have clued him in that he was making an error in his statement here. I will charitably assume he misspoke and didn’t really mean to say that.
Otherwise, he is confusing two completely different probabilities, and to help anyone else from making that mistake even if he didn’t mean to, remember this:
The probability that the evidence exists given that we are observing it, and the probability that that evidence would exist given that a particular event happened in the past, are not the same probability.
So, for example, if assessing the evidence of a murder, FoE found blood on the accused, he could rightly say “the probability that the accused is bloody, given that I observed and verified the accused is bloody” is 1 (or near enough; there is always some nonzero probability of still being in error about that, but ideally it will be so small a probability we can ignore it). But that doesn’t answer how the blood got there. What we want to know is: What is the probability that the accused would be bloody given that they murdered the victim? And then, what is the probability that the accused would be bloody (= that they will test positive for meningitis / that the accounts of Jesus we have would be written when and as we have them) whether or not they murdered the victim (= whether or not they have meningitis / whether or not Jesus existed)?
That is not going to be 1. The blood could be their own; it doesn’t follow that the blood is from the victim. Or the blood could be there because they tried to rescue the victim, not because they murdered them. It doesn’t even follow that every time someone murders someone, they get or keep the victim’s blood on them. Like a positive meningitis test, many people test positive, whether or not they have meningitis. Moreover, many test negative, whether or not they have it. Similarly, many a biography is written of men, whether or not those men existed. So P(e) is frequently not 1. And in fact whenever it is 1, that means there is no evidence for the hypothesis at all.
You heard me right.
Just as if everyone tested positive for meningitis: that test would give us no evidence whatever whether you had meningitis or not. All we would then have to go on is the base rate. The test contributed nothing. Similarly, if the evidence for Jesus is 100% expected “whether or not Jesus existed,” then there is no evidence for Jesus. All we have to go on then is the base rate. The “evidence” in that case tells us nothing at all about whether he existed or not. Though of course P(e) can never be 1, when P(e|h) is not. P(e) can only ever be 1 when not only P(e|h) is 1, but also P(e|~h) is 1. So a P(e) of 1 in his meningitis example would be impossible anyway, since he assigned a false negative rate of 0.005, which entails a P(E|M) of 0.995, making that the upper limit for P(e), the probability of a positive test “whether or not one has meningitis.” (Try it. Play around with the numbers and see if you can get higher. Remember, P(e) = [P(e|h) x P(h)] + [P(e|~h) x P(~h)], and always P(~h) = 1 – P(h).)
So either FoE means there is no evidence for Jesus. Or he screwed up his math. Or he didn’t mean to say P(e) in the case of Jesus “of course equals one because we have the evidence that we have.” I’ll charitably assume the latter.
Getting to the Point
At timestamp 4:33 Fishers of Evidence says something about generating a prior probability from the Rank-Raglan data, but presses no criticism there I could discern, unless when he says a prior must be generated “without reference to these bodies of evidence” he means the RR data comes circularly from the evidence I put in e, or he means a prior can’t be based on evidence. Both would be incorrect. A prior is always based on evidence—everything you put in b. And you can put anything in b that you exclude from e (and vice versa). As I explicitly state in OHJ, I put the RR data into b, and then properly excluded that data when I examined the remaining content of the Gospels (see my discussion of this mistake when made by Tim Hendrix). But since FoE developed no clear criticism from this, I’ll move on to his central point…
At timestamp 4:55, FoE starts on about the difference between dependent and independent probabilities. He confusingly frames this in terms of summing vs. multiplying probabilities. But he never shows what he means. To understand what he means, see my better steelman summary above. Although he is confusing probability equations using odds ratios and probability equations using percentages. Though even using percentages, you can multiply dependent probabilities. You just have to make sure you are doing it correctly (see my previous discussion of this).
For example, the lottery example: if we didn’t know whether I had won a lottery (or whether I was rich, either), the probability that I’d be both is then indeed 2/3 x 1/1,000,000 (not 2/3 x 1/1; nor, either, 1/100 x 1/1,000,000). In percentage form, that’s (roughly) 0.67 x 0.000001. In notation: P(rich & lottery) = P(rich|lottery) x P(lottery). Note that multiplying is perfectly fine. As long as I multiply P(lottery) by P(rich|lottery) and not P(rich). Because P(rich|lottery) is 2/3, while P(rich) is 1/100. The latter would be incorrect if we intended to derive the total probability by multiplication. But we can still derive the total probability by multiplication. You just have to use the correct probability. FoE seems not to know this.
FoE thus commits two errors here. First, he incorrectly claims you can’t derive a total probability from a series of dependent probabilities by multiplication. To the contrary, that is in fact how you usually do it. And second, he never identifies any examples of dependent probabilities in my series of estimates in OHJ. He correctly notes that there are causal connections between groups of evidence, e.g. the Epistles had causal effects on the Gospels, and the Gospels had causal effects on Acts and the Extrabiblical evidence. But he doesn’t identify a single probability I produce that is affected by these causal relations. For example, in Acts, I don’t just come up with “a probability of Acts given historicity/ahistoricity.” I actually divide Acts into three pieces of evidence, none of which reflect the dependency relations FoE alleges, except one, and for that I give a dependent probability, exactly as required by the math (see below). In other words, it does not appear FoE actually examined any of my estimates for any actual dependence relations. If he did, he would have discovered (a) many don’t have such relations to account for and (b) the ones that do, I already accounted for!
For example, Acts is dependent on the Gospels only for the material it shares with the Gospels, which I don’t explicitly assign any probability to! I assume that content is 100% expected on either theory, because it’s 100% expected given “the Gospels exist,” and the Gospels exist, regardless of whether Jesus did. I assign it odds of 1/1. In other words, no effect. That’s a dependent probability, like the 2/3 chance of being rich given having won a lottery. FoE seems not to have noticed. His criticism is thus completely inapplicable here.
Meanwhile, for material pertaining to the history of the church not recorded in the Gospels, Acts is independent of the Gospels. And that’s the only material I assign a probability to. Although that material is causally dependent on the Epistles, I already account for that in my estimations of the likelihoods of those contents of Acts given what’s in the Epistles. Indeed, the fact that Acts definitely used the Epistles as a source entails it deliberately contradicts the Epistles when indeed it frequently does so, which entails the probability that Acts is a reliable source is near zero. Although that isn’t a probability I use in OHJ. I only estimate how likely it is that Acts would lack the things it does, even given the fact that its author knew and was influenced by (and aiming even to contradict) the Epistles. So the probabilities I put in are already dependent probabilities as required. FoE’s criticism is again wholly inapplicable.
And so on throughout.
It’s especially weird that FoE doesn’t know you can derive total probabilities with dependent probabilities by multiplication. Because he briefly references independent probability calculations in gambling as an example of doing it wrong, and even shows a dependent probability calculation in there: correctly, as a multiplication. Though he doesn’t discuss this. So what is going on here? Either he doesn’t really know how to do that math, or he is actually trying to deceive his viewers by omitting his explanation of it, since that would refute his entire claim that I’m supposed to add and not multiply (a concept he never illustrates with any example).
Take poker. Suppose we are asking about the conjunction of two events, drawing a king from a full poker deck and immediately drawing a second king from that same deck. The probability is not, indeed, as FoE implies, 4/52 x 4/52 (there being 4 kings in a deck and 52 cards in a deck), because the second probability is indeed a dependent probability, the very kind of thing FoE is talking about: the second probability is dependent on what was drawn first. If the first draw was a king, then the probability of two kings drawn is 4/52 x 3/51 (since now, after the first king is drawn, there are only 3 kings and 51 cards), which is 12/2652, or 0.0045 (rounded), aka 0.45%, or a nearly half of one percent chance. Notice we are still multiplying, not adding. Contrary to what Fishers of Evidence claims. And we can do this for the evidence of Jesus as well. And that’s exactly what I do. All throughout OHJ.
For example, what is the probability Acts would have faked up the trial speeches for Paul to match the street sermons, and thus included references to a historical Jesus, instead of as we have it now, trial speeches that bizarrely omit any references to a historical Jesus, and street sermons that include such references? This would be, one might argue, a probability dependent on the existence of Luke’s Gospel. In other words, Luke certainly knows the material that evinces a historical Jesus, and it could have caused him to fake evidence everywhere in Acts. But in this case, he didn’t. He kept this bizarre incongruity between Paul’s trial speeches and street sermons (OHJ, pp. 375-80). That fact by itself is more likely if Jesus didn’t exist than if he did—though as I argue, the difference is extremely small: a likelihood ratio of only 9/10 against historicity on the a fortiori side. And that’s what that ratio would be even considering the fact that Acts was written by someone who knew the Gospel material. In other words, this 9/10 is the same figure as the 3/51 in the poker example above, and the 2/3 in the lottery example. It is thus not invalid. It’s perfectly correct. FoE’s argument has no relevance here.
My taking this dependence into account is even explicit: I estimate that even if Jesus didn’t exist, the fact that Luke would add a reference to historicity in Stephen’s speech given that Luke wrote the Gospel under his name is just as likely as that such a reference would be there if Jesus did exist. So the ratio is 1/1. Which means it has no effect. Stephen’s speech argues neither for nor against historicity. As I wrote: “when Luke inserts into Stephen’s speech a brief reference to the historicity of Jesus…this could obviously be Luke importing his own narrative assumptions,” among other possibilities I enumerate (OHJ, p. 383). Thus, I’m fully taking into account the fact that this is a dependent probability. And my incorporation of it is mathematically correct. I don’t literally assign either outcome a 100% probability (see OHJ, p. 605; cf. pp. 288-89, n. 18 and p. 357, n. 122); it could be 50% likely Luke would fake evidence and 50% likely he’d include a reference if Jesus really existed. What I estimate is that whatever those two probabilities may be, they are going to be so nearly the same that there is no measurable difference between them, and therefore Stephen’s speech argues for neither theory. And this is so even given Acts’ reliance on the Gospels and Epistles. So their being dependent probabilities makes no difference to the math. Contrary to what FoE claims.
I similarly adduce that the Gospels, deriving material from the Epistles (as I explicitly admit they do in OHJ), have the same probability of including Epistle-connected material whether Jesus existed or not, because of their dependence on the Epistles for that material. In other words, the Gospels cannot corroborate that material, if they are not independent of it. Thus, again, I am doing exactly what Fishers of Evidence says I should: accounting for the interdependence of some of the evidence when I assign probabilities.
All my probabilities thus are just like the poker draw probabilities: they are already factoring in the causal relations among the different categories of evidence. This is indeed so explicit in my discussion of the extrabiblical evidence (in Ch. 8) that it should have been obvious to FoE that that is what I was doing. My finding that no external sources corroborate the Gospels is derived from their dependence on those Gospels. Thus, that Tacitus should mention a Gospel claim about Jesus (if in fact he ever did) is already 100% expected on the existence of the Gospels, regardless of whether Jesus existed or not. That reference in Tacitus thus has no effect on our final probability of historicity. That’s how dependent probability works. And ironically, here it’s Christian apologists who typically don’t grasp the point that Fishers of Evidence is making: that the probability the extrabiblical sources would mention Jesus, even if he didn’t exist, is dependent on the Gospels having already done so (and their Christian informants subsequently relying on the Gospels, as we know they did).
If Fishers of Evidence thinks I have mis-estimated a conditional probability anywhere in OHJ, he should explain which probability, and what the correct value of it is and why. In other words, if dependence changes anything, he needs to identify what it changes, and by how much. You can’t just say it will change something. Because you don’t know. In any given case, I might already be factoring in the dependence, just as I have demonstrated I did in several cases here, and just as I did here when producing the multiplication (not addition) that correctly predicts the probability of drawing a pair of kings straight off a full poker deck.
So, am I factoring in dependence correctly, or am I not? If you think not, show me where, and why you think I’ve inadequately accounted for the effect of dependence. If you can’t show me where, then you don’t know I did. Therefore, you can’t claim I did.
Fishers of Evidence never shows I did. Anywhere. He shows no sign in fact of even knowing I did. Or of even checking. It doesn’t even seem apparent to him that he is supposed to check.
How Did He Figure That?
FoE spends some time on the famous Roy Meadow example of failing to correctly use the dependent rather than the independent probability in his evidence calculations. But this has no relevant parallel in OHJ. And Fishers of Evidence shows no parallel. If he could, that would be something, and indeed something I’d most like to see found, if any such effect is there.
But moving on to his conclusion, at timestamp 8:40, Fishers of Evidence says he recalculated my result by using a sums method. Or more precisely:
I have recalculated Carrier’s probabilities arithmetically. There are different ways of doing this, by using different weightings for example. But I have simply weighted his groups equally. My arithmetic calculation gives an upper bound probability of historicity of 46% and a lower bound of 18%.
