The Bible, History, and Bayes' Theorem
(This is posted in Biblical Errancy because it is sparked by discussion from these threads: The awkward fact of the baptism of Jesus, What Abe finds wrong with "The God Who Wasn't There", and The failed prophecies of the historical Jesus)
The first very big issue concerns the use of "evidence." You keep repeating your demand for evidence, and I repeat my answer that the primary evidence is contained in the set of the early Christian sources. I know you think this is very bad, very unreasonable, so I would like to explain.
To me, evidence isn't even the final destination. The final goal is to find the best explanations for the evidence, or to find explanations for the evidence that are most probable. The methods of finding the best explanations are such criteria as explanatory power, explanatory scope, plausibility, consistency, and minimum ad hoc. This set of criteria is known as the "Inference to the Best Explanation" or "Argument to the Best Explanation." It is designed for critical New Testament scholarship, but it can actually be applied to anything, and it is a great way to model debates in any subject of deciding beliefs concerning the objective reality.
Anything can be evidence if it relates to the origins of Christianity. The origins of Christianity are best reflected in the manuscripts of the Pauline epistles and the synoptic gospels (earliest compositions). Therefore, the most relevant evidence for any explanation relating to the origins of Christianity are the Pauline epistles and the synoptic gospels.
There is a continuing widespread error in the way we talk about "evidence." We conflate two very different principles of empiricism in our everyday language: evidence and explanations. When we talk about evaluating the evidence, it should be about evaluating the various explanations for the evidence. The "evidence" is actually somewhat fixed. It is the directly-observed objective reality. The evidence does not change, unless it is destroyed or if a researcher discovers something new about it. The evidence does not actually have values on a scales of quality. We should not be ranking some evidence as better than other evidence (though some evidence can be more relevant than others). If evidence in any form exists, then we are terribly misleading ourselves if we think of the evidence as non-existent, even if a hypothesis is extraordinary and the conclusions are extremely unlikely. I know that is generally the way everyone talks, but it can lead to a lot of unnecessary confusion, especially in a subject where evidence is scarce and ambiguous. Instead, the explanations for the evidence have values on a scale of quality. We can rank some explanations for the evidence as better than others.
So, when you say, "(By the way, have you ever wondered why John the Baptist, who by all accounts is a much less important and historically significant person than Jesus, nevertheless has more evidence for his existence than Jesus himself???)", I think it is a big mistake to quantify evidence like that, because the sources found in the Christian canon count as evidence! They are evidence that can have a bunch of explanations, and our goal should be to find the best explanation.
Let me give you an example. When Earl Doherty makes his case that the earliest Christians believed that Jesus was merely spiritual, he can't possibly do it by starting out with, "The Christian sources are not evidence." Anyone who says that can not possibly have any knowledge about how Christianity may have begun. Instead, he does and should use the Christian sources--the Pauline epistles and the synoptic gospels--as evidence. That evidence directly reflects what the earliest known Christians believed. Based on what Christians apparently believed, then we can try to find the best explanations for those beliefs.
Point taken, and I basically agree with your essential point that 'anything can be evidence'. I admit I was using the sloppy language of colloquial English when making my points about your evidence 'not being evidence'.
It is true that the way most people usually speak about evidence is problematic and can introduce confusion and ambiguity.
Also, I agree with you that what must be evaluated are the 'explanations' of the evidence that we have. The evidence usually stands on its own and doesn't change much (except in certain cases, such as forgeries, when previously accepted evidence must be directly questioned).
So, for the purpose of clearing the debris, I would like to introduce to you the idea of Bayesian probability theory, since it directly applies to the topics you raised in a crucial, fundamental way.
I hope to do this as a discussion, rather than an essay with a wall-o-text (you can be the judge of whether this prophecy has failed ). However, this topic will almost inevitably require a little extra reading work on your part. I'll try not to over-load you, though. I'm good at responding to questions, too, so if any material here seems too dense or abstract, let me know and I think I'll be able to bring it down to Earth and cut it down to size for you. That's why I think a discussion will work best, as I will tailor my points to your specific concerns and areas of interest.
(Also, forgive me if at times I sound pedantic. I'm writing this assuming that there will be some people reading this who are averse to math and/or statistics, even if you personally might be very comfortable with them. If the math is too slow and easy for you, just skim along and you'll do fine. It IS actually very easy math, to be honest.)
Introduction to Bayes' Theorem with An Easy Example
I believe the best way to start on this issue is to simply jump the first hurdle which is to understand the basic idea behind Bayesian probability calculation. So, here goes.
It's your birthday, and a friend Alice has just given you a present. "Guess what it is before you open it!" she says.
Now, it just so happens that you had made a wish list before your birthday, with only two things on it: A watch, and a keychain.
It also just so happens that your other friend Bob, who is 100% reliable and trustworthy, has told you that, "Alice definitely got you one of the two things on your wish list."
