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 \mu x \nu,   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 \mu,   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 \mu such that P_{X|y} << \mu for all y.   Furthermore, given random variables X and Y with a conditional density p(y | x) w.r.t. \nu and a marginal density p(x) w.r.t. \mu, we show that Bayes' formula p(x | y) = p(y | x)p(x) / \int p(y | x)p(x)d\mu(x) yields a conditional density p(x | y) w.r.t. \mu if 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. 
Janne V. Kujala
math.ST stat.TH