Bayesian evaluation for the likelihood of Christ's resurrection (Part 28)

(Continued from the previous post)

Miscellaneous thoughts


Here's a few more assorted thoughts:

I still think that you're too afraid of large odds. For example, my gut feeling is that 99.99% is far too small a limit on how certain we can be in history. I mean, we can make meaningful, almost empirical statements about everyone who existed. That should tell you that odds of something like 1e11 is not extraordinary. In fact, if we are quite certain in our statement about everyone who lived, we can add a few more orders of magnitude. I still mostly stand by my statement that a probability is not "too large" unless its log is "too large", although I have adjusted this "too large" value downwards somewhat. Or maybe a probability is not "too large" unless you can no longer meaningfully get at it empirically.

On a related note, you see that using the skeptical prior value of 1e-22 for the resurrection is not all just showmanship. Clearly, a prior value of 1e-11 is justified, in that Christianity is essentially claiming that Jesus was unique on all of world history. You'll probably want another 2 orders of magnitude on that to future-proof the claim for the possible increases in the human population. You'll then want another 4 orders to at least reach the 99.99% for "certain within limits of history" mark. That already brings us to 1e17 as the necessary Bayes' factor - which, incidentally, seems to be about what you come up with in your own summary. I'm just requiring another 5 orders of magnitude on top of that to really put the nail in the coffin, and to cover any unseen contingencies. You can call that "just showmanship" if you'd like, but it serves the honest purpose for which I have constructed it in my series - a prior smaller than any that a skeptic can justifiably ask for, which can still be overcome by the evidence.

There's also a few things to say about conspiracies - for example, I think that at sufficiently large N values they scale worse than independence. Everyone telling the truth scales exponentially, as you said, but a conspiracy has an additional factor where everyone's story has to match with everyone else's, and everyone has to get along with everyone else - meaning, there's an N! factor against conspiracies. This is why we rightly consider any hypothesis involving "a vast conspiracy" to be a crackpot theory. Fortunately, we don't have to worry about this kind of calculation, given that we have the historical data.

Lastly, I'm in total agreement with you that we should follow Truth wherever it leads. So I'm very glad for this discussion.

Summary of the issues


Here's a summary of the issues you brought up, and where I stand on them:

I agree that numbers like 1e300, 1e100, or even 1e54 are too large as the final, overall probability values, as they're beyond the limit of how certain we can be. I'm also backing up from my final value of 1e32 as the probability for the resurrection, which I thankfully did not state too forcefully too often.

I still think 1e54 or 1e44 can be used as Bayes' factors, given that they are the factors between two specific hypothesis and not the factors for the whole "true" or "false" hypothesis in aggregate. They should not be thrown out just for being too large.

I still think that 1e8 is a good estimate for the Bayes' factor for a human testimony, and in fact our discussion here has only strengthened my belief. I can perhaps be talked down to 1e7, or 1e6 in set circumstances, but as a rough, order-of-magnitude estimate, 1e8 is a perfectly serviceable value. My confidence in this has actually been strengthened considerably as a result of our exchange.

I agree that the "license plate effect" is real, and it has a number of fascinating and important implications. But it does not really affect my calculated Bayes' factor of 1e54. Instead, it's main function is to increases the Bayes' factor for a human testimony when it applies. This is what allows us to do things like believing that the Gospels, as a whole, are reliable.

Many of the testimonies for the resurrection can be considered independent, until they successfully knock out all "reasonable" priors, leaving only things like conspiracy theories as the leftovers. I agree that at this point, the possibilities for strongly correlated testimonies must be considered.

I agree that the odds for the resurrection is still very high even after considering things like conspiracy theories.

Farewell


Thank you again for taking the time to read my series and replying to me, Aron. Your reply was intellectually stimulating, and very useful!

(The series on the resurrection will continue on with this post.)

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 27)

(Continued from the previous post)

Planning out the stages of the whole argument


So, all that gives us that Bayes' factor of 1e54. Now, as I've said I'm okay with its large magnitude, under the specification that this is for a certain model evaluated under some well-justified degree of independence. Given its large value, Jesus would have clearly risen from the dead under these conditions. I also think it's clear that nearly no amount of partial-dependence hypotheses, like ordinary social pressures, can nullify this value. So I do think that this large Bayes' factor is useful in these broad cases.