The present is in a small, non-descript box, wrapped in bright paper with a bow on it.
Having no further clues, you ask yourself, "What is the probability that she got me a watch vs. a keychain?"
Again, having no further clues, you decide that it is reasonable to give both options equal odds, and so you calculate correctly that this would imply that the probability that the present is a watch is 1/2 or 50%, and likewise the probability that it's a keychain is equally 1/2 or 50%.
You decide to see if you can figure out what's inside by shaking the box and listening to whether it rattles or not.
But first, prior to shaking the box, and based on 100% reliable information from Bob, you know that IF the box contains a watch, THEN there is only a 20% chance that the box will rattle. You likewise know that IF the box contains a keychain, THEN there is a 60% chance that the box will rattle.
You shake the box, and it rattles! The question is, given this new clue, what is the actual probability that the box contains a keychain?
It is intuitively obvious that a rattling sound should more strongly indicate that the box contains a keychain, so the probability should be more than 50%, but how much more?
You almost want to say that it should be 60%, but quickly catch yourself, because you realize that by the same reasoning there would be a 20% chance of a watch, but 60% + 20% doesn't add up to 100%, so that can't be right.
It's also intuitively obvious that the best explanation for the rattling sound is a keychain, since that has the highest likelihood of making a rattling sound. But it doesn't prove it's a keychain, because a watch could technically also make a rattling sound. If someone asked you to bet on it, you would bet on a keychain, but it would depend on the gambling odds of the wager, wouldn't it? If the pay-off was stacked heavily in favour of the watch, it might be worth betting on the watch even though it is not as likely as the keychain. Would it be worth, say at a 10 to 1 pay-off, taking the risk on that bet?
What would be fair odds for this bet? Given only the clues above, what is the best estimate of the actual probability that it really does contain a keychain and not a watch?
You have an intuitive feeling that there should be a way to figure out this probability question, but intuition alone is not enough to give a solid, definitive answer, even if that solid answer happens to be a probability estimate. Is it 60%? 70%? 80%? 90%?
Human intuition alone can only give a rough answer. But there's a better way.
How to Solve It!
State the Assumptions
There are two competing 'explanations', or 'hypotheses'. Trusty Bob's clue told you that either the box contains a watch, or it contains a keychain. Let's give these hypotheses names/symbols to make it easier to talk about them:
W - The box contains a watch.
K - The box contains a keychain.
Now, before you shook the box, you had a nice, clear idea that both hypotheses were equally likely, and so you (correctly, according to the assumptions of this example) determined these probabilities are also equal:
P(W) = 50% = 1/2 = 0.5
P(K) = 50% = 1/2 = 0.5
And also before you shook the box, you had reliable information about the likelihoods of hearing a rattle in each of the cases where one hypothesis or the other are true.
If W is true (a watch), then the probability of hearing a rattle (let's give a 'rattle sound' the name/symbol R) should be 20%. In the lingo of probability, you can state this as, "The probability of R, given that W is true," or:
P(R|W) = 20% = 0.2
And likewise, if K is true (a keychain), then the probability of R (a rattle sound) should be 60%. Or, "The probability of R, given that K is true":
P(R|K) = 60% = 0.6
Following so far? These are just symbols. I haven't said anything new yet. I've only put the assumptions of the example into math lingo. Now let's look at how to use these math symbols to get a solid answer to the probability question.
Make the Predictions
We don't know which hypothesis, W or K, is actually true yet, but we do have some information about what we should expect if the alternative hypotheses were true. (Side-note: Another word for 'expectation' is 'prediction'.)
Let's examine the possible cases:
1. There is a 50% chance that W is true. If W is true, then there is a 20% chance that the box will rattle when shaken, and an 80% chance it won't.
2. There is a 50% chance that K is true. If K is true, then there is a 60% chance that the box will rattle when shaken, and a 40% chance it won't.
Hmm. The 50% and 50% add up to 100%, the 20% and 80% add up to 100%, and the 60% and 40% add up to 100%.
Case 1 and Case 2 are mutually exclusive. Either W is true or K is true, and not both at the same time. And, within Case 1, where W is true, the possibilities are also mutually exclusive. Either the box rattles (R) or it doesn't (which we'll call ~R, or 'not R'). So within Case 1, the likelihoods 20% and 80% add up to 100% of the possible outcomes.
Likewise, within Case 2, where K is true, again R and ~R are mutually exclusive, and 60% plus 40% adds to 100% of the possible outcomes.
So, there are only four possible, mutually exclusive outcomes in this example:
W and R - There is a watch, and the box rattles.
W and ~R - There is a watch, and the box doesn't rattle.
K and R - There is a keychain, and the box rattles.
K and ~R - There is a keychain, but the box doesn't rattle.
