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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:

  1. Notice the dependent structure between \(\mathcal{M}\) and \(\mathcal{F}_+\). The measure restricts the set of events that we can condition on.
  2. Notice also that if we rearrange the signature of the function, as done below, \(C \ \Omega\) would be the identity function.
\[C ::\mathcal{F}_+ \to \mathcal{M} \to \mathcal{M}\]