Parameterized Distribution Graphs

Definition

Let $G$ be a directed graph on $\set{1,\dots , n}$. A parametric distribution graph family (or parametric distribution network familiy, parameteric conditional distribution network family) is a family of conditional distribution networks $\set{G, \set{g_i^{(\theta )}}_{i=1}^{n}}_{\theta \in \Theta }$. We call the index set $\Theta $ the parameter set. $G$ does not depend on the parameters.

In the case that $\pa_i = \varnothing$ in $G$, $\set{g_i^{(\theta )}}_\theta $ is a parametric distribution family on $A_i$ and in the case that $\pa_i \neq \varnothing$, $\set{g_i^{(\theta )}}_{\theta }$ is a parametric conditional distribution family on $A_i$ from $\prod_{j \in \pa_{i}} A_j$ (for both these terms, see Parameterized Distributions).

A parametric distribution network family is functionally parameterizable if each of the conditionals is functionally parameterizable (again, see Parameterized Distributions).