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In Schervish's Theory of Statistics (1995) and again in A Measure Theoretic Formulation of Bayes' Theorem by @ArtemMavrin, the following equation was proved in detail

$$ \mu_{\Theta \mid X}(A \mid x) = \frac{\int_A f_{X \mid \Theta}(x \mid \theta) \, d\mu_\Theta(\theta)}{\int_\Omega f_{X \mid \Theta}(x \mid \theta) \, d\mu_\Theta(\theta)} $$

but this only calculates the conditional probability $P(A|x)$. What is the equation for $\mu_{\Theta \mid X}(A \mid B) $?

user3911153
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  • Do you mean $\mu_{\Theta|B}(A|B)$ in your last line? – jbowman Aug 13 '22 at 22:02
  • @jbowman, both $\Theta$ and $X$ are random variables here, while $A$ and $B$ are measurable sets. It is explained in ArtemMavrin post. – user3911153 Aug 13 '22 at 22:38
  • @jbowman I am interested in the equation for $P(A|B)$ while ArtemMavrin post only seems to qive an equation for $P(A|X=x)$ – user3911153 Aug 13 '22 at 22:41
  • Does $x \subseteq B$ in place of just $x$ do the job? – jbowman Aug 14 '22 at 02:43
  • @jbowman, do you mean $x\in B$? – user3911153 Aug 14 '22 at 08:21
  • No, $x$ can be a set of points, not just a single point. You can condition on, for example, $1 \leq x \leq 3$, not just on $x = 2.71$. I really meant $x = B$, though, not sure why I thought of the other notation. – jbowman Aug 14 '22 at 15:29
  • @jbowman, I think $x$ here means $X(\omega)=x$ where $\omega \in S$ the underlying probability space and $X$ is a measurable function (random variable). It is explained clearly in the OP's link. – gbd Aug 14 '22 at 19:33
  • @jbowman, $x\in \mathcal{X},\theta\in \Omega,B\subseteq \mathcal{B}$ and $A\subseteq \Omega$. – gbd Aug 14 '22 at 20:21

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