Conditioning
Conditioning
Given a measurable space, \((\Omega, \mathcal{F})\), let \(\mathcal{M}\) be the set of probability measures defined on this space. Then conditioning can be defined as follows:
Events
\[\begin{align*}
&C :: \mathcal{M} \to \mathcal{F}_+ \to \mathcal{M} \\
& C \ \mathbb{P} \ A \ B = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(A)} \end{align*}\]
Random Variables
\[\begin{align*}
&\tilde{C} :: \mathcal{M} \to (\Omega \to \mathcal{R}) \to \mathcal{F}_{\sigma(X)}\to \mathcal{M} \\
& C \ \mathbb{P} \ A \ B = \frac{\mathbb{P}(A \cap B)}{\mathbb{P}(A)} \end{align*}\]
Two things to note here:
- Notice the dependent structure between \(\mathcal{M}\) and \(\mathcal{F}_+\). The measure restricts the set of events that we can condition on.
- Notice also that if we rearrange the signature of the function, as done below, \(C \ \Omega\) would be the identity function.
\[C ::\mathcal{F}_+ \to \mathcal{M} \to \mathcal{M}\]