The Agnostic Inquirer | Bayes/Inference to the Best Explanation

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This is part of a series of posts in response to "The Agnostic Inquirer" by Menssen & Sullivan.
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One objection Menssen and Sullivan bring to the table is that perhaps revelatory claims cannot be evaluated because no good method is suitable for doing so. They resist the use of Bayes' theorem and put forward what they call the "Inference to the Best Explanation." I was quite perplexed by this, being an ardent follower of LessWrong, a site devoted to improving human rationality using a number of methods including, to an extremely high degree, Bayes' Theorem. Thus, I'm aware of a highly devoted community of thinkers who do believe that Bayes' Theorem is the way to go.

In any case, one of the objections was that since we need to establish the probability of an infinite number of possible worlds, no matter how small of a probability we assign to each, the individual probabilities will sum to infinity, which is intractable. But it's generally accepted that infinities don't exist in real life anyway. Thus, it would make no sense to apply Bayes' Theorem to an infinite number of infinite universes, anyway, since the only place the concept of an infinite number of universes can exist is in one's mind (and not even there, except as the simple words, "infinite number of universes").

They also object that the principle of parsimony is a valid one for examining theistic hypotheses:

When Swinburne sets up his Bayesian argument for theism, he takes the prior probability of h to be the intrinsic probability of h, and he vigorously defends the thesis that the intrinsic probability of a hypothesis is a function of its simplicity, which h takes to be an a priori matter... Yet, on close examination, the history of science might reveal that the simplest hypotheses have not proved in the main to be true.

It doesn't seem like the authors are familiar with the concept of parsimony. The best explanation is always the simplest one that accounts for all the evidence. Thus, it's not a matter of literal simplicity such as a theory about circular orbits of the planets. Sure, this has not proved to be true, despite being simpler (r² = x² + y² is simpler than 1 = x²/a² + y²/b²) precisely because such a theory failed to account for all of the evidence. But now imagine a much more complicated formula that also happens to draw the path of a particular ellipse (one escapes me). The standard elliptical formula would still hold, as it explains all evidence and is simpler. This is what is being referred to by simplicity... not just simplicity for simplicity's sake. Adding extra terms necessarily lowers prior probability, for A is always more probable than (A&B).

Next they bring up the often touted point that the numbers used are arbitrary.

The artificiality of trying to calculate the values required to apply the Bayesian approach in some areas can be illustrated by reflecting on the following question: Were the five books of the Pentateuch written by a single author, Moses?

I get the point of this question. The authors are trying to pick a complex issue filled with input from sociology, history, literary criticism, etc. to try and show how Bayes' Theorem fails to work. The problem is that it still holds. We can take all of that input and make our best estimates for probabilities and still come out with an answer. I think a great example of this is how Richard Carrier uses Bayes' Theorem to discuss the probability of the resurrection in Why I Don't Buy the Resurrection Story. He doesn't state that he knows exactly the values to use. He simply pulls facts together to support using values we could say are at the low or high end of the spectrum, and thus executes his calculations with all involved terms being "on the safe side." The same is possible for Moses and authorship of the Torah.

As their alternative, Menssen and Sullivan propose Inference to the Best Explanation (IBE) as follows:

1) If a hypothesis sufficiently approximates an ideal explanation of an adequate range of data, then the hypothesis is probably or approximately true.
2) h₁ sufficiently approximates an ideal explanation of d, an adequate range of data.
3) So, h₁ is probably or approximately true.

As you can imagine, they go one to discuss what they mean by an "ideal" explanation: valid logical form, true premises, the explanation uses a causal property in virtue of which the effect is observed, the explanation bottoms on fundamental substances and properties (unclear on this), and each of the above is satisfied to a high degree (they even state, for example, that something "may not only be true; [it] must be known to be true.").

Now, I had a quite difficult time discerning what, intrinsically, was different between this and Bayes' Theorem except that this confuses things with words and definitions rather than relying on the math. The above seem fairly obvious to me with respect to an explanation. If it's not logically valid, it's not really an explanation. If the premises aren't even true... then it clearly has issues.

And Bayes' Theorem appears to be doing exactly what Bayes Theorem is trying to do.

$$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$$

This says, we want to know the probability of A (some hypothesis) given the evidence, B. This is equal to the probability that the evidence, B, would be present given the hypothesis, A, multiplied by the probability that A would hold true in general, all divided by the probability that B would occur in general. The "ideal" explanation is one in which P(A|B) ends up being very high! That's it, folks.

The formula is intended to make sure we take into account the IBE term, "an adequate range of data." If the evidence has a low probability of occurring even if we grant that the hypothesis were true (P(B|A)), this will lower the value of the numerator. If the hypothesis if far fetched on it's own... this lowers the numerator. If the evidence has a high probability of occurring on it's own, with or without the support of the hypothesis, A, (P(B)), then the denominator increases, thus decreasing the final probability that B says anything at all about A's validity.

It's unclear what issue Menssen and Sullivan have with Bayes' theorem other than their objection that number assignments are arbitrary. But if that holds... then how aren't our thought experiments and armchair ponderings arbitrary when it comes to figuring out if h₁ sufficiently approximates an ideal explanation? One might as well take a stab at some min/max values for the various probability assignments in Bayes' Theorem to determine best/worst case results. This, again, is why I linked to Carrier's article above. If Carrier can appeal to theists to grant his various values for low/high estimates for various possibilities (surviving a crucifixion, a body being stolen, etc.) and Bayes' Theorem still comes up pretty bad even with a worst case analysis favoring theism, then we've learned something.

I inquired with LessWrongers on whether Menssen and Sullivan had anything with their IBE, and the result wasn't very favorable. I get that they are already predisposed to favor Bayes, but the comments I received at least contain some food for thought.