Conditional Distributions
Why
We want to speak of the pairwise conditional distributions of a particular joint distribution.1
Definition
Suppose $A_1, \dots , A_n$ is a list of finite sets and $p: \prod_{i = 1}^{n} A_i \to [0,1]$ is a distribution on the (finite) product $\prod_{i = 1}^n A_i$.
For $i \neq j \in \set{1, \dots , n}$, the conditional distribution of $i$ on $j$ is the function $p_{i \mid j}: A_i \times A_j \to \R $ defined so that that $p_{i \mid j}(\cdot, b)$ is the conditional distribution induced by conditioning on $\Set{a \in \prod_{i = 1}^{n} A_i }{a_j = b}$.
For $i,j = 1, \dots , n$ and $i \neq j$,
$p_i$, $p_{ij}$ and $p_{i\mid j}$ satisify
\[
p_{i \mid j}(b, c)p_{j}(c) = p_{ij}(b, c)
\quad
\text{for all } b \in A_i, c \in A_j
\]
- Future editions will rework this sheet. ↩︎