Consider a probability space [itex](\Theta, \Sigma_\Theta, P_\Theta)[/itex], where [itex]P_\Theta[/itex] is a probability measure on the sigma-algebra [itex]\Sigma_\Theta[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

Each element [itex]x \in \Theta[/itex] maps onto another probability measure [itex]P_{\Omega | x}[/itex], on a sigma-algebra [itex]\Sigma_\Omega[/itex] on another space [itex]\Omega[/itex].

In this situation, one should (as far as I can see) be able to write write a measure-theoretic generalization of Bayes' rule

[tex]{P_{\Theta |y}}(A) = \int\limits_{x \in A} {\frac{{d{P_{\Omega |x}}}}{{d{P_\Omega }}}(y)d{P_\Theta }} [/tex]

for any [itex]A \subseteq \Theta [/itex], given an observation [itex]y \in \Omega[/itex] where

[tex]{P_\Omega } = \int\limits_{x \in \Theta } {{P_{\Omega |x}}d{P_\Theta }} [/tex]

and [itex]{d{P_{\Omega |x}}/d{P_\Omega }}[/itex] is the Radon–Nikodym derivative of [itex]P_{\Omega |x}[/itex] with respect to [itex]P_{\Omega}[/itex].

The problem is that I cannot see how to prove it (I;m sure the proof is fairly simple). Anyone wants to help?

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# How to prove Bayes' rule for probability measures?

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