He doesn’t show his calculation or even explain what he means. How he derived these numbers is mysterious. I try to imagine how he’d do it with a poker deck, and can’t think of any way to “arithmetically” get any result other than the same one anyone would get with multiplication. So this sounds like taking that correct 4/52 x 3/51 calculation and “redoing” it somehow to get a different result: whatever he is doing, it can’t be mathematically valid.
At best he must be assuming—without ever confirming or observing or demonstrating the fact—that I didn’t take into account dependence when I assigned my probabilities in each category—even though I explicitly state I am assuming those dependencies, and even rely on that fact repeatedly to get the results I do (like the 1/1 result for Stephen’s speech). He seems to think I was doing 4/52 x 4/52. But he never explains where I do that. And I don’t. I’m doing 4/52 x 3/51. Every single time. My math is correct. I have no idea what his math is. As he doesn’t show his math, there’s no way to be sure what he’s doing wrong. He claims to be summing weighted probabilities, but I used ratios, not probabilities in the way he describes.
Indeed, as best I can tell, whatever his alternative method, he says he assumed an equal weighting for every item of evidence’s dependency. Which implausibly assumes the effect of every item of evidence on every other (such as the weirdness of the trial speeches in Acts on the omission of James from the entire public history) is exactly the same. He just assumes that. Without argument. But even before attempting an argument, I can already tell you, it would be cosmically bizarre for that to actually be the case. And it’s clear FoE has no argument for it being the case, that he did not even consider the matter. Which renders his alternative result completely meaningless. It has no connection with any argument for dependence or causality among the items of evidence I demarcated. His own result is thus, in fact, far worse than even he thinks mine is.
Using Acts to Illustrate the Point
Let’s talk about how mathematicians actually combine dependent probabilities. And what I actually did in OHJ.
I’ve already discussed above how I already make estimates of dependent probabilities in OHJ with respect to Acts. More specifically, I break that down into three things: the missing family (and other personages who should have had a causal effect on the history of the church); the oddness of the trial speeches (which read as if there were no historical but only a cosmic Jesus); and the rest of the content of Acts (OHJ, p. 386). I find that the rest of the content of Acts is just as expected on either historicity or not, and thus assign a ratio of 1/1. This is actually already a dependent probability: because Luke is being caused by the Gospels to repeat Gospel-derived historicity material, the dependent probability that he would do so even if Jesus didn’t exist is 1 (at least, as near to one as makes all odds), once we grant the Gospels causally influenced him. That’s exactly like realizing the probability of drawing a second king is 3/51 and not 4/52. There is no other way to do this math. Any other method he uses, if it gets him a different result than multiplying 4/52 by 3/51 in the poker case, he’s doing it wrong. And likewise if he makes the same mistake with my assignment of 1/1 here.
But since 1/1 multiplied by anything has no effect, we can ignore that now and look at the other two ratios. For those other two items in Acts I assign on the a fortiori side a likelihood of 4/5 for the missing people and 9/10 for the weird trial speeches. (Which are already super weak evidence ratios, BTW…good evidence should weigh 1/4 or 4/1 or more or even a million or a billion to one, so I am not making anything like a strong claim of effect here. But moving back to the point at hand…) For FoE to claim that these should not be multiplied against each other (4/5 x 9/10), he needs to show two things: that those two things are dependent on each other; and that that dependence changes the ratios to something other than I assigned. For example, imagine Luke’s not thinking to ever mention James ever led the Church, in a supposedly researched history of that Church, is somehow caused by his trial speeches also not seeming to know about a historical Jesus, such that wherever there is the latter, there will always be the former. In that case, the dependent probability of the omission of James is 1/1, and it no longer has any effect. We are then left with simply the 9/10 oddness of the trial speeches. Just as if drawing a king from the deck, at odds of 4/52, causes the next draw always to be a king, such that the probability of drawing two kings right off the top is simply 4/52, not 4/52 x 3/51.
Unfortunately, Fishers of Evidence never does this. For anything in OHJ. He doesn’t explain how, for example, the trial speeches omitting a historical Jesus, affects the probability of James being omitted from the entire history of the church. In what way are they dependent on each other? And even if we can come up with any plausible dependence (and they have to be plausible; wildly implausible dependencies have too low a probability to show up in the math at the resolution I employ), does it change my estimates at all? Is it really credible to say the weirdness of the trial speeches would always cause an omission of James? Certainly not. If they have any effect on each other at all, it’s extremely small. There are too many ways either could happen independently of the other. So is that tiny effect large enough to make the conjunction of both together any different from 4/5 x 9/10? A difference, that is, that shows up at that resolution? (For example, a difference of a millionth would be so small it would disappear when rounding to the nearest whole percentage point.) Such causal dependencies thus have no visible effect. (And even if they do, the effect may be in entirely the other direction: see comment.)
So if Fishers of Evidence wants to say it would have a visible effect, he has to show how it changes that conjunction’s probability (or the probability of any actual conjunction I score in OHJ), and to what—that is, what it changes the probability to. I fully welcome such revisions. If anyone can show that the probabilities I assign in OHJ should be different because of causal connections I overlooked or wrongly estimated the effects of, that would be progress, and well worth issuing a corrected edition. But you have to show that, before you can claim any such error exists. Otherwise, my 4/5 for the omission of Jesus’s family in Acts given the oddness, at the same time, of the trial speeches in Acts, is as valid as that 3/51 in the poker example. Fishers of Evidence has not shown otherwise, for any ratio I propose, anywhere in OHJ.
Conclusion
Fishers of Evidence also throws in at the end the completely undefended claim that, even apart from the dependency issue he thinks I didn’t account for (even though I did), my probability assignments were arbitrary. He doesn’t explain why they are arbitrary. The cases I make for them are far from arbitrary. One should be able to refute those cases and make a case for a different estimate if in fact they were arbitrary. So calling them arbitrary is just a lame way to avoid having to address any of the arguments and evidence presented for every single estimate I include. That’s not rational argument. That’s hiding from rational argument. It’s like telling yourself the sky isn’t blue and hoping that mantra convinces you so you never have to actually argue the point.
But the overall gist of his video is culminated at timestamp 10:06, where he claims weighing evidence is always a matter of adding, not multiplying, probabilities. That’s false, as my example for the poker deck shows: dependent probabilities are still multiplied, not added. He seems not to understand the fact that when using the Odds Form, we are dealing with ratios, which are always multiplied, never added. As with drawing two kings at poker: the causal effect of drawing one king, is reflected not by addition, but by changing the probability multiplied in. It’s not 4/52 x 4/52, yes. But it is 4/52 x 3/51. It’s still multiplication.
Apart from that error, if FoE thinks the causal relationship between any two pieces of evidence I assign likelihood ratios to would make the outcome different than I estimate, he has to show what the new multiplication is, not play around with addition. For example, if I said the odds of drawing two kings straight off were 4/52 x 4/52, it would be mathematically false to say I’m wrong because I need to add, not multiply. A mathematically correct criticism would agree we need to multiply, but that the dependence effect changes the second multiplier from 4/52 to 3/51, and that the correct result comes not from any addition, but from multiplying the correct ratios: 4/52 and 3/51. So what ratios does he propose we should multiply, other than mine? He never once says. Nor does he give any reason to believe the ones I propose are incorrect. Even granting every plausible dependence among every item of evidence in my array, they may all be just as correct as the 3/51 we must multiply by 4/52 to get the correct result at poker. It’s quite clear he doesn’t understand that, and has nothing to say about the actual, correct way to correct my math—if anything in my math is incorrect.
But indeed, if anyone can find a dependence relation between any of my probabilities that I incorrectly or don’t account for, and it’s enough to make at least a percentile of difference in the final result, let’s hear about that! Because that could be an important correction needed to OHJ. But until you can show that, you don’t have any criticism to make here.
FoE is correctly doing a calculation of a dependent probability at 5:22:
“The chance of drawing two hearts from a deck of cards is 13/52 times 12/51”
I can’t make any sense of the claim that probabilities should be added either, I can’t imagine a way to make this valid, but I’d like to see FoE’s calculations.
I also think he is generally confused when it comes to dependent probabilities, he seems to believe that any sort of connection between two pieces of evidence would make them statistically dependent, which is not generally true.
Good catch. I missed his on-screen showing of a dependent probability multiplication. He doesn’t discuss it as such, so I overlooked it. I’ve emended my article to reflect this point. I’m astonished. Did he not know he did that?
And yeah: addition is typically only for summing probabilities with the equivalent of an “or” operator (either x happened or z happened; what then is the probability of y given either x or z), which is exactly what the denominator of a standard Bayesian equation is. Addition is not used for calculating the probability of the equivalent of an “and” operator (a conjunction of events). Sure, all multiplication is the iteration of addition (e.g. 1/2 x 1/3 = 1/6 because 1/(2+2+2)), but that’s the same as multiplication. So he shouldn’t be getting a different result with it.
Note: It just occurs to me there is another point I should make.
In the paragraph where I discuss the point, “Is it really credible to say the weirdness of the trial speeches would always cause an omission of James?” and I answer “Certainly not,” one could propose a scenario whereby Acts is using a source (let’s say, a pre-Acts history of the church) that omits any knowledge of a historical Jesus (hence simultaneously causing an omission of James from the events of the church and the omission of an earthly Jesus from the trial speeches of Paul). That would get one of those to be 100% expected on the occurrence of the other.
However, by positing that scenario, you just changed the probability substantially—in the other direction. Because the likelihood now that the preceding history of the church used by Acts lacked a historical Jesus, given that Jesus existed, is far less than 9/10. Indeed, it’s well below 1/4, even a fortiori. In other words, if Acts added a historical Jesus to an earlier history that lacked one, it is all but certain Jesus never existed, and a historical Jesus was later fabricated—by authors doing exactly what this scenario entails Luke did.
So you need to be careful in what you are proposing to create dependent probabilities among the evidence.
If you want to tie two pieces of evidence together by some hypothesis of how they both came to match the appearance of a “no Jesus” result, that hypothesis cannot be presumed without substantially reducing the probability Jesus existed.
It’s a Catch-22 for any historicity apologist. What you need is a scenario whereby that conjunction is expected even if Jesus existed. But then you have to compare the prior probability of that scenario, with the prior probability of the alternative (e.g. a pre-Acts that lacked a historical Jesus). The latter might actually be more likely. After all, how would some proposed lost evidence of a historical Jesus simultaneously cause Acts to omit both James (from history) and a historical Jesus (from Paul’s trial speeches)? Far more likely that would be caused by evidence against the historicity of Jesus, not by evidence for a historical Jesus. A pre-Acts that omits a historical Jesus would easily explain that conjunction. But you would have to come up with a pretty convoluted theory to get, say, a pre-Acts that includes a historical Jesus yet still causes him to disappear in our Acts at precisely those two points (the role and existence of James altogether, and the content of Paul’s trial speeches).
And here is where addition comes in: since the probability that either kind of pre-Acts existed is not 100%, the probability of each existing must be multiplied by the probability of the resulting conjunction of evidence, and then those two products added together, to get what you need for the final Bayesian calculation. The end result is not likely to favor historicity. At least, not any more than my estimates already do. And to gainsay that, you have to actually do the math. Correctly. Otherwise you can’t claim to know it will come out differently.
Oh Really, Dr Carrier [the following text also posted here]
In mathematics, we often take common words and give them highly specific meanings consistent with their use in mathematical reasoning with its high degree of rigour compared to other subjects.
Dependent, independent, hard assumption, softer assumption, etc. are such words. When we talk about independence being a hard assumption in probability theory we mean that probabilities are provably independent of each other. Provable independence is mainly restricted to theoretical models, such as the simple dice and playing card models of basic probability theory. When we set up these models we specify rules such as the dice are not loaded and the result a dice roll has no influence on subsequent rolls, and card decks are well shuffled.
So if we draw two cards from a deck, then the probability that the first card is a King is 4/52. The probability that the second card is a King is dependent on what the first card was. If the first card was a King it is 3/51. It was not it is 4/51. The probability of drawing two kings without adjusting for the first card is strictly 4/52×3/51. The probability of drawing a King is only a question of how many kings there are in the deck divided by the total number of cards. We are tacitly assuming that there is no other influence. This assumption may be wrong. It may be a deck of kings. But it is an assumption that we make and if that assumption is wrong we’re going to be way out with our probabilities.
In the real life forms of these models, when highly improbable events happen a natural reaction is to suspect that the rule of independence doesn’t apply. For example, if we roll six consecutive sixes we will suspect that the dice is loaded. If we draw a highly improbable hand from a deck of cards we will suspect that it has not been shuffled properly. Going beyond these into the more indeterminate situations of healthcare, psychology, economics or history, it is rare for independence of probabilistic variables to be provable. In certain situations we do assume independence. Then great care is taken to assess probabilities objectively and there is good reason to believe results are independent. An example is football pools where the results of simultaneous games are modelled, and geometric combinations are used. Often, though, when geometric calculations produce extreme probabilities we suspect a key assumption is not met. Roy Meadow’s 73,000,000 to 1 chance was suspiciously low and similarly the lower bound of Carrier’s probability range of 0.12% is also suspiciously low.