It's actually quite straightforward to calculate the probabilities of these four, mutually exclusive outcomes:
P(W and R) = P(W) x P(R|W) = 0.5 x 0.2 = 0.1 or 10% chance
P(W and ~R) = P(W) x P(~R|W) = 0.5 x 0.8 = 0.4 or 40% chance
P(K and R) = P(K) x P(R|K) = 0.5 x 0.6 = 0.3 or 30% chance
P(K and ~R) = P(K) x P(~R|K) = 0.5 x 0.4 = 0.2 or 20% chance
Now we're getting somewhere. Note that 10% + 40% + 30% + 20% = 100%. We've covered all the possibilities.
But wait! We know something specific about these possibilities. We did shake the box, and it did rattle. So, using the process of elimination, we can rule out the 2 cases where the box didn't rattle. We are left with only two cases:
P(W and R) = 0.1 = 10%
P(K and R) = 0.3 = 30%
We already know that the box rattled, so in fact, R is a given. R is a clue, a piece of information that narrows down the possibilities.
Hmm... Unfortunately, these probabilities don't add up to 100% either.
But wait! They must somehow 'add up' to 100% because we already know that the box rattled! We are already in a universe where a non-rattling box is no longer possible, because that test was already done, and the results are in: It rattled. R is true. It no longer makes sense to talk about the 'possibility' of ~R. Those possibilities applied to the universe before we shook the box.
R is a given. What we want to determine now is the probability that the box contains a keychain (K) given that the box rattled (R)! This is the question in math-speak:
P(K|R) = ???
The probability that K is true, given that we know R is true, is what exactly?
Bayes' Theorem: The Better the Prediction, the Better the Hypothesis
Another way of thinking about the four mutually exclusive outcomes before we shook the box is that they are predictions, and perhaps it is intuitively easier to think about the problem in terms of what the hypotheses predict. The W hypothesis has predicted that R will occur 20% of the time, and prior to that, we predicted that W itself has a 50% probability. So, naturally, we predict that 'W and R' together should have a 50% x 20% = 10% joint probability.
Likewise, the K hypothesis predicts R with a probability of 60%, and K itself is predicted with 50% probability, so we predict that 'K and R' should have a 50% x 60% = 30% joint probability.
Well, we shook the box, and the prediction of R has come true! But wait, compared to K's prediction at 30%, W's prediction was pretty darn weak at a mere 10%.
It stands to reason that the hypothesis that made the best, most accurate prediction should be the one we favour at the end of the day. And not only that, but the stronger and better its prediction was, compared to the other competing hypotheses, the more favourably we should judge the hypothesis.
And that's where Bayes' Theorem comes in. We are only one step away from getting the correct answer to our probability question.
Bayes' Theorem basically says that we should update our initial estimates of the probabilities of competing hypotheses in a way that is proportional to the strengths of their predictions: The better the prediction, the better the hypothesis.
The proportion between 10% and 30% is the same proportion as the ratio 1:3. It is 3 times more likely that the box contains a keychain (K) than that it contains a watch (W). Converting the ratio 1:3 into a fractional probability for K, we divide P(K and R) by the total probability of P(K and R) plus P(W and R):
P(K|R) = P(K and R) / [ P(K and R) + P(W and R) ]
= 30% / [ 30% + 10% ]
= 30% / 40%
At last, we have a solid answer. After hearing the rattling sound, we can conclude that the probability that the box contains a keychain is 75%, and likewise the probability that it contains a watch is only 25%.
Prior to shaking the box, we were justified in claiming that the probability of a keychain was 50%, but after shaking the box and hearing the rattle, we realize that the keychain hypothesis had predicted that outcome 3 times better than the watch hypothesis. And so, we are justified in updating our probability estimates in a proportion of 3 to 1. Whereas, prior to shaking the box the probabilities were 1 to 1 even, after shaking the box they are now 3 to 1 in favour of the keychain hypothesis, which works out to a 75% chance of a keychain, which is 3 times bigger than the 25% chance of a watch.
The only thing left to do for this example is to wrap it all up in a nice little package (oh god, not another crappy metaphor....). Okay I won't go there. But seriously, the above example makes sense, but it's awfully messy. Seems like a lot of work for not much.
The nice thing about math is once you've worked out a solution to a problem, you can often generalize it so that it works for any problem of a similar kind. And this is what Bayes' Theorem is in practice:
P(H1|E) = P(H1) x P(E|H1) / [ P(H1) x P(E|H1) + P(H2) x P(E|H2) + ... + P(Hn) x P(E|Hn) ]
Which reads, basically, in English: The probability of a particular hypothesis H after observing some new evidence E is equal to the strength of H's prediction of E proportioned against the strengths of all the competing hypotheses' predictions combined. Where "the strength of H's prediction of E" is calculated as the joint probability of both H and E, which is equal to the prior estimated probability of H times the predicted likelihood that E would occur if H were true.
