Friday, July 15, 2011

A Primer on Bayesian Philosophy: 4- Interpretation and Subjectivity (part 2)

A brief summary is in order. The roughly common ground in the following post is that rational agents assign credences which obey the probability calculus (probabilism) and update probabilities using Bayes' theorem (conditionalization). Means of constraining `reasonable' probabilities using evidence are not universal, but some heuristics are accepted (calibration). The conjunction of these items is a decent description of empirical subjective Bayesianism, referred to simply as subjective Bayesianism.

There are two areas of dispute which characterize objective Bayesianism. The first is degree of calibration, and the second is equivocation.1 As far as I am aware, there is no clear consensus concerning the former; the category is rather soft. The more important dispute concerns equivocation.

There are two applications of equivocation: that which is applied in the absence of relevant evidence and that which is applied after calibration. This are features of the same rule, but the problems feel a little different. Equivocation, as applied in the absence of calibrating evidence, is the principle of indifference a.k.a. the principle of insufficient reason. You list your N (atomic) possibilities and assign each one a probability of 1/N (the discrete uniform distribution). If there the sample space is infinite, you assign the continuous uniform distribution.

The main problem is the most obvious. The goal of equivocation is to ascribe `informationless' priors in order to avoid unjustified prejudice for or against propositions in the absence of evidence.2 The uniform distribution is not an assumption, but is itself a result of assuming that one knows nothing about the distribution (!). (One derives this by comparing the unknown distribution to a null data set.) But admirable as this might seem, the results are absurd.

Suppose that you draw a ball from an urn which you are told contains white and coloured balls in some unknown proportion, and that the coloured balls are either red or blue. What sort of ball will you draw? Your data seem to be neutral, in the first instance, between its being white or coloured. Hence according to the Principle of Indifference, the probability that it will be white is one half But if it is coloured, the ball is red or blue, and the data are surely neutral between the ball's being either white or blue or red. Hence according to the Principle of Indifference, the probability of the ball's being white is one third (Howson and Urbach, p.45).

The underlying problem is that equivocation depends on partition choice, the way in which the possibilities are enumerated. For a common and more devastating example, suppose that you are told that a square fence with side length between 10 feet and 20 feet is being constructed. By equivocation, you should assign prob(`side length is below 15 feet')=1/2. But the question is logically equivalent to the following: what is the probability that the area enclosed by a square fence is below 225 given that the area is between 100 and 400 feet squared? Equivocating, you should say that it is (225-100)/(400-100)=125/300. Oddly enough, the `informationless prior' depends on how the question is formulated. Modern reformulations of equivocation, including MaxEnt, run into similar problems.

So an equivocation norm would in fact prejudice us against certain possibilities and deems certain agents unreasonable on a priori grounds. The Principle of Indifference and MaxEnt have their uses, but these are subsumed within the more general subjective account.

There is of course much more to this dispute. Readers may consult Howson and Urbach (1989) and Williamson (2010) for details, but some further overview is provided by Elliott Sober in Bayesianism -- Its Scope and Limits.

1. See section F in the SEP article. Note also that Williamson and others do not accept general conditionalization.
2. See (Howson and Urbach, pp.45-54) for a discussion.