In reality we approach claims based on geometric probability combinations with suspicion unless there is positive reason to believe that there is no third factor that influences the probabilities we are considering similarly. In other words, we have to be confident that the dice is not loaded and that the cards are well shuffled, and in Roy Meadow’s case that cot death does not run in families, and in Carrier’s case that there is no factor influencing the probability estimates that trends them in the same direction.
For most real-life situations we cannot assume independence and therefore non-independence becomes the default position which we assume unless there is a compelling reason not to.
Can dependent probabilities be combined geometrically?
Absolutely. In the card example above the probability of drawing a second King is dependent on whether the first card drawn was a King and this can be fairly accommodated using geometric combination. But it depends on knowing the dependency exactly. It becomes dangerous when the dependence is not known exactly as in the case of Roy Meadow where the evidence didn’t seem to suggest that cot death ran families, but to assume that was a major error. The case of Jesus is totally different from the dice and gambling models of probability theory. In those we can calculate theoretical probabilities exactly. We just can’t do that with Jesus.
Dr Carrier’s probabilities are not like poker draw probabilities. Those probabilities are deterministically defined in the rules of the game and can be calculated exactly. Everyone will come up with the same result. This is not true of the probabilities relating to Jesus.
This point is readily illustrated from the current debate. Johan Rönnblom agrees with me that the chance of drawing two hearts from a deck of cards is 13/52×12/51. Of course he agrees with me. It’s true and anybody examining the same problem will come up with the same result. This one is solid enough to use in geometric probability combinations.
On the other hand in Richard Carrier’s comment of 19 March 2017 11:55 am second paragraph he says. “Because the likelihood now that the preceding history of the church used by Acts lacked a historical Jesus, given that Jesus existed, is far less than 9/10. Indeed, it is well below 1/4”. Is that so? Is it below 1/5 or 1/6. What is it? 1/10? 1/8? And if one why not the other? And do all observers agree on this? Are assertions of this kind strong enough to use in geometric probability models? I think not.
The reason that this is so important is that for geometric probability combinations, any errors in probability estimates will be multiplied and have a large effect on the outcome. If we estimate the chance of rolling a 6 as 1/6 when in reality for the dice at hand it is 1/5, our probability of rolling 6 sixes will be out by a factor of 3. This means that we must know the probabilities with a high degree of accuracy in order to be confident in the result. Relatively small errors in the probabilities we estimate, that we are unaware of, can have a large bearing on the result as was the case for Roy Meadow.
So in probability the bar is high for the accuracy of probability estimates as it is for independence. So high it can never be met with a topic such as historicity v mythicism and that means that geometric probability models should not be used.
The error that Carrier made in his books, and his rebuttal to my video, both indicate that he doesn’t get it. He is using the term independent in the way that a historian would not a mathematician. The lack of objectivity in his probability estimates, his attempt to arbitrarily compensate for what he sees as dependency, and his failure to appreciate that independence can never be established to the satisfaction of the mathematical definition for this kind of problem in history, all point to his not understanding what mathematicians mean by the term.
Are Carrier’s probability estimates arbitrary?
This is not a criticism but an inevitable consequence of the system under study. It is not possible to calculate precise probabilities and the estimates used by Carrier necessarily involve a degree of judgement. That means the estimates will vary from one person to another and that means that they are arbitrary. That is what the word means. If a staunch historicist addressed the same data, it is possible that they would revise their views and become a mythicist but it is also possible that they would come up with different probabilities.
Carrier asks:
“So, am I factoring in dependence correctly, or am I not? If you think not, show me where, and why you think I’ve inadequately accounted for the effect of dependence. If you can’t show me where, then you don’t know I did. Therefore, you can’t claim I did.
Carrier’s factoring of dependence is no worse than anyone else’s would be, his error is not that he has done a poor job of it. His error is failing to appreciate that no one, not he or I or anyone else, can do it with the rigor necessary to satisfy the assumptions required to justify geometric probability models.
Carrier’s model is simply too vulnerable to factors that he and in particular his readers doesn’t know about. He has not made any attempt to address even the usual suspects. For example, given that the probability estimates required a degree of judgement, and the same person made these judgements, how do we know that these judgements were not being systematically influenced towards historicity or mysticism? Further, was an effective Chinese Wall system employed between the probability estimates and the final output of the model? Those are modern unknowns. There are ancient ones too. Who amongst ancient people believed historicity, who didn’t believe it, who didn’t know or care and when? We don’t know this for sure and it is not the same thing is historicity but it would be expected to influence all the evidence we have.
Other Points
Should the probability of having the evidence we have be 1?
The calculations in Carrier’s rebuttal of my video are correct. Disputes about mathematical modelling invariably turn not on the calculations themselves but the interface between the model and reality. So it is here. The two examples in play are my meningitis example and Carrier’s murder suspect. In both of these cases we can make a meaningful estimate of the prior probability. In the case of meningitis, it is the probability of having meningitis without knowledge of the test result. In the case of the murder suspect, we can formulate Bayes’ theorem as: the probability that the accused is guilty given that he has blood on him is equal to the probability of having blood on him given that he is guilty times the probability of being guilty all over the probability of having blood on him.
The probability of having blood on him is a meaningful number that we can estimate. We can observe what proportion of people have blood on then and therefore decide how likely it is that that he would turn up with blood on him without reference to the rest of the evidence.
Similarly in my meningitis example. The likelihood of having meningitis is something we can observe from epidemiological data and estimate without reference to the test result.
In the case of Jesus, the probability of historicity given the evidence we have, is equal to the probability of the evidence we have given historicity times the probability of historicity all over the probability of having the evidence that we have.
In this case the probability of having the evidence that we have is not something that we can meaningfully estimate. It is not analogous to the meningitis and blood examples.
There are things that we can meaningfully estimate. These include the probability of having the evidence given historicity, also the probably of having the evidence given not historicity but add them together in you end up with the same problem.
The estimate can be made meaningful but to do so it becomes trivial. We could construct a hypothetical situation with a number of different ancient histories and consider how many of them gave us the evidence we actually have. But as we don’t have these other ancient histories the matter gets us nowhere.
Carrier’s misunderstanding is illustrated by him emblazoning this statement:
The probability that the evidence exists given that we are observing it, and the probability that the evidence would exist given that a particular event happened in the past, are not the same probability.
This is true, but neither of these probabilities are the probability at issue. What is at issue is the probability that the evidence would exist irrespective of what happened in the past.
I assume we can agree that the probability that the evidence exists given that we are observing it is 1. But the probability that the evidence would exist irrespective of what happened in past is either 1 or not meaningful. Specifically what does it mean to say that the probability that: [the evidence would exist given historicity + the evidence would exist given not historicity] is less than 1? Or that the probability that the evidence would exist irrespective of what happened in past is less than 1?
Carrier notes that if the denominator is one it defeats the purpose of using Bayes theorem at all. Yes, it does!
How did I figure that?
Very simple really. I converted Dr Carrier’s odds into probabilities for and against historicity for the four bodies of evidence and for the Rank Raglan hero class, and then simply took the average of the resulting 5 probabilities. So does this give a formal probability? No it doesn’t. And is it open to challenge? It certainly is. The method has a weakness that a compelling argument will be significantly diluted by trivial arguments. An improvement would be a points-based system where an argument favouring one side is allotted that side and given a weighting according to its strength. This is an attempt to model the decision-making process, as it occurs in scholarly discourse, with numbers. The reason for using it has nothing to do with the internal mechanics of the model, it has to do with the ubiquitous vulnerability of all mathematical models of the real world, and that is they may not describe the real processes accurately. My contention is that the geometric models so familiar from probability theory get the wrong result because of their failure at this interface, not because of any internal inconsistency.
Weighing evidence
Carrier is seduced by the elegance of probability theory and applies it without sufficient regard to whether his model fits the purpose. If you really want to develop a mathematical model to suit this question then what you need to do is to describe the actual process of decision-making by scholars in this field in mathematical language. That’s not going to lead to a model with the same elegance as theoretical probability but at least it will have a chance of describing the situation fairly. The way to do this is simply to weigh the evidence. There are multiple arguments in favour one side or the other. We need a method of assessing the strength of these arguments and comparing the two sides. The most obvious way of doing this is to collect all the arguments that favour of mythicism on one side and all those that favour historicity on the other. A necessary elaboration would be to account for the fact that strong arguments carry more weight than weak ones and therefore a point system springs to mind where strong arguments are allotted more points than weak ones. Then in the final analysis the points for historicity are added up as are the points for mythicism. Multiplying points is in no way analogous to the scholastic process.
This is not the method I use which was simply to average the probabilities Carrier calculated from his five groups. I did this because it was the simplest way of arithmetically combining the probabilities that he estimated.
Attractive theories are often shot down by inconvenient truths and here, as in many cases, the exponents of those theories recognise their limitations belatedly if at all. Review of previous discourse on this matter shows that I’m not the first person to point this weakness out to Carrier but he sticks to his position doggedly.
Odds v probabilities
Odds and odds ratios are extensively used in statistics and the more elaborate models are essentially geometric in nature and are not entirely convertible into arithmetic form. The Bayesian odds model Carrier uses can be expressed with exact equivalence as a probability model. His argument that I’m confusing to different things is fallacious. I chose to express his concepts with the probability version of Bayes’ theorem purely because I find this to be more accessible to people not accustomed to probability calculations.
The version of Bayes’ theorem I expressed on screen is Bayes’ theorem. It can of course be made longer by expanding the various terms but it is in no way incomplete. There is nothing that this version hides from the lay observer. There is a word that describes the hidden subtleties that can only be appreciated by the initiate using an expanded version with extra variables and esoteric symbology and that word is bamboozle.
We all know how this is going to go. Carrier is not going to back down. I know this for three reasons. If he understood the issue he would not have made the error in the first place. Secondly I’m not the first person to point this problem out to him and he has not in any way altered his position. Thirdly he has too much invested in it and can’t afford to concede the point. His acolytes will jump to his defence and we won’t see much discussion on where geometric models should and should not be used.
But consider this. The next time you’re making a complex decision, with multiple input considerations, which method are you going to use? Carrier’s or mine?
Fishers of Evidence
Holy crap. You were averaging the probabilities!? And when you did that you even just averaged the prior probability in with the likelihoods??
I could drop the mic right there.
I’ll let all the mathematicians of the world explain why you don’t know what you are doing.
You also betray your ignorance of the fact that P(e) = [P(e|h x P(h)] + [P(e|~h x P(~h)]. You claim “the probability that the evidence would exist irrespective of what happened in past is either 1 or not meaningful” is simply not even remotely true. The probability that the evidence would exist irrespective of what happened is [P(e|h x P(h)] + [P(e|~h x P(~h)]. Which can never be 1, unless both P(e|h) and P(e~h) are 1. Which would entail there is no evidence for the existence of Jesus. Exactly as I explain in my article. That you don’t know that, proves you don’t know Bayes’ Theorem.
As to the rest:
I actually give data-based reasons for all my estimates of probability. I account for the uncertainties in the data, and possible biases and subjectivities, with margins of error so wide it would be unreasonable to believe the probabilities deviate further. If you wish to gainsay that, you have to actually make a case. You can’t just insist the probabilities are different, and then simultaneously insist no one can know what the probabilities are. If you can never show how they could be different, you can’t claim to know we aren’t using reasonable margins.
And even if we granted no estimates of probability were defensible in this case, you then can’t claim to know the historicity of Jesus is probable—if you say no one can assign and calculate any probabilities of anything pertaining to answering that question, then you are saying no one can know if Jesus existed.
So the only way to get a different result than “historicity agnosticism” (which means fully doubting the existence of Jesus) is to say what you think these probabilities are, and explain why that’s what you think they are, and not something else. You can’t say “no one can know what they are, therefore Jesus probably existed.” That’s full on illogical.
So either you are just arguing for agnosticism about the existence of Jesus, by saying no one can ever know anything about the past (or at least the ancient past of human events). Or you are trying to argue for a different probability than I am, without giving a single argument for a single instance of that being true, while simultaneously (and illogically) insisting you can’t know that. I have given reasons in OHJ. For every single probability I assign. Good reasons. And indeed, good reasons for doubting it’s reasonable to put the probabilities any further out than I do. You are refusing to address any of those reasons. Nor, in any single case, are you presenting any comparable reasons for putting the probabilities any further out than I do!
