Markov random field

Remarks

Two weaker properties are also used. In the local Markov property, each $X_v$ is conditionally independent of all non-neighbours given its neighbours $X_{N(v)}$. In the pairwise Markov property, $X_u \perp X_v \mid X_{V \setminus \set{u, v}}$ for all non-adjacent $u \ne v$. Global implies local implies pairwise, and all three are equivalent when the joint density is strictly positive. Example: on the path $1 - 2 - 3$, a Markov chain $X_1, X_2, X_3$ is a Markov random field, since $\set{2}$ separates $1$ from $3$ and $X_1 \perp X_3 \mid X_2$. Non-example: let $X_1, X_2$ be independent fair coins and $X_3 = X_1$. Then $X_1$ and $X_3$ are dependent given $X_2$, so $X$ is not a Markov random field on this path.