Religious Testimony and Prior Probability
March 31, 2010 — 16:03

Author: Josh Rausmussen  Category: Uncategorized  Comments: 4

I’d like to share some recent thoughts I’ve had about testimony in relation to prior probability. In particular, I’d like to motivate the claim that in many important cases of testimony (including religious testimony, depending upon one’s background beliefs), one’s estimate of the the prior probability of a claim has no bearing at all on the credibility of the testimony expressing that claim. Disclaimer: The discussion to follow is tentative and expressed by one who hasn’t studied the literature on the epistemology of testimony.
I used to suspect that one important reason why certain testimonies are not credible is that the prior probability of the claim reported is very low–perhaps lower than the prior probability that the testifier hasn’t lied or made a mistake. But here’s some evidence against thinking that testimonies are defeated merely by virtue of their having a low prior probability. My wife says to me, “I just bought a book. The book says ….” She reads me the first page of the book. Suppose she reads 30 sentences and suppose for simplicity that for each sentence, the prior probability that the book really records that sentence or something similar is Â½. (Naturally, it would be much lower). Then the prior probability that the book contains all those sentences (or similar sentences) is 1/2^30 = about 1/(1 billion). That’s pretty low. If she reads another page of 30 sentences, the prior probability drops to 1/(1 trillion * 1 trillion * 1 trillion * 1 trillion * 1 trillion). That’s ridiculously low. It’s so low that if low prior probabilities can defeat testimony, my wife’s testimony should be defeated well before she completes the second page. But surely her testimony isn’t defeated. Sure, she may have gotten a word or two wrong, but most likely the book really does contain sentences approximately like the ones she reports. Her testimony isn’t defeated by the remarkably low prior probability of her claim.

After thinking about the matter carefully (using ven diagrams to illustrate probabilities), I’ve come to suspect (perhaps even believe) that when evaluating the credibility of a claim, a useful question to ask is this:
Q. What’s more likely given my background knowledge: that subject S claims C without C being true or that subject S claims C with C being true?
A crucial point about Q is that that it compares probabilities of conjunctions, not conditional probabilities, such as P(A/B).
Consider how Q applies. It’s pretty unlikely (given my background knowledge) that my wife would claim that the book reports such and such without it actually reporting such and such. So her claim is credible by the following fairly simple reasoning: there seems to be a greater overlap of probability space between her making the claim and her claim being true than there is between her making the claim and her claim being false. Therefore, given that she has actually made the claim in question, it is more likely that the claim is true than that it is false. (In other terms: (i) P(testifying X and X is false) + P(testifying X and X is true) = P(testifying X) = 1 given what I hear; (ii) P(testifying X and X is false) < P(testifying X and X is true) according to my answer to Q; therefore, P(testifying X and X is true) > .5 given what I hear (from (i) and (ii)); therefore, P(X is true) > .5 given what I hear.)
Now one might object that one can’t very well compare the two conjunctions in question unless one already knows the prior probability of the claim C. But what has become apparent to me is that the reason we don’t have to estimate the prior probability of C is that it–whatever it turns out to be–is approximately the same as the prior probability of the proposition that my wife claims C. They are about the same because they have about the same internal complexity. My wife’s uttering the exact sequence of sounds that she does has about as low a prior probability as the book’s contains the exact sequence of sentences that it does. We might say that my wife’s testimony is a mirror of the situation (or proposition) she is reporting, where a mirror of X is something whose prior probability mirrors (is approximately equal to) the prior probability of X. Given that the proposition that my wife claims C is a mirror of C itself, the low prior probabilities cancel each other out; all that remains of significance is the degree of overlap between the probability space of her making the claim in question and the claim’s being true.
Consider a different case in which the testimony is not a mirror of the proposition testified. My wife tells me that she just finished flipping a quarter heads up by chance 60 consecutive times in a row. Should I believe her? I think not. For in this case, her making that claim is not a mirror of the claim itself. It’s not a mirror of it because she is much more likely to report a pattern than a random sequence, yet the prior probability of a recognizable pattern turning up isn’t in the slightest more likely than the prior probability of a random sequence. As a result, the ridiculously low prior probability of her claim in this case does indeed call into question the credibility of her claim.
So, to answer Q, it suffices to estimate two things: (i) whether or not the proposition that S claims C is a mirror of C, and (ii) the degree to which the respective probability spaces overlap.
The important point is this: If I’m right that Q is the question to ask, then (one’s estimate of) the prior probability of the claim is completely irrelevant to assessing its credibility as long as the proposition that a claim is made is a mirror of the claim itself (given one’s background knowledge).
This may have some useful implications concerning religious testimony. Consider the slogan, “extraordinary claims require extraordinary evidence.” The term, “extraordinary” might signify low prior probability. If it does, then the slogan is false since I don’t require extraordinary evidence to believe what my wife says about what the book says. But if prior probability doesn’t matter, then testimonies of (say) seeing Jesus alive days after he was crucified shouldn’t be doubted merely by virtue of the low prior probability of the claims.
“Extraordinary” might mean in conflict with other things we know or have good reasons to believe. And in that case, the slogan seems correct.
A question, then, is this: are claims of miracles (e.g., resurrected saviors) in conflict with other things we know or have good reasons to believe (such as of natural laws governing biological decay and death)? It all depends upon who the “we” refers to (different people have different background beliefs and experiences) and upon the historical/sociological/psychological/theological data surrounding the claim being made. Someone might think that if there were a God, it wouldn’t be unexpected for Him to perform miracles on occasion for certain good purposes. Suppose this person doesn’t have reasons to think that God does not exist, and nor does she have positive evidence to think that people who report acts of God are usually liars or confused about such claims. Then for her, a claim of a miracle (or sequence of events that wouldn’t happen without an act of God) performed for a good purpose would not be in conflict with her background beliefs. Thus, the slogan wouldn’t apply to her: she may take the miracle claim at face value; ordinary evidence will do just fine.
The important point, again, is that prior probability is not a relevant factor when it comes to evaluating the credibility of a claim so long as the making of the claim is a mirror of the claim itself. In general, claims about historical events mirror the events that occur (because the more complex the claim, the more complex the making of the claim).
Therefore, claims about historical events, be they miraculous or not, are not defeated merely by their having low prior probability.
One might think that when it comes to miracle claims passed down to us from thousands of years, credibility is diminished over time. Maybe so. But there are plenty of claims today of miracles–including claims of being visited by Jesus (see, for example, this testimony ). What I’m suggesting is that the credibility of a claim–religious or not–should not be diminished by the claim’s low prior probability if the making of the claim is a mirror of the claim itself.
At least, that’s my epistemological hypothesis to account for the case of my wife testifying to me what she read. I’m open to further considerations.

