November 1, 2021

Borel and Bertrand

Snow Zhang, New York University

Abstract: The Borel-Kolmogorov paradox is the phenomenon that, sometimes, when P(E)=0, the probability of H given E appears to depend on the specification of the sigma algebra from which E is drawn. One popular diagnosis of the paradox is that it reveals a surprising fact about conditional probability: when P(E)=0, the value of P(H|E) depends on the choice of a sigma algebra. As Kolmogorov himself put it, “[the paradox shows that] the concept of a conditional probability with regard to an isolated given hypothesis whose probability equals 0 is inadmissible” (p.51, 1956).

This talk has two parts. The negative part raises some problems for the Kolmogorov-inspired relativistic conception of rational conditional probability. The positive part proposes an alternative diagnosis of the paradox: it is an instance of a more familiar problem — the problem of (conditional) priors. I conclude by applying my diagnosis to a different debate in the foundations of probability: the issue of countable additivity.