You aren’t discussing the actual probabilities of anything or why they should be different. You are illogically trying to get a probability out of multiplying arbitrarily invented “points” (??), and then you claim using actual probability mathematics is less valid, and more arbitrary? And then you are proposing a mathematically incorrect procedure of averaging the probabilities of individual events to determine the total probability of a conjunction of those events! And averaging the prior in as if it were another event in the conjunction! And then claiming P(e) is 1, which entails no evidence exists for Jesus!
It’s clear you have no idea what you are doing. It’s also clear by your own assertions that you do not know, and indeed are declaring you cannot know, whether Jesus probably existed.
Pick a lane.
Then learn how to actually do the math.
Then try actually critiquing the actual evidence and arguments presented in OHJ for the assigned probabilities.
Anything else is a waste of everyone’s time.
I’m assuming this was a typographical error, because if it was not then we do have a problem:
“The probability that the evidence would exist irrespective of what happened is [P(e|h x P(h)] + [P(e|~h x P(~h)]. Which can never be 1, unless both P(e|h) and P(e~h) are 1.”
What I presume you mean is this sum can never be 1 unless P(e|h) and P(e~h) add up to 1. If they were both 1 then it would add up to 2.
I guess we can all see what it means for them to add up to 1. It means that the probability if the evidence given historicity plus the probability of the evidence given not historicity adds up to a certainty, in other words the probability of the evidence irrespective of what happened in the past adds up to a certainty.
The question I will put again is what is your understanding of the meaning of the statement “The probability of the evidence irrespective of what happened in the past adds up to less than 1”. Or to put it another way what is the meaning of (the probability that the evidence exist given that we are observing it) minus (the probability of the evidence irrespective of what happened in the past).
What I presume you mean is this sum can never be 1 unless P(e|h) and P(e~h) add up to 1. If they were both 1 then it would add up to 2.
That’s also true (P(e|h) and P(e~h) add up to 1).
But the point of [P(e|h) x P(h)] + [P(e|~h) x P(~h)] is to always get you a total probability (not a P(e) of 200%, but at most of 100%).
For example, suppose P(h) is 0.1. Then P(~h) is 0.9. And suppose P(e|h) and P(e|~h) are both 1 (in which case, the evidence is 100% expected whether h happened or not; which means e cannot be evidence for h).
Then:
[P(e|h) x P(h)] + [P(e|~h) x P(~h)] = [1 x 0.1] + [1 x 0.9] = 0.1 + 0.9 = 1
See how that works?
Now try any other numbers. You’ll never get higher than 1.
The question I will put again is what is your understanding of the meaning of the statement “The probability of the evidence irrespective of what happened in the past adds up to less than 1”. Or to put it another way what is the meaning of (the probability that the evidence exist given that we are observing it) minus (the probability of the evidence irrespective of what happened in the past).
That’s not an intelligible question.
The probability that the evidence exists given that we are observing it, has nothing whatever to do with the probability that the evidence would exist given that h happened. What on earth you mean by subtracting the one from the other, I have no idea.
What makes e evidence for h, is when the probability that the evidence would exist given that h happened is high and the probability that the evidence would exist given that h didn’t happen is low. The first probability can be 1; but the question then is what the other probability is. It is how much the first probability exceeds the second probability, that determines how much e supports h over ~h.
Frequently, the first probability is not even 1 (e.g. a meningitis test can have false negatives). But it’s precisely how much lower the second probability is that determines how worthwhile the evidence is (e.g. a meningitis test whose false positive rate is so high it is no more likely to detect actual meningitis than the base rate, is a useless test). Since P(e) combines the false negative and false positive expectancies, P(e) needs to be low for the evidence to be good. Not 1. But as far away from 1 as one can hope. The closer P(e) gets to 1, the more useless e is. And at 1, e is 100% completely useless.
That is all agree. Furthermore, it is agreed that if P(e) = 1 this entails that P(e) is not dependent on h and therefore P(e|h) = P(e|-h) = 1. However, this is not the same as saying there is no evidence for Jesus because we still have P(h|e) and P(-h|e) which are not 1 (but sum to 1).
I don’t disagree with your assessment of the question. To make it simpler: What is your understanding of the meaning of the statement “The probability of the evidence irrespective of what happened in the past adds up to less than 1”?.
You say
“What makes e evidence for h, is when the probability that the evidence would exist given that h happened is high and the probability that the evidence would exist given that h didn’t happen is low.”
Are these two exhaustive and exclusive? If so, then they must sum to 1. If they don’t sum to 1 they are not exhaustive and exclusive. They certainly appear to be exclusive. Therefore, if they don’t sum to 1 they must not be exhaustive. If they are not exhaustive then there must be something else in the probability space. What is that something else? Is it a probability that the evidence would not exist? And if that is so how is applying a probability >0 to it justified when we know the evidence does exist?
If there is nothing else in the probability space then they must be exhaustive as well as exclusive and sum to 1 and therefore my conclusions on the redundancy of Bayes’ theorem apply.
This is curiosity not bombast. Where is the flaw in this argument?
That is all agree. Furthermore, it is agreed that if P(e) = 1 this entails that P(e) is not dependent on h and therefore P(e|h) = P(e|-h) = 1. However, this is not the same as saying there is no evidence for Jesus because we still have P(h|e) and P(-h|e) which are not 1 (but sum to 1).
If they sum to 1, then e is not evidence for h.
See my other comment.
That you don’t know this, is why we know you don’t know what you are talking about. You don’t even understand how Bayes’ Theorem works.
What is your understanding of the meaning of the statement “The probability of the evidence irrespective of what happened in the past adds up to less than 1”?
My understanding is the mathematically correct one: P(e) = [P(e|h) x P(h)] + [P(e|~h) x P(~h)]
You will find this stated on every website, in every textbook, anywhere, on Bayes’ Theorem.
If you don’t know this, you don’t know at all what you are talking about.
If so, then they must sum to 1.
See my other comment.
You clearly don’t know what P(e) is or the role it plays in the equation. You need P(e) to be lower than 1. The more below 1 it is, the stronger e is as evidence for h. And the closer to 1 P(e) is, the less e evidences h at all. And when P(e) = 1, e is no longer evidence for h (it does not increase the probability of h any more than if e didn’t exist).
So for you to insist P(e) must always be 1 is profoundly ignorant.
If they don’t sum to 1 they are not exhaustive and exclusive. They certainly appear to be exclusive.
Hypotheses are exclusive, not evidences.
Once again, you evidently don’t know what you are talking about.
What is that something else?
The probability that the evidence would be different.
P(e) is always equal to 1 – P(~e).
Is it a probability that the evidence would not exist? And if that is so how is applying a probability >0 to it justified when we know the evidence does exist?
The same reason it’s justified to say the probability >0 that we’d observe a positive meningitis test result even when you don’t have meningitis.
But that’s P(e|h). Not (e). For the test to be useful at detecting meningitis, observing a positive result must lower P(e) substantially below 1, closer to 0.5 (the closer P(e) is to 0.5, the more reliable the test is). And it does that by getting P(e|~h) to be as low as possible.
Again, see my other comment.
I’m sure you could drop the mic. Your entire position is based on discussing the branches of a tree without realising that the roots are rotten and you are barking up the wrong tree. Your tree is geometric probability theory and it doesn’t fit the situation. The avoid those problems you need to look at another model.
If you really want to justify the methods you have used, you should not focus on your specific probability calculations and the mathematics involved in Bayes theorem, on which I have not challenged you, but rather you should focus on the assumptions that such calculations rely on and justify how these assumptions are met in the case of historicity in that ever difficult interface between mathematical models and reality.
So am I alone in my woeful misunderstanding of mathematics and the question of historicity? Amongst your disciples I’m sure I am. Outside of that group I would point out that if your model was justified, your case would have been proven. Not only that but proven to a higher degree of rigor than virtually anything else in ancient history.
You are the first person to rely so heavily on such a method in an ancient historical issue. This may be because of the unique insight you bring to the topic. On the other hand it may be because most people examining ancient history and probability abandon the attempt when they recognise what assumptions probability theory entails.
That doesn’t make any sense. Conjunctions of events always require multiplication of their probabilities.
You don’t know what you are talking about. Clearly. And until you do, I can’t help you.
(Also, no, I cite archaeologists who used my method before me. Likewise CIA analysts. You should know that if you’ve read Proving History. Likewise, Tucker proved all historians are doing what I am doing even when they don’t realize it.)
If you have followed the comments on this site you may have noticed that I frequently disagree completely with dr Carrier, including being wholly unconvinced of his single strongest (by his estimate) argument in favor of mythicism. So you can hardly claim I’m not open to persuasion.
However, the reasoning behind your objections appears to be clearly fallacious. Carrier’s methods do not have the flaws you suggest, and your suggested ‘improvements’ have fatal flaws.
Carrier’s claims are not generally accepted, but the reason for this has nothing to do with maths, it is rather because most scholars expressing an opinion do not agree with Carrier’s estimates of the various probabilities. Sadly, none of the other experts in the field (that I know of) have clarified what they believe those probabilities to be, or which other probabilities they believe are relevant. Even more sadly, the vast majority of experts simply have not expressed any interest in the question whatsoever. For instance, despite searching for years, I have still not found any single accredited historian besides Carrier who has published anything, even as popular science, within the last half century or so. The possible exception being a Swedish historian, although not focusing on ancient history, who broadly agrees with Carrier. I do not write this to dismiss scholars of other related subjects, but there is clearly something missing in a debate about history, without historians!
Fishers of Evidence wrote:
You give absolutely no reasons for this claim that are related to the case at hand. Your only argument is that the number is “low”. If that was a valid argument, you would have to agree that *any* low probability outside of theoretical models is suspicious. For instance, if someone says that the probability that your mother was a hampster is 0.12% you would have to complain that surely we cannot say that it is that low. But we can!
You may have a “hunch” that it is low, because you believe you have relevant knowledge that makes this seem improbable. But then you have to examine your knowledge and motivate why it should be higher.
This is correct, but this is not a flaw in the method, quite the opposite! If a serious historicist actually did this, we would first of all get a combined range for what serious scholars consider plausible. Perhaps that range would be very large, perhaps it would be small enough that we can say they are pretty much in agreement. And if we get a large range, we will see which arguments are the main causes of this disagreement. Those would be the areas where further research should be focused. That is how we make progress in science, by focusing on areas of disagreement and examining the facts until most reasonable scholars are more or less in agreement.
It means that it is not certain that the evidence would exist. For instance, let’s take a spam detector. The probability that a spam email contains the word “viagra” is much lower than 1. The probability that a legitimate email contains that word is even lower. The probability that an email, whether it is spam or not, contains the word “viagra” is much lower than 1. Spam detectors often use Bayes’ theorem to calculate the probability that an email is spam given many indications, where the occurrence of the word “viagra” is one factor. And yes, they will, of course, multiply these probabilities. Even though they are not necessarily independent.
Indeed. Using that method, we can add irrelevant arguments until the probability of *anything* becomes 50/50. Vaccines are poisoning the children? Lizards are controlling the world? It would all be 50/50. That’s obviously completely useless, which is why you will not find this method in any textbooks. Really, you should be more careful about thinking that you can invent something that is superior to what all the mathematicians have been able to do.
There is, I think, a risk involved with combining subjectively judged probabilities. This risk is that if the probability of any single argument is judged to be very low, it can easily overcome arguments pointing in the other direction. To avoid getting mislead by this, we should be careful of assigning very low probabilities, especially for hypotheses we are arguing against. You will notice that Carrier has not assigned any such very low probabilities. Instead, he ends up with a low combined probability because he believes there are many facts that are at least moderately improbable on historicity.
Johan Rönnblom, really good comment all the way through. I concur. Thank you.
so: “It means that it is not certain that the evidence would exist. For instance, let’s take a spam detector. The probability that a spam email contains the word “viagra” is much lower than 1. The probability that a legitimate email contains that word is even lower. The probability that an email, whether it is spam or not, contains the word “viagra” is much lower than 1.”
Obviously we all know what is meant by the probability of emails containing the word Viagra but that’s not the question.
So it means it not certain that the evidence would exist? Really? under what circumstances would it not exist given that this probability is irrespective of what happened in the past?
Patrick, P(e|h) = 1 – P(~e|h). Thus, the question is not whether the evidence “would exist given that anything whatever caused it” but how likely it is that that evidence would exist if h were true (compared to how likely it is that that same evidence would exist if h were false).
Thus, if I had in the past been infected by meningitis, how likely is it that that event in the past would cause the evidence now of a positive test result for meningitis? As long as a false negative has a nonzero probability, the answer is never “100%.”
And what we need to know is not what the probability of a false negative is (though we need to know that), but also what the probability of a false positive is. Thus, if in the past I was not infected by meningitis, how likely is it that that not being an event in the past would cause the evidence now of a positive test result for meningitis? In other words, the false positive rate. It’s the ratio between the two that determines how much a positive result weighs in favor of my having in the past been infected by meningitis.
As for historically past meningitis tests, so for all historical events whatever.