Translation from math to English
P(H1|E) -Probability of H1 given E. The probability that H1 is true, given that E is true. The posterior probability of H1, given E. The estimated probability of the hypothesis after having observed the new evidence.
P(H1) - Probability of H1. The probability that H1 is true. The prior probability of H1.The estimated probability of the hypothesis before having observed the new evidence.
P(E|H1) - Probability of E given H1. The probability that E is true, given that H1 is true. The conditional probability of E given H1. The likelihood of the new evidence under the hypothesis. The predicted probability that the new evidence would be observed, in the case where the hypothesis is assumed to be true.
P(H1) x P(E|H1) - The joint probability of both E and H1. The "strength of H1's prediction of E".
H1, H2, ..., Hn - The competing, mutually exclusive hypotheses.
[ P(H1) x P(E|H1) + P(H2) x P(E|H2) + ... + P(Hn) x P(E|Hn) ] - The marginal probability of E. The sum of the joint probabilities of E across all of the Hs. The "combined strengths of all the competing hypotheses' predictions".
How Is This Relevant?
The thing about Bayes' Theorem is that it is a theorem. Given the extremely basic postulates of probability, it can be and has been proven as axiomatically true as any other fundamental mathematical theorem.
The basic postulates of probability cover a lot of ground: All science, all statistics, all profitable casinos and lotteries, all applications of math to any form of speculation. This includes the study of history, and specifically the study of the Bible and the historicity of Jesus.
The above discussion of a mundane situation like guessing a birthday present is not just a matter of speculation, it is a strictly logical consequence of the assumptions of the problem. It is a logically valid argument: If the premises are true, then the conclusion is inescapable.
It may sound strange to speak of probability as having 'inescapable conclusions', and this is why I took the time to explore this simple example in detail and why I am emphasizing this point right now.
Imagine that you got the present, shook the box, heard the rattle, and concluded that the box contained a keychain with a probability of 75%. And then you open the box, and sure enough, it's a ... WTF!? It's a fucking watch! I thought you said it would be a keychain!
The inescapable conclusion of Bayes' Theorem is not that there is definitely a keychain in the box. It is that there is a 75% probability of a keychain in the box. There is still a chance that it's a watch, it's just not an even money bet. In fact, it's a bet with 3:1 odds in favour of the keychain. But 3:1 favoured horses can still lose the race. In fact, they will lose the race about 25% of the time in the long run.
Again, how is this relevant? Just that Bayes' Theorem is the axiomatically correct tool to use to judge how we should modify our probability estimates of various competing hypotheses.
Historical studies of the Bible have gotten along fairly well with the standard, accepted criteria for judging explanations (hypotheses) of different types of evidence. But for the particularly thorny and politically-tinged issue of the historicity of Jesus, they have so far failed to resolve the issue definitively. Although most scholars will dismiss the competing hypothesis of mythicism, they really do not have solid grounds to do so. While some of the standard historicity criteria are indeed good criteria (and they are supported by Bayes' Theorem when they are), others are weak and/or problematic (and Bayes' Theorem can help explain why).
In the case of Jesus' historicity, it doesn't require one to hold a mythicist position in order to see clearly that the standard criteria have failed miserably. All one need do is look at the plethora of conflicting and/or contradictory 'historical' Jesus hypotheses put forth by various supporters of Jesus' historicity.
Is Jesus a doomsday cult leader? Yes! Is Jesus a freedom fighter? Yes! Is Jesus a teacher? Yes! A rabbi? Yes! Unknown? Yes! Famous? Yes! Crucified? Yes! Not crucified? Yes!
You name it, Jesus was it, depending on which historian you're asking.
This is, not surprisingly, exactly the same situation that theists are in when they try to tell us who 'God' is. And it is exactly as suspicious as theism is. When the evidence does not clearly point in one direction, it is easy to imagine it pointing in whichever direction you happen to be looking. Jesus 'is' whoever you want him to be.
It doesn't require positing a mythicist hypothesis to see that this confusion is a problem that cannot be resolved with the intuitive tools of the standard historicity criteria, precisely because they are based only in intuition, and different historians with different perspectives (and undoubtedly different agendas in some cases) cannot agree based on intuition alone.
There is really only one way out of this in the long run, and that is to take a more principled, rational approach to the evidence and hypotheses in question. Hypotheses must be judged according to consistent, justifiable criteria, not according to fuzzy rules of thumb and personal guesses, however educated those guesses may be.
Bayes' Theorem can be applied to this situation to sort the wheat from the chaff.
Recommended reading is An Intuitive Explanation of Bayes' Theorem by Eliezer S. Yudkowsky
I think I'll stop here for now. I have much more to say, but it's better if I wait for your response on this one tidbit, I think.