On two arguments for Bayesian regularity

Standard Bayesianism requires regularity: it requires that I not assign prior probability zero to any contingent proposition. There are two main reasons for this: one technical and one epistemological.

The technical reason is that it is difficult to make sense of conditionalizing on events with probability zero. (Granted, there are technical ways around this, but there are also problems with these.) But the difficulty of conditionalizing on events with probability zero does not give one any reason to prohibit assigning probability to zero to events that one would never conditionalize on.

The Bayesian agent conditionalizes on evidence. But while the question of what constitutes evidence is highly controversial, there are some plausible things we could say about what could and could not be evidence for beings like us. Thus, the proposition that it’s looking like the multimeter is showing 3.1V seems like the sort of thing that could be evidence for a being like us, but the conjunction of the propositions constituing Relativity Theory does not seem like the sort of thing that could be evidence for a being like us (maybe it could be evidence for some supernatural being that has an infallible vision of the laws of nature; and maybe God could make us be such beings; but we don’t need to adapt our epistemology to such out of the world possibilities).

If this is right, then the technical difficulties with conditionalizing on events with probability zero do not give us a good reason to assign a non-zero prior probability to Relativity Theory, or any other proposition that is not of the right sort to constitute a body of potential evidence (where a body of potential evidence is a consistent finite conjunction of individual pieces of evidence).

There is, however, a second reason not to assign prior probability zero to any contingent proposition. If we assign prior probability zero to some hypothesis H, say Relativity Theory, then the only way a body of evidence E could raise the probability of H to something non-zero would be if P(E)=0 (for if P(E)>0, then P(H|E)=P(HE)/P(E)≤P(H)/P(E)=0). Thus, if we assign prior probability zero to a hypothesis, it seems that we will be unacceptably stuck at probability zero for that hypothesis no matter what evidence comes in. This is not a merely technical reason: it is an epistemological one.

Note that this formulation of the second reason for regularity depends on the first, though in a subtle way. The first reason gave us reason to have regularity for evidential propositions, i.e., propositions reporting a body of evidence. The second reason, if formulated as above, tells us that if we should have regularity for evidential propositions, then we should also have regularity for contingent hypotheses that are not themselves evidential propositions.

But now notice that the second reason for regularity seems to show rather more if we think it through. The reasoning here is that propositions like Relativity Theory should be confirmable but if we assign credence zero to them, they are not confirmable (assuming the first reason successfully shows that all bodies of evidence have non-zero probability). But now notice that the requirement of confirmability for a hypothesis shows something a lot stronger than that the hypothesis have non-zero probability. For surely it is not merely our view that Relativity Theory should be confirmable given infinite time. Rather, Relativity Theory should be the sort of proposition that would be confirmable by observation prior to the heat death of the universe, or maybe even within a single human lifetime. But the number of potential pieces of observational evidence for a being like us is finite (there are only finitely many perceptual states our brain can distinguish), and gathering a piece of evidence takes a minimum amount of time, and if Relativity Theory starts with a sufficiently low prior probability, we have no hope of confirming it before the deadline.

Hence, the confirmability intuition, if correct, yields a lot more than regularity: it yields substantive non-formal constraints on the priors. We shouldn’t assign a prior of, say, 10⁻¹⁰⁰ to Relativity Theory, at least not if our priors for observational evidence propositions are anything like what we tend to think they are. I am not, however, claiming that every contingent proposition should be confirmable before the heat death of the universe. We would not expect the proposition that there have been 10¹⁰⁰⁰⁰ fair and independent coin tosses made over the lifetime of the universe and that they all turned out to be heads to be confirmable in this strong sense.

In any case, here is what I think has happened. The first reason for regularity, the technical one, only applied to potential bodies of evidence. The second, on the other hand, shows more than it claims: it yields non-formal constraints on priors that go beyond regularity. In particular, I think, the subjective Bayesian is on thin ice if they want to require regularity.