Conditional Densities

Definition

Suppose $f: \R ^d \to \R $ is a probability density. For $i, j = 1, \dots , d$ and $i \neq j$, let $f_{i \mid j}: \R ^2 \to \R $ satisfy

\[ f_{ij}(\xi , \gamma ) = f_{i \mid j}(\xi , \gamma )f_j(\gamma ) \]

for $\xi , \gamma \in \R $. We call $f_{i \mid j }$ a conditional density of $f$.