Evidence can be produced by things other than the event happening (the false positive rate) and that same evidence can fail to be produced by the event happening (the false negative rate). You have to factor both into any estimation of the weight any piece of evidence has for an event happening.
That’s why evidence can never have a probability of 1 regardless of what happened. Unless it isn’t evidence of anything. Because the only way the evidence can have a probability of 1 is if the evidence would exist regardless of whether the event happened. And if the evidence would exist anyway, it can’t be evidence of anything.
Richard, you are still not getting the question that I have in mind. All this is granted but in the analogy with meningitis, the probability of having meningitis, which is the prior, is a meaningful less than one probability. If we are assuming that I have meningitis, then this becomes 1. In that case the probability of testing positive versus testing negative is simply a matter of the sensitivity and specificity of the test, making Bayes’ theorem redundant.
I have addressed this above in terms of probability space. In terms of the meningitis analogy it is meaningful to say that I have meningitis and my probability of having it is one and its also meaningful to say that I may have it and my probability of having it is 1 in 1000. In terms of historicity it is meaningful to say that we have the evidence and the probability of having it is 1. You argument appears to imply that there may be circumstance under which we would not have the evidence but any such possibilities are quickly dismissed by observing that we do have it.
This is not the same in the meningitis example, because there is a meaningful probability that I don’t have meningitis that cannot be dismissed in the same way.
All this is granted but in the analogy with meningitis, the probability of having meningitis, which is the prior, is a meaningful less than one probability. If we are assuming that I have meningitis, then this becomes 1.
You don’t get to assume you have meningitis. Assumptions in = assumptions out. Assumptions aren’t knowledge.
The question is always do you actually have meningitis, just as the question is always, did Jesus actually exist.
It’s the same as “Did Jon Frum exist?” “Did Ned Ludd exist?” “Did Betty Crocker exist?” “Did Hercules exist?” “Did Homer exist?” Or likewise even “Did Julius Caesar exist?” As even that has a nonzero probability of being ‘no’; just a really small one, owing to the vast evidence we have. Or other cases.
In that case the probability of testing positive versus testing negative is simply a matter of the sensitivity and specificity of the test, making Bayes’ theorem redundant.
This statement makes no sense.
You have right now a prior probability of having meningitis. You do not “know” for certain you do or don’t have it. That’s why Bayes’ Theorem is not redundant. Especially when you want to know how much a test result changes that base rate expectancy.
In terms of the meningitis analogy it is meaningful to say that I have meningitis and my probability of having it is one…
Meaningful. But false. It is never and will never be the case that your probability of having meningitis is 1. Because omniscience doesn’t exist. That’s why we need Bayes’ Theorem. And tests for meningitis.
In terms of historicity it is meaningful to say that we have the evidence and the probability of having it is 1.
You are now confusing the base rate (the prior) with the likelihood.
You clearly don’t know what you are talking about. And I’m starting to think you might be insane.
BT is not about what’s meaningful. It’s about what we know. Do we know that the probability of having a positive test result if you caught meningitis in the past is 1? No. Neither we nor you know that. Nor will you ever. Because the false positive rate will never be absolutely zero.
Nor if it were 1 would that be P(e). The false positive rate is P(e|~h). P(e) is the sum of P(e|~h) x P(~h) and P(e|h) x P(h).
If P(e|~h) were 1, then the test would never detect meningitis—because everyone who doesn’t have it, would test positive!
This is why you need to know P(e|~h). That’s the only value that determines the effectiveness of the test—in other words, that tells you how much the evidence supports the conclusion (in this case, that you have meningitis when you test positive for it). Because it’s the difference between P(e|~h) and P(e|h) that measures the strength of any evidence e.
Thus, P(e) actually needs to be far from 1. You want it to be as close to 0.5 as possible (that’s the lowest value it can ever have)—if you want e to prove h true.
You unintelligibly keep thinking it’s 1. Worse, you seem to think that’s good! To the contrary, a P(e) of 1 establishes e is not evidence of h. A P(e) of 1 is the worst e conceivable. It is, in fact, equivalent to not having any e.
You argument appears to imply that there may be circumstance under which we would not have the evidence but any such possibilities are quickly dismissed by observing that we do have it. This is not the same in the meningitis example, because there is a meaningful probability that I don’t have meningitis that cannot be dismissed in the same way.
Just as there is a meaningful probability that you will test positive even though you don’t have meningitis.
You are again confusing the probability of having evidence (a positive test result) with the probability of having meningitis.
It is true that there are circumstances under which we would not have the evidence. That is precisely why we lack tons of evidence! (For example, the court record of Pontius Pilate for the execution of Jesus—if Jesus existed and was thus executed, such a record existed; but obviously the probability of our having it was not 1, otherwise we’d have it! And indeed, if the probability of our having it were 1 if Jesus existed, that would entail the probability Jesus existed was 0, because we don’t have it!)
But that’s not simply the issue here. What we want to know is not just the probability Jesus’s existing would cause certain evidence to exist. But what the probability is that Jesus’s not existing would cause that same evidence. That’s P(e|~h). And P(e), again, is the sum of P(e|~h) x P(~h) and P(e|h) x P(h).
Some evidence does not have a high probability of existing if Jesus did. For example, we should have much clearer data in the Epistles of Paul. The Epistles are unexpected, and require some ad hoc assumptions to get them as they are if Jesus existed, assumptions that don’t have a 100% probability of being true. And you can’t say what we have in Paul is 100% expected, because that would entail the probability is ZERO that Paul would ever have said anything else about Jesus, like that he was crucified by Pilate or his mother’s name was Mary or that he chose disciples before he died, or literally any and every other possible thing Paul could have mentioned. That’s ridiculous. And also wholly unsupported. (Why would the probability be literally 0 that Paul would never mention any detail about the life of Jesus?)
That’s debate number 1. You seem not to understand even the mathematical logic of that debate. But then there is debate number 2:
Indeed some evidence might have a virtually 1 probability of existing Jesus existed, e.g. that wild legends about him built on kernels of truth would exist (e.g. the Gospels) is pretty much 100% expected if Jesus existed (all else being equal). However, the probability that those legends would exist if he didn’t is also virtually 1 (just as it is for Hercules, Osiris, Romulus, Ned Ludd, John Frum, etc.). Again, all else being equal—see Proving History, index, “coefficient of contingency.”
For example, plenty of heroes like Jesus didn’t get such legends written about them, that is, there are no (at least surviving) Gospels for many other holy figures of the same period (e.g. none of the “Josephan Christs” I catalogue in OHJ; none of the famous Rabbis or sages of the era; not the Qumran Teacher of Righteousness; not even John the Baptist). So the probability that that would happen to Jesus is not literally 1. It’s 1 x c (the coefficient of contingency), where c is the probability of such legends being written eventually about a religious hero and surviving for us to have or know of them. But likewise, for a mystery cult savior deity, the probability of earthly myths being invented for them (despite never having actually lived) is the same probability (1 x c). It’s just as expected. Because it happened to nearly every other example of that category of person.
Since the Gospels are thus just as expected whether Jesus existed or not, P(e) for the Gospels is virtually 1 (close enough to round to 100% by the nearest percentile). Which means the Gospels do not make Jesus any more likely to have existed than not. They are therefore not evidence of his historicity. Nor are they evidence against his historicity (the key exception being the Rank Raglan data, as explained in OHJ, p. 395).
To get the Gospels to be evidence for Jesus, they need to entail a P(e) much lower than 1. For example, if we had a biography of Jesus written by a non-Christian eyewitness observer (as we have for another holy man, Alexander of Abonuteichos, written by Lucian of Samosata, who met and observed the man in what is now northern Turkey and wrote on his career and deeds there in the mid-second century). That would be highly unlikely if Jesus didn’t exist (just as it is highly unlikely if Holy Alexander didn’t exist). It therefore has a low P(e|~h). Let’s say, 0.1 (a 10% chance Lucian would write such a text if Alexander didn’t really exist). And it would have a P(e|h) of virtually 1 (after canceling out the coefficient of contingency, which is the same for both h and ~h).
That then entails (if we assume a neutral prior of 50/50): P(e) = [P(e|~h) x P(~h)] + [P(e|h) x P(h)] = [0.1 x 0.5] + [1 x 0.5] = 0.05 + 0.5 = 0.55.
So this e is strong evidence for historicity (of Holy Alexander; so also it would be of Jesus), because it has a P(e) not of 1, but of 0.55. If P(e) were 0.5 then e would be even better evidence for historicity than that. But already a P(e) of 0.55 gets you a really good P(h|e). To wit:
[P(e|h) x P(h)] / P(e) = [1 x 0.5] / 0.55 = 0.5 / 0.55 = 0.91 (rounding up)
If P(e) were 0.5, the lowest P(e) can ever be, then:
[P(e|h) x P(h)] / P(e) = [1 x 0.5] / 0.5 = 0.5 / 0.5 = 1 (a 100% chance of historicity, which would make that the very best possible evidence there can ever be)
Whereas if P(e) were 1, as you irrationally claim it must always be, then:
[P(e|h) x P(h)] / P(e) = [1 x 0.5] / 1 = 0.5 / 1 = 0.5, a mere 50/50 chance Jesus / Holy Alexander existed—in other words, simply the prior probability (so the evidence changed nothing, and thus is not evidence of historicity—which is a bad thing, not a good thing, if you want e to make the existence of Jesus more likely). Your insistence that P(e) is 1 is therefore a perversion of the entire logic of Bayes’ Theorem and makes zero sense.
Do you see why I mean when I say you clearly don’t know what you are talking about?
In this case, it would not exist in most emails. We have four groups of emails. Let’s assume they have the following distribution:
1) Spam not containing the word ‘viagra’ (45%).
2) Spam containing the word ‘viagra’ (5%).
3) Legit emails (“ham”) not containing the word ‘viagra’ (49%).
4) Legit emails containing the word ‘viagra’ (1%).
So the denominator would be P(viagra|spam)*P(spam)+P(viagra|ham)*P(ham)
= 5/(45+5)*1/2 + 1/(49+1)*1/2 = 6/100. As Richard noted, the only way it could reach 1 is if all emails contained the word ‘viagra’. But then, we obviously could not use the presence of that word to determine if a mail is spam.
Re P(e)=P(e│h)P(h)+P(e│-h)P(-h)
This formula is derived by writing Bayes’ theorem in 2 ways:
P(h│e)=(P(e│h)P(h))/P(e)
and
P(-h│e)=(P(e│-h)P(-h))/P(e)
And then stating that for both of these it is simultaneously true that P(e) is fixed and constant. An example would be the meningitis case where the probability of getting meningitis is fixed and constant and independent of the properties of various tests. In that case we can note that the 2 left hand side probabilities are exhaustive and exclusive so must sum to 1 and therefore
1=(P(e│h)P(h)+P(e│-h)P(-h))/P(e)
so
P(e)=P(e│h)P(h)+P(e│-h)P(-h)
This is true but does not preclude the case where P(e) =1. It simply means that in that case P(e) is not dependent on whether or not h is true and therefore P(e|h) = P(e|-h) = 1. I does not mean there is no evidence for h as we still have P(h|e) and P(-h|e) which make perfect sense but as I previously said it does make use of Bayes’ theorem redundant.
This is all false. And it’s clear you aren’t even checking your math to notice what you are saying isn’t true.
Patrick, I think where you go wrong is that you are unable to see that in a real case, you need to reason counterfactually: to weigh the importance of some evidence, we need to be aware of not just the case where we have that evidence (because we have it), but what the likelihood of having that evidence would be, if we did not know that we had it.
This is probably easier when you think about a hypothetical case like the meningitis example, because you have not actually performed the test. But the only reason the meningitis test is meaningful is that it was not certain before you took the test that it would test positive.
Likewise, with evidence for or against Jesus, or anything else, we need to pretend that we have not done the test (eg we do not yet have the evidence) and calculate the probability that we would have it on the hypotheses we compare. Just like we can do for the meningitis test – the importance of that evidence is obviously the same regardless of whether we do the calculations before or after we have taken the test.
Ok so we start on meningitis before we know the test result. We allocate probabilities to p(m|t), p(m|-t) and p(-m), the probability that the you don’t have meningitis. We can then check the completeness of our probability space noting that p(m|t) + p(m|-t) + p(-m)=1, we have not left any possibilities out.
Then on the bottom we have P(m)=P(m│t)P(t)+P(m│-t)P(-t). The complexity of the t dependency is redundant as all we need is the prevalence of meningitis among headache sufferers which we look up from epidemiological statistics.
Then we need p(t|m) and p(t) on the top.
So that gives us the Bayes formulation we need.
Are we agreed so far?
So turning to Jesus we start by allocate probabilities to p(e|h), p(e|-h) and p(-e), the probability that the evidence is different or absent. We can then check the completeness of our probability space noting that p(e|h) + p(e|-h) + p(-e)=1, we have not left any possibilities out.
