Symmetry and Probability

Anubav Vasudevan
Columbia University
Judgments of symmetry lay at the heart of the classical theory of probability. It was by direct appeal to the symmetries exhibited by the processes underlying simple games of chance that the earliest theorists of probability were able to justify the initial assumptions of equiprobability which allowed them to compute the probabilities of more complex events using combinatorial methods, i.e., by simply counting cases. Nevertheless, in spite of the role that symmetry played in the earliest writings on the subject, in light of the fact it is only in highly contrived settings that a direct appeal to symmetries can suffice to determine the probabilities of events, many philosophers have been led to conclude that the concept of symmetry itself has, at best, a limited role to play in a general theory of probability. In this essay, I argue that this view of the matter is mistaken, and that judgments of symmetry, in fact, have an indispensible role to play in all probabilistic reasoning. In chapter 1, I provide a detailed account of symmetry-based reasoning and argue against the view that the judgments of relevance on which such reasoning is based must be construed in subjective terms if symmetry-based reasoning is to be applied to deterministic processes. In chapter 2, I argue that the two most plausible proposals for how to avoid an appeal to symmetry in the assignment of probabilities (viz., those which are based on a priori principles of epistemic conservatism or the observed frequencies of events) must themselves rely on implicit assumptions of symmetry if they are to defend themselves against the charges of incoherency and arbitrariness. In chapter 3, I consider a decision-theoretic example of symmetry-based reasoning, in which the appeal to symmetry arises in the context of an agent's choice of a deliberative methodology. In this context, the principle of symmetry amounts to the requirement that the agent avoid adopting a biased deliberative methodology, i.e., one which treats two equally legitimate sources of information differently. In the specific context of the exchange paradox, I propose an account of how biased information is to be handled, which, despite suffering from some important defects, does, I believe, capture some of our general intuitions about how a rational agent ought to adjust his expectations to correct for the effects of bias
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