But as you've pointed out, we now need to worry about low-probability hypotheses that would wipe out that Bayes' factor by asserting a near-total dependence of the evidence. That is to say, we need to consider hypotheses which are specifically constructed to allow for ignoring the evidence, like conspiracy theories. I had initially planned to simply dismiss things like conspiracy theories at the end, but this conversation with you has convinced me that I need to address them.

So here's my plan. I'll group the spectrum of prior possibilities for the "no resurrection" hypothesis into the following regimes, in the order of decreasing probabilities involved:

1. Largely independent testimonies
2. Partially dependent testimonies
3. Near-total dependency in testimonies (conspiracy theories, other theories specifically designed to allow for ignoring the evidence)
4. Epistemological obliteration (brain in a vat, multiverse, all just a dream, you can never know anything, possibilities you can't even think of, etc.)

I do plan on just dismissing that last one at the end. Possibilities 1 and 2 are taken care of with the enormous Bayes' factor of 1e54. So now, we must start considering possibility 3.

Future of the series


I think I can argue convincingly against hypotheses like conspiracy theories in the upcoming posts of my series. In particular, the part of the series on resurrection stories from non-Christian sources are not just a "Christianity is better than these others" compilation. It's a way to double check the Bayes' factor, and to provide protection against hypotheses like a conspiracy theory. If, in fact, a conspiracy (or alien interference, or malicious spirits, or whatever) can produce results like Christianity, then over the course of world history you can expect for it to have done so once before, or at least come somewhat close. But empirically, no such results exist. None even come remotely close. Christianity is a distinct outlier.

You said, in attempting to estimate a Bayes' factor from historical data, that "For the kinds of skeptical reasons I stated above, it would be hard to get this much above 10^11 by itself since then we run out of the ability to check how many potential parallels there are." But you can check not only how many parallels there are, but how close they come. Assuming independence, you can even put precise numbers on the Bayes' factor involved by measuring the degree to which Christianity is an outlier. Even without independence you can definitively say that that the Bayes' factor was at a minimum around 10^11, and likely a good deal larger.

That's the great thing about arguing from empirical, historical records. You can bypass all the calculations about the probability of conspiracies or exactly what kind of dependence the testimonies might have had, or whether it was some other kind of hypothesis that generated this dependence. All of that automatically gets incorporated into the historical data at their actually correct historical values, and all we have to do is to read off the final result.

Anyway, that's my plan for the future of the series. I'll look at the historical comparisons, then use it to argue that the Bayes' factor of 1e54 is about correct assuming mostly independence. Furthermore, the historical comparisons allow us to say that Jesus is very likely to have risen even after all the other hypothesis, such as a conspiracy, are taken into account. That would probably be a good time to re-iterate that the 1e54 was for a specific model, that the final probability value would be not so extreme in reality, but still plenty high.

If you looked at how I write for my blog, I generally make a final compilation post at the end of a long, multi-part series, where I clean things up and maintain it as the final link. There, I'll probably rearrange the material so that the issues you brought up are resolved from the beginning. I do want that final post to be good, and clear of sloppiness and error.

So if you would share your thoughts on the future of the series, I would be grateful.

(To be continued in the next post)

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 26)

(Continued from the previous post)

The "license plate effect", and its applicability to my calculations


Now, I acknowledge that your "license plate effect" is in fact real - that 1e8 can be split between the "license plate effect" and the remaining "human honesty factor". But for the examples that I provided, I disagree that the split between these factors is as extreme as 1e6 to 1e2. I mean, if you've won the lottery, there generally aren't 1e6 other things that you could have chosen to lie about that would be equally interesting.

But more importantly, the exact split doesn't matter. The examples from which I calculated that 1e8 value were all specifically chosen to be similar to the disciple's testimony for Christ's resurrection. They are all special, interesting, positive claims - indeed in each case they're probably one of the most interesting things that the person could talk about. So, as long as this similarity holds, my full value - both the "license plate effect" and the "human honesty factor" - is applicable to the disciple's testimony. As Bayes' rule says, posterior odds is prior odds times Bayes's factor. And that Bayes' factor doesn't care whether certain parts of it were from certain effects. The entire Bayes' factor applies, as long as it was calculated for a similar situation. This is why I'm not too concerned about, say, your counterexample of "[naively slapping] 8 orders of magnitude on a 1:1 odds proposition": because that does not correspond to the scenario that the disciples faced.

Now, this "license plate effect" and the "human honesty factor" does interact in interesting ways in cases of multiple testimonies. I'm not sure if this is what you were getting at when you said that "if I'm right about the 8 = 6 + 2 split, you can only discount that 6 once". But I believe something like that can happen, depending on the degree of dependence between the multiple testimonies.

