2011/03/31 by Janne V. Kujala, Kujala, Janne V.
Computer Science · Mathematics · #60A05 #60A10 #Bayesian Modeling and Causal Inference #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #msc:60A05 #msc:60A10 #stat.TH
paper · pdf · doi:10.48550/arxiv.1103.6136
10 pages
arxiv created 2011/03/31 · openalex publication_date 2011/03/31 · arxiv updated 2011/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. PX,Y << PX x PY, (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 PX|Y such that PX|y << PX for all y, 5. X has a conditional distribution PX|Y and a marginal density p(x) w.r.t. a measure μsuch that PX|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) / ∫ 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.