Let $(S, \mathcal{S}, \Pr)$ be a probability space. Let $(\mathcal{X}, \mathcal{B})$ be a standard Borel space and a measurable map $X : S \to \mathcal{X}$. Let $(\Omega, \tau)$ be a standard Borel space and a measurable map $\Theta : S \to \Omega$. Then

$$ \mu_{\Theta \mid X}(A \mid x) = \int_A \frac{f_{X \mid \Theta}(x \mid \theta)}{\int_\Omega f_{X \mid \Theta}(x \mid t) \, d\mu_\Theta(t)} \, d\mu_\Theta(\theta) $$ for all $A \in \tau$ and $x \in \mathcal{X}$. I have seen the above equation in Theory of statistics (Schervish, 2005) and in this post.

What I don't understand is that a measure deals with sets but $x$ is not a set here.

Does $A|x$ represent a set here?

Shouldn't the conditional measure $\mu_{\Theta \mid X}$ be a measure on a set difference $A|B$ where $A \in \tau,B \in \mathcal{B}$?

  • 2
    $\begingroup$ Conditional probability is defined in terms of the conditional expectation of the indicator variable. For the measure-theoretic account of that, please see stats.stackexchange.com/questions/230545. $\endgroup$
    – whuber
    Aug 17, 2022 at 16:28
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    $\begingroup$ @whuber, I don't think this answers why we use $A|x$ instead of $A|B$. $\endgroup$
    – Isaac
    Aug 17, 2022 at 16:38
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    $\begingroup$ I had hoped it would be clear that the conditional expectation is a function. $\endgroup$
    – whuber
    Aug 17, 2022 at 17:00
  • $\begingroup$ @whuber, I think OP is asking whether $x$ here is a single value or a set of values since measure functions are defined on subsets and not on single values. $\endgroup$
    – gbd
    Aug 17, 2022 at 17:08
  • $\begingroup$ @gbd Once again, conditional expectations are functions. Although not all the symbols in that integral are defined in this post, it is evident $f$ is some kind of a density function, whence $x$ does not designate a set: it is manifestly an element of $\Omega.$ $A$ explicitly is a set (it's in the sigma algebra $\tau$). $\endgroup$
    – whuber
    Aug 17, 2022 at 17:25


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