Chance and credence

Mon 29 May 2006 12:02 AM

In his paper at the SEP, Alan Hajek argued for this analogy: One's degree of belief in P being equal to the objective chance of P is like one's categorical belief that P being true. That is, a degree of belief getting the world right consists in it matching the objective chances.

This position requires that there be objective chances, he argued that there are, and I'll presume it for now.

A problem for his position arose in various forms during the question and answer period. Consider a fair coin.* The chance that it will come up heads is .5. I flip it, and it does in fact come up heads. Before flipping it, I have degree of belief .5 that it will come up heads on that flip. After flipping it, I can either change to degree of belief to 1 or continue with degree of belief .5. Neither option seems very good. The former has me believing the truth about the flip, but I no longer acknowledge the fact that the coin flip was a chancy event. The objective chance that the flip would come up heads is still .5, even after it did actually come up heads. The latter would leave me ignoring evidence about how events actually turn out, in favor of intermediate degrees of belief which are presumed to preserve information about how they might have gone.

It seems to me, however, that this is not especially a problem with Alan's account. Rather, it is an artifact of representing his account in terms of the usual probabilist framework. The framework treats agents' degrees of belief as probabilities assigned to propositions in first-order logic, with probabilistic judgments represented as intermediate degrees of belief. That is clearly inadequate in this case. I do believe that the chance that the coin would come up heads is one-half, and I also do believe that it did come up heads.

There is a difference between a degree of belief which is meant to correspond to an objective chance and a degree of belief that is merely an intermediate belief about a matter that does not admit of degrees.

There are formal ways of representing this difference, but notice that it will not help to consider belief to be a probability distribution over probability assignments. If I am less than certain that the coin is fair, then there will be a distribution around .5. If I am less than certain that the coin came up heads, then there will be a distribution close to 1. A bimodal distribution with peaks at .5 and 1 would not represent either situation.

We might instead introduce an operator Ch(P,p), meaning that the chance that P is p. After the coin toss, my degrees of belief are DoB(H)=1 and DoB(Ch(H,.5))=1. This handles the distinction, but it also means that the probabilist story about degree of belief is doing no work here. We might just as well represent my beliefs as H and Ch(H,.5).

Although we might try other expedients, representing degrees of belief as probabilities is de rigeur. Philosophers often do it presumptively nowadays, just as it was usual to write sentences in first-order logic in the 1960s. Why? Here are a few possible explanations:

1. The usual probabilist framework works for many purposes. There is no need to use a more complicated formalism if you don't need to.

2. Subjective Bayesians deny that there are any such things as objective chances, except insofar as they can be recovered from convergence in subjective degrees of belief. They have no incentive to try and represent beliefs about things that don't exist. (I am not sure whether the Subjective Bayesian should say that no one actually has beliefs in objective chances or just that beliefs in objective chances are wrong-headed.)

3. There is no clearly correct way to accommodate the distinction. This is a problem, because many probabilists say that probability just is the logic of confirmation. This imperialism requires that there is one specific formal system and that it captures the relevant structure. Once we admit that belief involves structures that cannot be readily represented in the usual probabilist idiom, then we must admit that there is more to the logic of science.

* If determinism makes you think that you can't attribute objective chances to coins, suppose instead that it is a radioactive atom and that the event is it decaying during a period equal to its half-life.