Graphical Models

Definition

The notation we shall use is defined in Table 1, and some core definitions are presented in Table 2. In each of the examples presented in Fig. 1, we are simply asserting that

$$\underbrace{p({x}_{A},{x}_{B}\vert {x}_{C})}_{\text{ function of three variables}} =\underbrace{ p({x}_{A}\vert {x}_{C})p({x}_{B}\vert {x}_{C})}_{\text{ functions of two variables}},$$

(1)

which arises by a straightforward application of the product rule (Definition 1), along with the fact that X _A and X _B are conditionally independent, given X _C (Definition 3). The key observation we make is that while the left-hand side of (Eq. 1) is a function of three variables, its conditional independence properties allow it to be factored into functions of two variables.

Graphical Models. Table 1 Notation