This is how it would work out (using your numbers): if Peter testifies that Jesus rose from the dead, then you should give his testimony the full 1e8 = 1e6 * 1e2 Bayes factor - exactly as it worked out in my numerous examples. But if you then turn to John and ask "hey, is Peter telling the truth?" and John answers "yes", John should only get the 1e2, because he has now been put in a different kind of situation than Peter - more akin to answering an 1:1 odds proposition.

But if you then have someone completely random burst into the room afterwards - let's say Paul - who says "guys! Jesus rose from the dead!", then that testimony should again get the full 1e8 Bayes' factor, because it was made independently from the other two.

So this again turns into a question of independence. Fortunately, I do think there is a strong case to be made for independence among the three named witnesses I used - Peter, James, and Paul. I mean, you yourself gave a 1e6 factor to Paul's conversion because it was so unexpected that an enemy of Christianity would have such a drastic turnaround by the encounter with the risen Christ. That's a factor you gave out on top of the fact that people claimed to have seen the risen Christ, as an expression of the strong anti-correlation (beyond mere independence) you'd expect between Paul's testimony and the other's. That was an additional factor on top of what's in my calculations, and it amply covers any possible dependence between Peter and James. As for the remaining testimonies in 1 Corinthians 15, I did severely discount them to account for dependence. So I'm quite comfortable with my 1e54 value, with the aforementioned caveat that we're not yet considering things like conspiracy theories.

The main effect of the "license plate effect"


Now, as I said, I disagree with your 1e6:1e2 split for the "license plate effect", and the implication that this effect mainly serves to weaken the witness testimonies. I think that its more important function, by far, is to immensely strengthen the testimonies to which it applies. It works to "cancel out any amount of low prior probability", in your words. Or to empower the testimony with a Bayes' factor of something like 1e120, in my example of recording a chess game.

So I'm very glad you mentioned the effect, because it was somewhat foggy in my own mind, and it allows us to do things like justify the remainder of the stories in the Gospel accounts. These other stories just get filed under "more details" once the resurrection is accepted, whereas there's no way to cover the prior on all those stories on just a Bayes' factor of 1e8.

The Bayes' factor for a human testimony


But at the end of this discussion on the strength of a single testimony, I still pretty much stand by my 1e8 number. I think 1e7 is also quite reasonable, and it can drop to something like 1e6 in circumstances where the possibility of lying is distinctly real (e.g. claiming to an aid agency that a loved one died on 9/11, or when calculating how many times you've been lied to). Maybe I could be convinced to use 1e6 as a lower bound, instead of 1e8 as a likely value. In any case it would not really affect my series, as I've said that a human testimony is "within a couple of orders of magnitude of my answer", and even the lower value is large enough to overwhelm any possible prior against the resurrection.

As one more confirmation of that 1e8 number, take a look at this video - it show's a woman's reaction to an acquaintance claiming to have won the lottery. Now, did that woman seem like a gullible idiot to you? I didn't feel that way. She starts off quite skeptical, but not dismissively skeptical. You can then see the man's sincerity working on her. Her degree of belief is clearly somewhat close to even odds right before the numbers are confirmed. I think her overall reaction is pretty rational. Now, there are some small differences between the video and my examples. For instance, she knows that there's a winner out there, and the man making the claim is already an acquaintance - but on the other hand, this result is achieved with little effort on the man's part, taking only minutes of insistence. The man being an acquaintance also reduces the "licenses plate effect". On the whole, you can see her mind being pulled through a Bayes' factor of something like 1e6 within mere minutes, in good accord with rationality, in a situation pretty similar to what I described in my examples. So 1e8 for something like the disciple's testimony about the resurrection is quite reasonable, and remains the best value to use.

(To be continued in the next post)

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Bayesian evaluation for the likelihood of Christ's resurrection (Part 25)

And now, time for a short interlude in this series.

I've been in communication with Aron Wall of Undivided Looking. It's a great blog that people should check out, which covers much of the same subject matters as my blog. I've asked him for feedback on my series, and he graciously replied back with a very lengthy post. There are some thing in my series that he disagreed with, and there are some things in his reply that I disagreed with, and there are things that came up that merited further discussion - so I thought it best that I reply back in a few of my own posts.


Greetings


Aron,

Thanks for taking the time to read over my series and post your reply. It's been very helpful. Your reply was in fact a great deal more than what I had anticipated, so I thought it appropriate to respond in a few of my own posts.

