Many contemporary statisticians who use Bayesian methods characterize a prior probability distribution as “just another assumption” of a model (e.g. Gelman and Shalizi, p. 19), by which they mean that they choose priors in order to make their methods work well (in some sense) rather than to represent any individual’s prior beliefs.

I suspect that using priors in this way is a good practice, although I’m not in a position to form an independent judgment on this matter. One thing that strikes me as a bit odd about it is that prior distributions cannot be “assumptions” in the usual sense. I am used to scientists and statisticians using the word “assumption” for statements that they know are false, such as the “assumption” of a frictionless plane or of a likelihood function that contains the true distribution generating the data. In this case, however, the word “assumption” is being used to refer to a statement that is typically neither true nor false—probability distributions that cannot be thought of as propensities or long-run frequencies are generally taken to lack objective truth values. The claim that a given probability distribution represents a given agent’s degrees of belief may have a truth value, but the probability distribution itself typically does not.

I make this point not to be fussy about language, but because I think it might be illuminating. Many frequentists have objected that Bayesian methods are inappropriately subjective for use in science. Bayesians have responded to this concern in at least three ways: developing what they call objective Bayesian methods, arguing that subjectivity is to be embraced, and arguing that frequentist methods are equally subjective. They claim that frequentist methods are equally subjective because such methods also make assumptions that are not known to be true and in many cases known to be false. I have not known any frequentists to respond to this objection by pointing out that his or her assumptions, unlike a Bayesian’s, are at least objective in the sense that they have truth values. As a result—and here’s the essential point—his or her assumptions could themselves be investigated in the course of future inquiry, at least in many cases. (A Bayesian’s priors might change with further inquiry, but they would never take on truth values in typical cases. An objective Bayesian is no better off than a subjective Bayesian in this respect.) Disagreements about the assumptions of a frequentist analysis thus tend to lead to deeper inquiry, whereas disagreements about prior probability distributions can be washed out but not resolved.

This point has a mild Kuhnian flavor—science succeeds in part because scientists manage to avoid fruitless disputes about fundamentals. It also fits well with Norton’s picture of scientific knowledge as being like a medieval cathedral. Assumptions provide temporary scaffolding on which the structure is initially assembled. Each assumption eventually becomes unnecessary as pieces of inductive knowledge come to be supported by other pieces of inductive knowledge. The whole is ultimately grounded in the bedrock of experience, but not in the rigid, hierarchical manner that epistemological foundationalists typically imagine.

My one point of dissatisfaction with this metaphor is that medieval cathedrals can reach the point where they rest entirely on solid ground with no scaffolding at all. Because of Agrippa’s trilemma, I don’t see how even a small part of our knowledge could be completed in the same way. The only non-skeptical options I see are to insist that certain bits of scaffolding shouldn’t bother us (foundationalism), to build a freestanding cathedral in space (coherentism), or to insist that it’s “cathedral all the way down” (infinitism). I prefer the skeptical option of learning to live with the fact that although the proportion of the structure that is scaffolding might shrink toward zero, it can never reach that point.

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