Transition Kernel and Conditional distribution
Let [math](E,\mathcal{E})[/math] and [math](F,\F)[/math] be two measurable spaces. A transition kernel from [math]E[/math] to [math]F[/math] is a map
- [math]\nu(x,\cdot)[/math] is a probability measure on [math]\F[/math] for all [math]x\in E[/math].
- [math]x\mapsto \nu(x,A)[/math] is [math]\mathcal{E}[/math]-measurable for all [math]A\in\F[/math].
Example
Let [math]\rho[/math] be a [math]\sigma[/math]-finite measure on [math]\F[/math] and let [math]f:E\times F\to\R_+[/math] be a map such that
Then
is a transition kernel. An example for [math]f[/math] would be
The following two hold.
- Let [math]h[/math] be a nonnegative (or bounded) Borel function on a measurable space [math](F,\F)[/math]. Then
[[math]] \varphi(x)=\int_F h(y)\nu(x,dy) [[/math]]is a nonnegative (or bounded) measurable function on a measurable space [math](E,\mathcal{E})[/math].
- If [math]\rho[/math] is a probability measure on a measurable space [math](E,\mathcal{E})[/math], then
[[math]] \mu(A)=\int_E\nu(x,A)d\rho(x), [[/math]]is a probability measure on a measurable space [math](F,\F)[/math] for all [math]A\in \F[/math].
Let [math]X[/math] and [math]Y[/math] be two r.v.'s with values in a measurable space [math](E,\mathcal{E})[/math]. The conditional distribution of [math]Y[/math] given [math]X[/math] is any transition kernel [math]\nu[/math] from [math]E[/math] to [math]F[/math] such that for all nonnegative (or bounded), measurable maps [math]h[/math] on a measurable space [math](F,\F)[/math] one has
If [math]\nu[/math] is the conditional distribution of [math]Y[/math] given [math]X[/math], we get for all [math]A\in F[/math]
Assume that [math](E,\mathcal{E})[/math] and [math](F,\F)[/math] are two complete, separable, metric, measurable spaces endowed with their Borel [math]\sigma[/math]-Algebras. Then the conditional distribution of [math]Y[/math] given [math]X[/math], exists and is a.s. unique.
No proof here.
General references
Moshayedi, Nima (2020). "Lectures on Probability Theory". arXiv:2010.16280 [math.PR].