4

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 $A \in \tau,B \in \mathcal{B}$?

Isaac
  • 199
  • 2
    Conditional probability is defined in terms of the conditional expectation of the indicator variable. For the measure-theoretic account of that, please see https://stats.stackexchange.com/questions/230545. – whuber Aug 17 '22 at 16:28
  • 1
    @whuber, I don't think this answers why we use $A|x$ instead of $A|B$. – Isaac Aug 17 '22 at 16:38
  • 1
    I had hoped it would be clear that the conditional expectation is a function. – whuber Aug 17 '22 at 17:00
  • @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. – gbd Aug 17 '22 at 17:08
  • @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 $\tau$). – whuber Aug 17 '22 at 17:25
  • @whuber,, but then shouldn't $\mu_{\Theta \mid X}(A \mid x)$ be $\mu_{\Theta \mid X}(A \mid {x})$ so that $A \mid {x}$ is a set and $\mu$ a measure in a set. – gbd Aug 17 '22 at 17:58
  • @gbd It is conventional in mathematics to drop the set brackets in such instances. The context usually makes it clear what is meant. But I believe you don't need them anyway: I could be wrong, because I don't have a copy of the book, but I read this notation as a shorthand for $\mu_{\Theta\mid X}(A\mid X)(x).$ – whuber Aug 17 '22 at 18:15
  • @whuber, but if you check the wikipedia article https://en.wikipedia.org/wiki/Probability_measure#Definition you find that $\mu(A|B)=\frac{\mu(A\cap B)}{\mu(B)}$. What is the relationship between $\mu(A|B)$ and $\mu_{\Theta|X}(A|x)$ here? – Isaac Aug 17 '22 at 18:50
  • Integration. All the subtleties pertain to conditioning on sets of measure zero, for which the Wikipedia formula does not apply. – whuber Aug 17 '22 at 18:56
  • @whuber, could you please expand on your comment. I not sure I understand the difference. – Isaac Aug 17 '22 at 18:59

0 Answers0