Things I learned


Your explanation for exceedingly small probabilities being unreasonable, due to the need to take even crackpot theories into account at such extremes, makes sense. I gladly abandon my suggestion, in the comments of your blog, that scientific laws or the existence of the Roman empire can be asserted with 1e100+ odds. I was actually pretty uncomfortable with saying those things, but I couldn't put my finger on why I felt that way at the time - so this is in fact a relief.

In my own series, I have not very much strongly asserted that the probability I obtained (1e32) is definite. So I will take your suggestion that I will not take this value seriously, instead using it to demonstrate that Jesus plainly rose from the dead if we make a set of likely of assumptions. I will handle the other, very unlikely cases separately.

Fortunately, I'm still in the middle of my series, so I can add in things like this without too much trouble. For that reason, I'm glad we're having this discussion now, rather than after the end of the series.

Probabilities and Bayes' factors


As for a great deal of your other comments - I think there's some confusion between a probability and a Bayes' factor. Both of us have used words like these somewhat imprecisely, and clarifying our usage will resolve a number of things. So:

That 1e54 factor I cited is a Bayes' factor, not a probability. It is a ratio of in-model probabilities - and each model may just be a simple, computable hypothesis (e.g. "a random shuffle") or a complex aggregate thereof ("all the different possible ways that a deck of cards may be shuffled"). As such, I more or less stand by that number and am not bothered at all by its large magnitude - with the caveat that the value is for a specific model, and not for the resurrection hypothesis in the aggregate. Essentially, I'm saying that I'm ignoring crackpot theories for the time being (conspiracy theories, malicious spirits playing jokes on us, etc.). As a Bayes' factor, that 1e54 should not be thrown out simply because it is too large, although it is unreasonable to use it to calculate a final, aggregated probability. If I understood your point correctly, this should not be a problem.

The five sigma probability of 1e-6, although I called it a probability, should also really be interpreted as a Bayes' factor. The words get in the way here, because in null hypothesis significance testing, that five sigma value is a probability. But in the Bayesian framework it should really be a Bayes' factor between the null hypothesis and the "whatever result you got" hypothesis.

1e8 as the Bayes' factor for a human testimony


Furthermore, the 1e8 factor for the strength of a human testimony is also a Bayes' factor, not a probability. It is NOT saying that people lie only 1 out of 1e8 times, which would be clearly absurd as you pointed out. Rather, it's saying that your prior odds should be adjusted by that much based on a human testimony.

This difference resolves your counterexample of "how many times in my life have I been lied to?" Yes, there has been 1e5 situations where someone was tempted to lie, and you've maybe been lied to around 1e3 times. That works out to lying rate of 1e-2, or a truth to lie posterior odds of 1e2.

But each of these lies probably had a typical prior odds of 1e-3 ("I got into a car accident", "I got locked out of my house", "I went to Harvard", etc). You yourself have said that one in a million events happen all the time, so a prior odds of 1e-3 is not at all out of the ordinary. This is especially the case when someone is making a positive assertion that something happened ("I once bowled a perfect game"), rather than trying to deny something or get out of something ("I have to wash my hair").

So, that's a prior of 1e-3, and a posterior odds of 1e2, which gives us a Bayes' factor of 1e5. The other 3 orders of magnitude can be made up for by adding the "earnest, sincere" condition, and the fact that we're specifically taking about scenarios where people are tempted to lie to you. So, the value of 1e8 as the Bayes' factor for a human testimony is in good accord with your lying example.

That 1e8 number also still bears out through your license plate example. Would you doubt someone who's earnestly, sincerely claiming to have a license plate like 6DVL666? If the Bayes' factor for a human testimony is really about 1e8, you should have some doubt, but not a very strong doubt, that this person was telling the truth - this seems to be about the right level of skepticism required here. I don't think, for example, you'd find 10,000 or even 1000 liars to a single truth-teller in this scenario.

I'm quite sure of this because all the examples I gave in my series - winning the lottery, being struck by lightning, having a PhD in physics from Harvard, having a loved one killed in 9/11, etc. - all involve cases like the 6DVL666 license plate, and not like the 4ZIW623 license plate. They're all special, interesting, positive claims that someone might want to choose to lie about, where all the improbability is in the main claim and not in the details - and I still get values around 1e8. So in every example that I have examined or you have brought up, this value seems to hold up.

(To be continued in the next post)

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