Then on the bottom we have p(e)=p(e│h)p(h)+p(e│-h)p(-h). So where do we get the number here to put in the formula? This is where Jesus differs from meningitis. We can’t look up epidemiological statistics on the prevalence of the evidence. We can do one of two things. We can look at the evidence and find the probability that it exists is 1. We can put this into Bayes’ theorem which consequently reduces to p(h|e)=P(h). That does not get us very far but it is true. It makes the use of Bayes’ redundant but not false.
The other option which is the one you’ve taken is to imagine a situation where we have multiple histories and consider what proportion of them would give the evidence we have under different circumstances. We imagine that we do not have access to the evidence but we can gain it. We look at the situation and then we decide what we think values should be for p(e|h), p(e|-h) and by implication p(-e) making p(-e) >0. This is a matter of hypothesizing about situations that don’t exist but could potentially have existed rather than observing any data on these situations. That’s fine but why not examine the data and make estimates for p(h|e) and p(h|-e), or better just for P(h) as most people would? If we’re talking about a psychological process rather than mathematical one, what’s the reason for favoring making estimates of p(e|h) over making estimates of p(h|e)? when simply estimating p(h|e) or P(h) is direct and avoids the complexity and ambiguities of applying Bayes’ theorem.
Of course there is an obvious reason to select this second option. That is that you’ve done it in order to exploit the complexities and ambiguities of applying Bayes’ theorem to obscure your use of an inappropriate geometric probability combination method. Actually I think bad faith is unlikely. More likely is that the issues surrounding Bayes’ theorem contributed to obscuring the problems with geometric combinations from yourself.
So turning to Jesus we start by allocate probabilities to p(e|h), p(e|-h) and p(-e), the probability that the evidence is different or absent.
No, that it would be different or absent.
We are not asking what the probability is that the meningitis test is giving a positive result (that’s always 100%, or near enough, once you observe the result). We are asking what the probability is that it would do so, if you have, and then if you do not have, meningitis. Ditto Jesus existing and not.
We can look at the evidence and find the probability that it exists is 1.
No, we need to know the probability that it would exist if Jesus existed, and then what the probability is that it would exist if he didn’t. The difference between those two probabilities determines P(e). And P(e) needs to be close to 0.5 to make the evidence strong; the closer P(e) is to 1, the weaker the evidence is (and at P(e) = 1 the evidence has zero effect on the probability—this is as true for Jesus as for meningitis or anything else).
We can put this into Bayes’ theorem which consequently reduces to p(h|e)=P(h).
Uh. You do realize that means e is not evidence for h, right? If e does not increase the probability of h without e,then e has had no effect on the probability of h.
This is a matter of hypothesizing about situations that don’t exist but could potentially have existed rather than observing any data on these situations.
Yes. That’s the whole purpose and design of BT. It’s about asking the P of an event if two mutually contradictory events occurred (h and ~h) and you don’t know which occurred (only that it can’t be both). It is thus always a tool of counterfactual reasoning. That is in fact the very problem Thomas Bayes was trying to solve, and his solving it is exactly what he is celebrated for.
…what’s the reason for favoring making estimates of p(e|h) over making estimates of p(h|e)?
Logic.
To know why you think p(h|e). In other words, to expose what your premises are, so we can do two things: confirm the conclusion follows from those premises (i.e. check for formal error), and verify that your premises are sound (i.e. check the factual basis for asserting those premises rather than others).
All the more important when the premises are numerous. As they are in complex ambiguous cases. Like Jesus. Or spam filtering.
This is actually exactly the opposite of what you allege. By making clear what my underlying assumptions are (what my premises are) I am being more honest than someone who conceals them, and I’m making it easier for critics to evaluate the merits of my argument. That’s why it’s weird that you don’t even try to use that advantage: I’ve made it possible for you to see what all my assumptions are, and to test whether they hold up on what we know about the world. But you refuse to vet or revise a single one of them. You are the one who seems to care more about an agenda, than actually making honest progress toward a secure and defensible probability Jesus existed.
This is something of a side issue as I do not think it alters the final result of an analysis, but it has an interesting feature so I will defer returning to my main point and pursue it.
I’ll use the term scholastic probability estimate to describe the situation where a scholar considers evidence that is not primarily numerical in nature to determine a probability relating to a hypothesis. This is distinct from a mathematical probability estimate which is drawn from numerical data either observed or calculated.
So, at issue are two alternative epistemological approaches for scholastic estimations. The conventional approach is to consider the evidence and estimate a probability of a hypothesis, based on the evidence.
Your approach, which is also reasonable, is to consider a hypothesis and estimate the probability that the evidence would be as it is, given that the hypothesis is true.
You say the reason to select your approach is logic and you explain
“To know why you think p(h|e). In other words, to expose what your premises are, so we can do two things: confirm the conclusion follows from those premises (i.e. check for formal error), and verify that your premises are sound (i.e. check the factual basis for asserting those premises rather than others).”
But you don’t show how this favours your approach over the conventional one. And don’t bother with “it’s obvious” and “you don’t know what you’re talking about”.
It looks more like a matter of scholastic preference than logic, particularly as most historians use the conventional approach.
And it makes this a key question: would a scholastic estimate of p(e|h) differ numerically from a scholastic estimate of p(h|e) and if so how?
And don’t bother telling me that p(m|t) is not equal to p(t|m). I know that holds for mathematical estimates like meningitis. I’m talking about scholastic estimates.
I’ll use the term scholastic probability estimate to describe the situation where a scholar considers evidence that is not primarily numerical in nature to determine a probability relating to a hypothesis. This is distinct from a mathematical probability estimate which is drawn from numerical data either observed or calculated.
Those are ultimately the same thing. One is just less informed by data than the other. But the first is always just an attempt to estimate the second.
See Proving History, pp. 265-80.
Your approach, which is also reasonable, is to consider a hypothesis and estimate the probability that the evidence would be as it is, given that the hypothesis is true.
That’s not my approach. It’s the only approach. All Bayesian reasoning adopts it. All.
You will not find any other approach used, in any other peer reviewed application of Bayes’ Theorem, anywhere.
And it makes this a key question: would a scholastic estimate of p(e|h) differ numerically from a scholastic estimate of p(h|e) and if so how?
Seriously. Read any peer reviewed example of Bayesian reasoning, on any subject whatever, and you will see not only that they differ, but why. (There would never be any reason to publish a Bayesian argument for anything, if the posterior probability was identical to the prior probability…except to demonstrate that there is no evidence for h.)
And don’t bother telling me that p(m|t) is not equal to p(t|m). I know that holds for mathematical estimates like meningitis. I’m talking about scholastic estimates.
You just made that up. There is no such distinction. And the method you claim conventional, does not exist. Not in any application of Bayesian reasoning in any peer reviewed book or article, anywhere, ever.
P(e|h) is always a counterfactual subjunctive: the probability that e if h. That means the probability we can expect an e, if h occurs.
Just as P(e|~h) is: the probability that e if ~h. That means the probability we can expect an e, if h doesn’t occur.
The ratio between P(e|h) and P(e|~h) is what determines how strongly e weighs in favor (or against) h.
If P(e|h) and P(e|~h) are both 1, then e is not evidence for h. It has zero effect on the probability of h.
Which gets me back to reminding you, again, that P(e|h) is not P(e)—and you were talking about P(e). That you don’t know the difference is just one more example of how you literally don’t know what you are talking about.
“would a scholastic estimate of p(e|h) differ numerically from a scholastic estimate of p(h|e) and if so how?”
Let’s assume that e is “a first century audiovisual recording found in the desert showing Jesus, introducing himself as Jesus of Nazareth, performing what is identifiable as the Sermon on the Mount from the Gospels”.
Here, p(e|h) is the probability that this evidence would exist given that Jesus was historical. We can probably agree that this probability is extremely low, considering that we have no knowledge of any video recording technology available at that time.
On the other hand, p(h|e), which is the probability that Jesus was historical given this evidence, must be extremely high. If indeed we found such evidence, we must assume all reasonable scholars would agree that Jesus was historical.
“We can probably agree that this probability is extremely low, considering that we have no knowledge of any video recording technology available at that time.”
This is a good example; but I fear some readers won’t understand the point and be confused. It is actually a confusing counterfactual, I admit.
The reason this P(e|h) being low nevertheless greatly increases p(h|e) is that the corresponding P(e|~h) is even lower. Namely, the probability of there being an authenticated 30 A.D. video recording of that kind, if Jesus didn’t exist, is even lower than the probability of there being such equipment at all (once we grant the recording genuinely dates so; since then, the tape is evidence the equipment existed).
And this is of course presuming the probability the video is a forgery is even lower still. Normally, the low probability of P(e|h) for this evidence means the most likely explanation of this evidence is that it is fake (the video was not made in 30 A.D.). So we have to assume enough additional evidence exists to lower the probability of that explanation, below the probability of there being such equipment then at all (i.e. it has to be more likely the tape is real than a fake, before it becomes evidence for Jesus, and then it becomes evidence for Jesus because P(e|~h) is then lower than P(e|h), since you couldn’t get such a tape in that case unless there really was someone claiming that name and saying those things; and all that entails a P(e) well below 1, exactly as we need P(e) to be for e to increase P(h|e)).
Whew.
Just explaining that is complicated!
So maybe not the best teaching tool. Hm.
Maybe. But I think it is important to highlight that in p(h|e), the “|” sign means “given”, which means that it is something that is true assuming that e exists (and therefore is real, not fake). When teaching logic, it is common to use logical statements that are implausible or nonsensical just to teach students to focus on one thing at a time. I think this is such a case. Whether we believe that claimed evidence is real or not is a completely separate issue from what our conclusions should be if the evidence is real.
Whether we believe that claimed evidence is real or not is a completely separate issue from what our conclusions should be if the evidence is real. [Etc.]
I agree. But a lot of people don’t know that. Hence it’s easy to confuse them if that isn’t explained first.
(In OHJ, of course, I properly exclude forged evidence from e.)
Okay I will accept that the ultimate objective of scholastic and mathematical probability estimates is the same and the distinction is that scholastic estimates are less informed by data, and consequently require a higher degree of expert judgement.
Secondly, we can agree that your approach is Bayesian and is entirely analogous to other examples of Bayesian reasoning.
But, just because “|” appears in a probability expression, does not mean that it entails Bayesian reasoning. In fact, a scholastic estimate of p(h|e) is the same as a scholastic estimate of the probability of historicity made from the available evidence, in other words , using non-Bayesian reasoning. This is what most historians do.
And the effect of making p(e)=1 is to make Bayesian reasoning reduce to non-Bayesian. This is the point of correspondence between the two.
So how would a non-Bayesian scholastic estimate of the probability of historicity based on the available evidence compare with a Bayesian estimate of p(e|h)?
Or, do you hold that that when historians make such non-Bayesian scholastic estimates they actually are using Bayesian reasoning without realising it?
But, just because “|” appears in a probability expression, does not mean that it entails Bayesian reasoning.
Um, what entails Bayesian reasoning is that P(h|e) always = P(h) x P(e|h) divided by [P(h) x P(e|h)] + [P(~h) x P(e|~h)].
And [P(h) x P(e|h)] + [P(~h) x P(e|~h)] is always what is represented by the expression P(e) in the short form of Bayes’ Theorem.
We are talking about that. Don’t change the subject.
In fact, a scholastic estimate of p(h|e) is the same as a scholastic estimate of the probability of historicity made from the available evidence, in other words , using non-Bayesian reasoning. This is what most historians do.
Actually, no. Most historians are doing Bayes, and just don’t know it. Because it’s the only logically valid way to get P(h|e). And what historians always do is estimate the relative base rates of all the competing explanations (= priors) and then estimate how much more likely the evidence is on one explanation compared to the others (= likelihoods). And there is only one valid way to derive a posterior probability from those two assumptions. It’s called Bayes’ Theorem.
See Tucker.
And my formal proof in Proving History, pp. 106-14.
Indeed all other forms of historical reasoning are just disguised or sloppier versions of Bayes’ Theorem: Proving History, pp. 97-106.
And the effect of making p(e)=1 is to make Bayesian reasoning reduce to non-Bayesian. This is the point of correspondence between the two.
No. It doesn’t. Good lord man. You are like talking to a wall.
Even if we granted that historians can just “guess” what P(h|e) is without any reference to plausibility (= prior probability) or the strength of any evidence (= likelihood ratios), that still wouldn’t get you a P(e)=1. Because a P(e)=1 means the evidence, e, has zero effect on the probability of h. The mathematical expression “P(e)=1” for any h means there is no evidence for h.
If you had evidence for h, then P(e) will be lower than 1. The stronger the evidence you have for h, the closer P(e) gets to 0.5.