• Mike Almeida

But what has become apparent to me is that the reason we don’t have to estimate the prior probability of C is that it–whatever it turns out to be–is approximately the same as the prior probability of the proposition that my wife claims C. They are about the same because they have about the same internal complexity. My wife’s uttering the exact sequence of sounds that she does has about as low a prior probability as the book’s contains the exact sequence of sentences that it does.
I’m not sure how the fact that C and the claim that T(C) have the same internal complexity leads us to believe that they have the same priors in the case you describe. Suppose for instance that the book in question is extremely rare, and by complete chance happened to be discovered. Imagine that what the book states is that ‘President Truman (followed by a precise and extremely detailed and accurate description of what he happened to be wearing) approved the bombing of Hiroshima’. The prior probability of that claim is, presumably, the claim as it stands in the text prior ot being uttered by the discoverer of the text. That probability pretty high. But the prior probability that this phrase is uttered by the discoverer of that book prior to the discovery is extremely low (recall the detailed description). That’s true despite the fact that what is uttered mirrors in internal complexity the proposition uttered. Maybe I’m missing something here.

March 31, 2010 — 19:01
• Joshua Rasmussen

Mike,
Good thoughts. Internal complexity may not be the only relevant factor. One clarification: in the case I described, part of the claim was “the books says”; so, it is a claim about the book, not about (say) President Truman. But I do see your point: there can be claims with high internal complexity that don’t have a low “prior” probability were the prior probability is the probability prior to its being uttered. In that case, internal complexity doesn’t seem to be the only factor.
Perhaps what’s important is that the prior probability of the claim be at least as great as the prior probability of uttering of the claim…

April 1, 2010 — 8:21
• The reading from the text example is a nice one. I may have seen an example using a newspaper report of the lottery winning numbers–the antecedent probability is low. But your example has even lower probabilities, so it’s more vivid.
Note, however, that (assuming all the probabilities are strictly between 0 and 1) Q is equivalent to:
Q’. What’s bigger: P(C is not true|background and S claims C) or P(C is true|background and S claims C)?
So one is doing something equivalent to comparing conditional probabilities when one answers Q.
Also, priors do enter in at least in the following way: The prior probability of C is an upper bound on the probability that S claims C with C being true.
The larger point, though, seems correct.

April 1, 2010 — 8:43
• Joshua Rasmussen

I do realize that Q is equivalent to Q’, which is a nice point. My thought–which may be wrong–is that if one estimates the conditional probabilities, one normally does that by first having a sense of the conjunctive ones (perhaps subconsciously…)

April 1, 2010 — 9:21