#### Michael I. Jordan概率图模型tutorial（一）

independence networks, Markov random fields, loglinear models, influence diagrams等很多模型也都属于概率图模型的范畴，下面从基本概念开始介绍：

## 1.Directed Graph

Define: Random variable $$X$$ is marginal independent of random variable $$Y$$ if, for all $$x_i \in dom(X), y_k \in dom(Y)$$ , we have

$P(X=x_i|Y=y_k) = P(X=x_i)$

Define: Random variable $$X$$ is conditionally independent of random variable $$Y$$ given random variable $$Z$$ if, for all  $$x_i \in dom(X), y_k \in dom(Y), z_m \in dom(Z)$$ , we have

$$P(X=x_i|Y=y_k, Z=z_m) = P(X=x_i, Z=z_m)$$ </span>

$P(S) = \prod_iP(S_i|pa(S_i))$

## 2.Undirected Graph

$P(A, B, C, D, E, F) = \Phi(A, B)\Phi(B,C)\Phi(C,D,E)\Phi(D,E,F)$

## 4.Inference

Inference的意思就是求边缘分布。

### 4.1.Inference of Directed Graph Model

$P(A, B, C, D, E, F) = P(A)P(B)P(C|A, B)P(D|C)P(E|C)P(F|D, E)$

$\phi(A, B, C) = P(A)P(B)P(C|A, B)$ $\phi(C, D, E) = P(D|C)P(E|C)$ $\phi(D, E, F) = P(F|D, E)$

$P(A, B, C, D, E, F) = \phi(A, B, C)\phi(C, D, E)\phi(D, E, F)$

We say a cycle in G is chordless is all pairs of vertices that are not adjacent in the cycle are not neighbors.

A graph is triangulated if it contains no chordless cycles of length greater than 3.