A Remark on the Assumptions of Bayes' Theorem

We formulate simple equivalent conditions for the validity of Bayes' formula for conditional densities. We show that for any random variables X and Y (with values in arbitrary measurable spaces), the following are equivalent: 1. X and Y have a joint density w.r.t. a product measure μx ν, 2. P_{X,Y} << P_X x P_Y, (here P_{.} denotes the distribution of {.}) 3. X has a conditional density p(x | y) w.r.t. a sigma-finite measure μ, 4. X has a conditional distribution P_{X|Y} such that P_{X|y} << P_X for all y, 5. X has a conditional distribution P_{X|Y} and a marginal density p(x) w.r.t. a measure μsuch that P_{X|y} << μfor all y. Furthermore, given random variables X and Y with a conditional density p(y | x) w.r.t. νand a marginal density p(x) w.r.t. μ, we show that Bayes' formula p(x | y) = p(y | x)p(x) / \int p(y | x)p(x)dμ(x) yields a conditional density p(x | y) w.r.t. μif and only if X and Y satisfy the above conditions. Counterexamples illustrating the nontriviality of the results are given, and implications for sequential adaptive estimation are considered.

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