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Bayes Theorem and Starting With Different Priors

01 Oct

One critique of using Bayes Theorem is the problem of subjective priors. The criticism is that, if multiple people start with different priors then the whole formula is useless for arriving at any sort of consensus due to the inherent subjectivity of the priors. The actual problem isn’t the priors but the conditionals, which is something you ideally should have no control over. Even if your priors are the same, if the conditionals are different then this is where Bayes’ Theorem would disintegrate into postmodernist subjective-silliness.

So for example, if we have two people who are dealing with the Synoptic Problem — one starts with a prior of 90% that Mark was written first and the other starts with a prior of 22% that Mark was written first — it almost doesn’t matter where you start at because accumulating all of the evidence will push the two priors towards each other. As long as your priors aren’t 0 or 1 since those aren’t probabilities. And actually, if your prior is really in line with reality, then it won’t move much no matter how much evidence is gathered. So for example:

Person 1 assumes the prior probability that Mark was written first, or P(Mark) = 90%. P(Mark) for the second person is 22%. The first evidence they analyze is the length of Mark. It is a fact that Mark is the shortest Synoptic Gospel (MSSG), so this is our evidence E (if Mark being the shortest gospel was only hypothetical, then the logic behind Occam’s Razor would apply, not BT). How likely is it that Mark would be the shortest gospel given that Mark was written first? A historian might look at this by doing a large survey of ancient works and see how many of the newer versions are shorter than the older versions. I’m not a historian so I wouldn’t know, but I would guess that the usual way it happens is that original compositions are shorter than the ones derived from them, though there could be shorter versions that came after the long version (like what’s argued is Marcion’s relation to Luke, at least until relatively recently). The word “usually” could be quantified in some way somewhere around 80% (since I don’t have the experience with historical documents that actual historians do). The third value we need is the false positive rate, or how many times a shorter document is actually a rewrite of a longer document. Again, not a historian, but this doesn’t seem like it happens a lot (5%). So for the sake of example BT would look like this:

Person 1: P(Mark | MSSG) = P(MSSG | Mark) * P(Mark) / [P(MSSG | Mark) * P(Mark)] + [P(MSSG | Not Mark) * P(Not Mark)]
: = .8 * .9 / [.8 * .9] + [.05 * .1]
: = .9931

Person 2: P(Mark | MSSG) = P(MSSG | Mark) * P(Mark) / [P(MSSG | Mark) * P(Mark)] + [P(MSSG | Not Mark) * P(Not Mark)]
: = .8 * .22 / [.8 * .22] + [.05 * .78]
: = .8186

Both priors moved up, which makes sense since shorter documents are usually earlier versions of longer documents (but again, not a historian here).

With our new priors, we look at some other evidence. Like the content only found in Mark such as Mk 8.22-26. How likely is it that this pericope would be in Mark given Mark’s priority? How likely is it that this pericope would be in Mark given some other Gospel’s priority, or restated, Mark added this pericope after reading the other Synoptics?

Again, just for example since I’m not a historian, P(Mk 8.22-26 | Mark) is “probable” or 80% and P(Mk 8.22-26 | Not Mark) is “extremely improbable” or 1%.

Person 1: P(Mark | Mk 8.22-26) = P(Mk 8.22-26 | Mark) * P(Mark) / [P(Mk 8.22-26 | Mark) * P(Mark)] + [P(Mk 8.22-26 | Not Mark) * P(Not Mark)]
: = .8 * .9931 / [.8 * .9931] + [.01 * .0069]
: = .9999

Person 2: P(Mark | MSG) = P(MSG | Mark) * P(Mark) / [P(MSG | Mark) * P(Mark)] + [P(MSG | Not Mark) * P(Not Mark)]
: = .8 * .8186 / [.8 * .8186] + [.01 * .1814]
: = .9972

As you can see, the two priors are starting to converge. And you would repeat the process for each piece of evidence both for and against Markan Priority, with the priors changed from the previous use of BT (i.e. the posteriors) functioning as the priors for the next line of evidence. Again, the driving force here is the conditional probabilities. Which makes sense since these two numbers are crucial for figuring out Bayes Factor, which determines how strongly the evidence either favors or disfavors your hypothesis.

Of course, in the example above, my conditionals actually are subjective since I’m not a historian.

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