If we have a posterior distribution $p(A,B|\theta)$, is it always true that $p(A,B|\theta) = p(A|\theta)p(B|\theta)?$
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1$\begingroup$ When everything is conditioned on θ, you can basically ignore it. (Or just pretend θ = "the sky is blue".) Is the statement true in that case? $\endgroup$– user541686Commented Apr 10, 2021 at 5:34
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No, it is not. In order for that to be true, $A$ and $B$ should be conditionally independent given $\theta$.