You still think a P(e)=1 is a good thing. It’s not. It’s the worst thing. It means you have no evidence; that whatever is in e does not increase the probability of h. That you still don’t know that, is why you don’t know how Bayes’ Theorem works.
So how would a non-Bayesian scholastic estimate of the probability of historicity based on the available evidence compare with a Bayesian estimate of p(e|h)?
Just p(e|h) is useless information. You need to know p(e|~h) before you can assess whether the p(e|h) you have is any good. Because it is precisely how much more p(e|h) is than p(e|~h) that determines how strongly e argues for h (how much e increases the probability of h). If p(e|h) = p(e|~h), you get a P(e)=1: e is not evidence of h; because e does not increase the probability of h.
This again you seem not to understand. This is why you can’t just have P(e|h). That is a meaningless term by itself. It tells you nothing about whether e is even evidence for h, or how strong an evidence of h it is.
You seem to have missed the main point but Johan has not so I will expand further in a reply to his comment. In any event you raise an issue.
You say “if p(e|h)=p(e|-h), you get p(e)=1”
That means that when p(e|h)=p(e|-h), in the probability space of e, there are only p(e|h) and p(e|-h) and no other possibilities.
You also say “the stronger the evidence you have for h, the closer p(e) gets to 0.5”
This implies that there is a component of the probability space of e that =0 when p(e|h)=p(e|-h), and is >0 when p(e|h)p(e|-h).
So what is that component?
That means that when p(e|h)=p(e|-h), in the probability space of e, there are only p(e|h) and p(e|-h) and no other possibilities.
That’s true by definition: ~h as a symbol means all possibilities that are exclusive of h.
If you want to expand the range to test more than two possibilities independently, you can expand ~h, i.e. you can do h1, h2, h3, and h4, as long as those four exhaust all possibilities. I show how to do expanded tests like that in Bayesian equations in Proving History, p. 283, middle bottom. The most formal version of the equation uses the sum operator to cover even infinite possible explanations (see Proving History, p. 283, bottom).
You also say “the stronger the evidence you have for h, the closer p(e) gets to 0.5.” This implies that there is a component of the probability space of e that =0 when p(e|h)=p(e|-h), and is >0 when p(e|h)p(e|-h).
I have no idea what you are talking about.
What do you mean by “the probability space of e”? And what do you mean by “a component” of it being “0”?
And why are you multiplying P(e|h) and P(e|~h)?
None of this makes any sense.
You also seem to be confusing the condition P(e|h) = P(e|~h) = 1 with P(e|h) = P(e|~h) /= 1. You’ve been insisting it’s always 1. I’ve been speaking only of that condition, because that’s the only condition you’ve been speaking of. If now you are allowing it to not be 1, the outcome is the same (the evidence never increases the probability of the hypothesis), but the ideal P(e) drops below 0.5.
Example:
P(e) = P(h)P(e|h) + P(~h)P(e|~h)
So when P(h) = 0.9, P(~h) = 0.1, and if P(e|h) = 1 = P(e|~h) then:
(0.9)(1) + (0.1)(1) = 0.9 + 0.1 = 1
And anything divided by 1 is itself and anything multiplied by 1 is itself. So P(h|e) = P(h) = 0.9. Since the full equation then gets you (0.9)(1)/[(0.9)(1) + (0.1)(1)], and that simply equals 0.9, the prior probability you started with before looking at or considering any evidence e.
Now, if we allow (which you have not until, I guess, now) the value of P(e|h) to be less than 1—let’s say, P(e|h) = 0.2 = P(e|~h)—then:
P(e) = (0.9)(0.2) + (0.1)(0.2) = 0.18 + 0.02 = 0.2
And the lowest that P(e) can get then is (0.9)(0.2) = 0.18 (that would indicate the best possible evidence for h; you can in that case have no lower P(e) than that).
Back to the full equation: (0.9)(0.2) / [(0.9)(0.2) + (0.1)(0.2)] = 0.18 / 0.2 = 0.9
Which is what you started with when you didn’t have any evidence at all: a prior probability of 0.9.
Thus, the evidence did not increase the probability of h at all; so the evidence had no effect on the probability of h.
This always results, when P(e|h)=P(e|~h).
And note, still P(e) is not 1 in this condition. It’s 0.2.
Richard Carrier wrote:
“If p(e|h) = p(e|~h), you get a P(e)=1”
I think this is mistaken or at least misleading. The conclusions are right (such evidence does not point in either direction). But it misses the possibility of e not existing at all, e.g.
p(~e|h) and p(~e|~h).
For instance, suppose e is “a letter from Paul discussing Jesus’ opinions on non-Jewish converts”.
Let’s say that in this case we think p(e|h) is 0.5 and that p(e|~h) is also 0.5.
Because it does not really matter if Jesus was historical or just a visionary being, Paul might equally well reference Jesus’ teachings on the matter either way.
But we cannot be sure that Paul would have written such a letter, it’s likely that he would given that he was very interested in this subject and also liked to reference Jesus, but it’s also quite possible that he would not have written about this.
In the Bayesian calculation, only the ratio between these probabilities matter, and in OHJ you mention that you are assigning these probabilities only to express the relative likelihood, but this can certainly be misleading, especially when not explained.
I think this is mistaken or at least misleading.
Yes, it’s confusing if we lose track of the fact that I’ve been talking about what Patrick’s been talking about, which is that P(e|h) is always 1, or P(e) is always 1 (he confuses the two a lot, so technically he has asserted both). When P(e|h) = 1 and P(e|h)=P(e|~h), then P(e) = 1.
If we change the condition to allow a P(e|h) below 1 (something Patrick has denied possible), then the numbers change (a neutral P(e) will equal P(e|h) and P(e|~h), which not being 1, will no longer be 1; likewise, the “best evidence” condition when P(e|h) /= 1 is no longer a P(e) of 0.5, but lower).
I just cleared this up in my most recent comment here.
It’s hard to keep track of Patrick’s confusions.
(P.S. Likewise, I addressed your point about equal expectancies often being below 1 with the “coefficient of contingency,” which can be canceled out to simplify the math. See index in Proving History. It’s one of the reasons I preferred to use the Odds Form in OHJ, where it’s always just ratios.)
By the probability space of e I mean all the exhaustive possibilities for e and -e, which of course can be broken down to e1, e2, … or simply stated as e and -e. Then the probabilities of these sum to 1.
You comment ending with “None of this makes any sense”. And you’re absolutely right, it makes no sense at all, I apologise for this. This seems to be a software problem with symbol recognition. I use a not equal symbol that was clearly not recognised. I’ve typed it again here between the colons : : to see if it disapers again.
What I meant to say was: and is >0 when p(e|h) is not equal to p(e|-h).
Anyway, I don’t think there’s anything wrong with either your use of Bayes’ theorem or your arithmetic but I remain of the view that it is equivalent to the conventional non-Bayesian approach. And I’m not persuaded that it is somehow more fundamental and rigourous than the conventional approach. You have contested my assertion that setting p(e)=1 is the point of correspondence between the non-Bayesian and the Bayesian approaches. Do you mean there is no correspondence or that the point of correspondence is under some other condition?
I remain of the view that it is equivalent to the conventional non-Bayesian approach.
There is no other approach. All approaches are Bayesian. Whether you notice that or not. As I formally proved in Proving History, Ch. 4. You can never validly talk about what happened in the past without reference to base rates and likelihood ratios. And there is no valid way to integrate base rates with likelihood ratios other than Bayes’ Theorem. That’s precisely what Thomas Bayes discovered and proved. That’s what the theorem is.
You have contested my assertion that setting p(e)=1 is the point of correspondence between the non-Bayesian and the Bayesian approaches. Do you mean there is no correspondence or that the point of correspondence is under some other condition?
P(e) is only a term in Bayes’ Theorem. It has no obvious meaning in any other context.
It can mean either P(e|observation), in which case e is just h (the hypothesis of what evidence exists, given what we observe), and thus it is just a stand-in for the entire Bayesian formula (and is only about what evidence exists, not what caused that evidence), or it can mean P(e|h)P(h) + P(e|~h)P(~h), and thus is simply the denominator of the Bayesian formula (for a causal hypothesis of why the e exists). Either way, Bayesian. You have not explained any other meaning or function for it.
So you are just getting even more confused than you already were.
Back to what we were talking about:
You said in your video critique, the one that I am responding to on my blog here, that in Bayes’ Theorem p(e) is always 1. As I point out in my article above, that’s false. And it’s not only false, it exhibits a complete lack of understanding of how Bayes’ Theorem works. Any claim for which there is good evidence, will never have a P(e) of 1 in Bayes’ Theorem.
I have said this and explained this to you at least three times now.
Figure it out.
“So how would a non-Bayesian scholastic estimate of the probability of historicity based on the available evidence compare with a Bayesian estimate of p(e|h)?”
If you want to continue that discussion, I think you need to define what you think this non-Bayesian reasoning would be. You would need to present a formula or algorithm for how it would work. I agree with Dr. Carrier that any such formula could either be rewritten as a Bayesian formula without losing anything, and that if it cannot be restated in such a way, there must be something that can objectively be shown to be inferior.
I can only think of obviously flawed alternate strategies, such as for instance using only one piece of evidence and discarding the rest, or using some arbitrary weighting as you alluded to earlier. But although the latter approach can work under limited circumstances, I think it is clear that it cannot give better results overall, and if you believe otherwise you really need to present exactly how you think that would be done.
The non-Bayesian approach is the a familiar one from scholarship and law. It is not usual to express it mathematically as these subjects are not inherently numerical but it can be done. One common way is iterative probability adjustment. You start not with a prior probability but with a default assumption which may be a null hypothesis, a presumption of innocence, a consensus position or a neutral position depending on the situation and your preference. You then look at the evidence sequentially and when each item is considered, you adjust probabilities away from or towards the default position. These adjustments are usually cumulative rather than geometric.
A difference is that when you look at the evidence sequentially, you can organise it in different way. It is clear that Richard organised 0HJ into bodies of evidence rather than arguments because this suites the Bayesian approach better but I’m sure you know that David Fitzgerald in Nailed organised the book into 10 arguments, as is more commonly done, as this suits the non-Bayesian approach better.
So the commonest non-Bayesian approach from non-numerical data, such as historical data, is to consider arguments in turn, assess which side of the issue they support, then allocate a numerical value to them proportional to their power and add that value to the side of the issue that they support, there are several ways of allocating numerical values. A common one is to consider arguments to be of value 1, 2 or 3 depending on weather they are weak medium or strong. Another is to rank arguments according to their persuasiveness and use their ranking. Then in the end weigh the two sides against each other.
So you will, of course, object that this kind of scheme has nothing like the precision and rigor of formal probability calculations which is perfectly true but beside the point. The point is rather that for Jesus, we cannot meet the exacting assumptions that formal probability calculations demand and therefore we do not have that option. If we ignore those assumptions and go ahead anyway, we’re liable to be seriously misled. Garbage in – garbage out is how programmers express the problem.
Patrick, you started with a video making claims about Bayesian reasoning and probability theory. Those claims were false. And worse, they betrayed an ignorance of even basic principles of Bayesian reasoning and probability mathematics. I pointed that out in my article.
Now you are making shit up about some other method no one has ever heard of and that I never used and that has no demonstrated validity anywhere. That’s not what you were talking about in your video.
So please admit you fucked up. Before continuing this tedious exchange. If now you want to change the subject and argue that you have a better method than Bayes’ Theorem for determining how likely a hypothesis is, you need to actually do that: prove to the field of mathematics that you’ve discovered a better way of doing that.
Otherwise, admit you have no idea what you are doing.
You then look at the evidence sequentially and when each item is considered, you adjust probabilities away from or towards the default position. These adjustments are usually cumulative rather than geometric.
That’s the Bayesian method. It’s called starting with a neutral prior and updating the priors as evidence is added. Each iteration converts the added e into the contents of b for the next run of the equation. I explain this in Proving History. Look at the index under “iteration, method of.”
Of course it will never give you a credible answer if you leave evidence out. For example, all human background knowledge has to go into the equation at some point. Otherwise, it will not give you a correct probability. This is indeed how Christian apologetics works: leaving evidence out, so as to get a probability they want. That’s invalid because we have the missing evidence, so we have to put it in—and when we do, the conclusion changes, to what we actually know it to be, because it is arguing from knowledge and not from engineered ignorance: see Bayesian Counter-Apologetics.
A difference is that when you look at the evidence sequentially, you can organise it in different way.
That cannot make any difference to the outcome. If it does, your method is formally invalid.
Thus, order is irrelevant in Bayes’ Theorem, too.
It is clear that Richard organised 0HJ into bodies of evidence rather than arguments …
Every body of evidence is an argument. Every argument is a declaration that some body of evidence changes the posterior probability. They are the same thing. You cannot have a relevant argument for any h, that does not reference any e, or make any assertion about how e increases (or decreases) the probability of h. And Thomas Bayes proved there is no valid way to make such an argument, but through what Laplace later articulated as Bayes’ Formula.
