Distribution Graph Selectors

Why

We want to select a distribution graph to summarize some data.

Definition

Let $(G, A)$ be a typed graph on $\set{1, \dots , n}$. Let $S \subset \set{1, \dots , n}$. Let $x^1, \dots , x^n$ be a dataset in $A_S = \prod_{j \in S} A_j$ (see \sheetref{function_graphs}{Function Graphs}).

A distribution graph selector for typed graph $(G, A)$, dataset of size $n$, and indices $S \subset\set{1, \dots , n}$ is a function from datasets of size $n$ in $A_S$ to distribution graphs on $(G, A)$.

In the case that $S \neq\set{1, \dots , n}$ we call $S$ the observable (or data) indices and $T = \set{1,\dots ,n} \setminus S$ the hidden (or latent, nonobservable) indices. It is common for many authorities to use the notational convention $Z$ for $A_T$ and $X$ for $A_S$.

Let $p: \prod_{i} A_i \to [0, 1]$ denote the full joint distribution of a distribution graph. In this case, we call $p_{S}: A_S \to [0, 1]$ the observable distribution (or evidence distribution).