You can’t escape Bayes’ Theorem. The moment you make any claim that the probability of h is P given everything we know, you are making a Bayesian argument. See formal proof, again, in PH.
because this suites the Bayesian approach better but I’m sure you know that David Fitzgerald in Nailed organised the book into 10 arguments, as is more commonly done, as this suits the non-Bayesian approach better.
Dude. That’s just an order of presentation. It’s still Bayesian.
He gives no other equation, and derives probabilities in no other way.
He uses no such baloney method as you describe.
He reasons the way all historians do: each chapter argues from a body of evidence, to a conclusion about the probability of h. And the reasoning he uses is Bayesian (whether he describes it that way or not, and whether he knows that or not).
So the commonest non-Bayesian approach from non-numerical data, such as historical data, is to consider arguments in turn, assess which side of the issue they support, then allocate a numerical value to them proportional to their power and add that value to the side of the issue that they support, there are several ways of allocating numerical values.
That’s arbitrary nonsense. It has no logical validity whatsoever. And isn’t used by any historian, ever, anywhere, in the history of peer reviewed history.
Indeed, the numbers you’d be using in such a method, and getting out of such a method, would be meaningless. And as such, their relationship to each other would be completely incoherent and incapable of producing any logically valid conclusion.
And any attempt to make it coherent and valid, will just convert it into Bayes’ Theorem. Hence I prove one version of the method you describe, the Inference to the Best Explanation—an argument historians actually use and thus one more coherent than this nonsense you just made up—is actually reductively Bayesian: Proving History, pp. 100-03.
You need to stop making shit up.
You have only two options.
Admit all historical methods are Bayesian.
Or show us a logical proof of a method that (a) doesn’t reduce to Bayes’ Theorem and which (b) gives us a logically valid conclusion as to how likely one hypothesis is relative to another. That is, show us the formula, and prove it is logically valid.
That’s it.
Do one or the other.
Or go away.
You are right in that this is a totally pointless debate. I hold that there are two alternative methods that give the same result. You hold that they boil down to the same method and so give the same result.
But, you have user this pointless debate to repeatedly avoid answering the issues that I have repeatedly raised. Your estimates are not independent for the reasons I previously given and also because they all involve an element of judgement and they were all made by the same person, you, and moreover there is reason to believe you have axe to grind and therefore there is concern about a systematic bias.
You point out that your treatment of historicity is being very generous and so if there is any bias it’s towards rather away from historicity. That is true but there’s a reason for that isn’t there Richard.
That reason is that you needed your probability results to be as they are. The lower bound needed to be low and you made it good and low. The upper bound was more of a problem. It had to come in under 50% or evens but if it was too far under it would make the overall result too far from the mainstream scholastic consensus be taken seriously. You felt that odds of 2-to-1 against historicity was about right. So you got the upper bound to come in just on the right side of 2-to-1 odds. This was easy to do because small adjustments to multiple probabilities using your model lead to large changes in the result. It had the happy side effect of making you appear generous towards historicity but that was not its purpose.
Now only you know whether or not you did this, but no amount of protestation and abuse will make any difference. You need to show that you didn’t do this and you can’t can you? Because if you could it would have been in the book.
Your approach is analogous to that of David Baggett when he used probability to “show” that the resurrection happened. The ease of doing this and the difficulty of detecting it is the very reason why you should not use formal probability calculations without addressing their assumptions.
It’s a pity, really, because otherwise you made a good case.
You can’t say my upper bound is too low, if you can’t explain how any of the premises that entail that are false.
That’s how logic works.
I’m not the one avoiding issues here. You are. You’ve avoided admitting all the errors I caught you at and called you out on. And now you’ve dodged, into a new whack-a-mole game, into some other argument about how you can prove Jesus probably existed without ever doing any logically valid probability reasoning. Which is inherently illogical. And here you just assert that my pro-historicity estimates are too low. Just “because.” Without any evidence or argument that they are too low.
That is dodging logic. Not using logic.
Patrick Mitchell wrote:
“You then look at the evidence sequentially and when each item is considered, you adjust probabilities away from or towards the default position. These adjustments are usually cumulative rather than geometric.”
I have seen this done in a reasonable way, but then it is not dealing with *probabilities*, but rather just with a very simple overview of available evidence. If you do this with probabilities, you are doing it wrong. I have an actual live example to illustrate where such nonsense leads:
A Swedish man, Thomas Quick, confessed to a large number of murders and was convicted of eight of them. After journalists investigated, it was found that he had been encouraged to give false confessions while under psychiatric care, and that the evidence presented in the trials had been skewed and deeply flawed. His convictions were all overturned, and most observers agree it is highly unlikely that he was involved in any of them.
However, Göran Lambertz, a justice of the Swedish Supreme Court, could not let go of the case, and has vigorously and publicly kept arguing that Quick is guilty. In a book, he gave probabilities for various evidence he believed were indicative of Quick’s guilt, and summed them. Just as you describe. He concluded that the probability of Quick being guilty was 183% (yes, he actually wrote that himself).
After being ridiculed for this, he updated his calculation using Bayesian calculations, arriving at a conclusion that is at least not *mathematically* insane.
“Garbage in – garbage out is how programmers express the problem.”
But that problem does not in any way diminish if you use bogus math. Lambertz above has absolutely ridiculous arguments, but his flawed math made the problem even worse, not better. Moreover, if you believe the arguments to be “garbage” then you need to provide arguments for this, rather than attacking the mathematics used.
All that is true and you cite an an amusing example that I shall look up. The problem here is that you need to have a method that reflects scholastic issues that are not numerical, a method where, if you must use numbers, you use categories that are broad enough to allow a fair consensus between scholars. Any method that allows minor adjustments to multiple pseudo-formal *probabilities* as you aptly wrote it, leading to wild swings in the result is never going to achieve that. And that’s basically the reason why most scholars don’t do it. There are areas of scholarship where precise mathematical analysis is appropriate and very powerful, such as in statistical textual criticism, and the growth of such methods has meant that scholars are aware of them. In general it’s not ignorance that stops them it but prudence.
If we came up with a method which allowed us to sit around a table with historicists and agree rules and values with them, and reach a conclusion that none of us could escape, then we’d be onto something.
You can’t say “Jesus probably existed” and not be making a mathematical statement. That the word “probably” is in English doesn’t make it not a mathematical statement. There is no escaping that fact. You cannot say Jesus probably existed, and not be able to show mathematically that he probably existed. If you can’t show mathematically that he probably existed, you do not know he probably existed. By definition. This is already obvious, from the definition of “probably.” But again, I formally prove it in Ch. 4 of Proving History. And yes, when historians refuse to admit they are making mathematical statements, they are rejecting logically necessary facts. And that’s a problem for the field as a whole. See the analysis, again, of Aviezer Tucker.
There is no logically valid way to reach conclusions about what probably happened in history without making assertions of mathematical probability. Even if those assertions are in English and vague.
Patrick Mitchell wrote:
“Any method that allows minor adjustments to multiple pseudo-formal *probabilities* as you aptly wrote it, leading to wild swings in the result is never going to achieve that.”
For that claim to make any sense, you would have to demonstrate an alternative method that, for the same input data, leads to less wild swings. You have not done that, and I do not believe it to be mathematically possible.
You are blaming the method for the fact that there is wide disagreement. That is not a flaw in the method, rather the method is simply reflecting – and clarifying – a disagreement that would be there regardless of the method used. Any method that did not “swing” wildly in the face of profound disagreement would be fatally flawed.
Thanks, Johan.
Superb point.
I posted a couple of weeks ago. I’m not sure if you receive the post.
Many mathematical models ultimately fail to be convincing. When they fail, they all fail in the same way. There is never a problem with the internal consistency of the model because that part of the modelling process is easy. The problem always arises with the interface between the model and reality. Models such as yours involve deriving numbers from non-numerical data, a process of numerisation. It is this numerisation which is the most vulnerable link in your chain of reasoning. This is such a significant issue that most modelling exercises concentrate on optimising the numerisation process and justifying it rather than on the internal mathematics of the model.
My objection to your use of Bayes’ theorem has nothing to do with its internal consistency, or the soundness of your reasoning when discussing the internal mathematics of it. My objection is that using Bayes’ theorem has led you to numerise the historical data in a specific way, to give numbers that can be the used as a substrate for Bayes’ theorem. Furthermore it has led you into a branch of mathematics which is particularly intolerant of errors and uncertainties in the numerisation process.
My position is that your numerisation process is so weakened by its inexactness, its variability from one observer to another, and its lack of repeatability that it cannot form the basis of any convincing model, especially one where your numbers are used directly in probability calculations.
Now Johan says I should demonstrate an alternative method that uses the same input data and produces less wild swings in the output. He also says he does not believe it is mathematically possible and you have agreed with his point.
I had previously assumed that a Scholastic estimate of p(e|h) would be the same as a Scholastic estimate of p(h|e). You rejected this assumption and I accept your rejection of it. That in turn also means that I too agree. It is not mathematically possible to do what Johan asks.
This leads me to think that neither of you understood what I’m actually suggesting which is not to develop a new method using the same input data but to develop a different method of numerisation and then build the mathematical model around that method. This is what we commonly do when building models for non-numerical data. We concentrate on modelling the weakest link in the process, numerisation, rather than on some underlying theory of epistemology about the data itself. And such methods lead us into areas of mathematics like ranked statistics or fuzzy logic where the models do not describe either the theories under test or any philosophical understanding of truth, but rather the numerisation process and its susceptibility to errors and bias.
The first task in such an approach would not be to develop a model but rather to agree a numerisation process. This would have to give at least some form of consistency between observers. There is no prospect of your method achieving this consistency but alternative approaches may do. One would be to consider arguments in turn and decide whether they support or detract from historicity. This first attempt may be easiest to gain consensus on but there are likely to be issues about how much weight to place on different arguments so it would be necessary to add some indicator of argument strength.
If that could be achieved, and I am not at all sure it could be, then an appropriate mathematical model could be easily developed to analyse the resulting numbers.
I posted a couple of weeks ago. I’m not sure if you receive the post.
Everything you’ve submitted has been cleared through the queue.
Models such as yours involve deriving numbers from non-numerical data, a process of numerisation.
False. I have countable data for my prior, and I frame all my likelihoods in terms of countable data and expectancies.
If you think there is something incorrect about my counting, then you need to pick an actual example and explain how the count is different than I estimate.
You have yet to ever do so.
My objection to your use of Bayes’ theorem has nothing to do with its internal consistency, or the soundness of your reasoning when discussing the internal mathematics of it.
This is a lie. You started saying exactly that, and I refuted you. Now you are pretending none of that ever happened. This is dishonest.
My objection is that using Bayes’ theorem has led you to numerise the historical data in a specific way, to give numbers that can be the used as a substrate for Bayes’ theorem. Furthermore it has led you into a branch of mathematics which is particularly intolerant of errors and uncertainties in the numerisation process.
False. It is very tolerant when you allow for those uncertainties in your margins of error. Exactly as I do throughout OHJ. Explicitly.
And that was not your objection originally. If it is your “new” objection, then you have to actually show it: pick a case where my numbers don’t fit historical data, and show what number range would better fit, and why.
My position is that your numerisation process is so weakened by its inexactness, its variability from one observer to another, and its lack of repeatability that it cannot form the basis of any convincing model, especially one where your numbers are used directly in probability calculations.
It’s entirely repeatable. And is explicitly inexact: the uncertainty is built in to my estimating. If you want to show you get different results from the evidence, you have to show you get different results from the evidence.
You have only two options here: show different numbers are more defensible on available data; or show no numbers are defensible on available data. The latter gets you historicity agnosticism: if no probabilities can be estimated, then the probability Jesus existed cannot be estimated. This is absolutely necessarily the case: as I demonstrate in Proving History, Chapter 4. As I have said to you, repeatedly. Stop ignoring me.
This leads me to think that neither of you understood what I’m actually suggesting which is not to develop a new method using the same input data but to develop a different method of numerisation and then build the mathematical model around that method.
You have to. You cannot say “Jesus probably existed,” which is a mathematical statement, and not be able to justify that statement. And as it is a mathematical statement, only a mathematical justification is possible for it.
So by what means do you derive the conclusion “Jesus probably existed”? If it’s not Bayes’ Theorem, then what is it? And how is it logically valid?
You can’t dodge this question. If you cannot show you have any logically valid way to get that conclusion, then you have no logically valid basis for asserting that conclusion. By contrast, I have shown I have a logically valid way to do that. So you have none, I have one. One beats zero.
These are your options.
There are no other options.
We have told you this repeatedly.
Stop dodging what we keep telling you.