pgm_Bayesian_Network_fundamentals

A Bayesian network is :

a) A directed acyclic graph (DAG) G, which nodes represent the variables X1,...,Xn. (有向无环图，每个结点表示变量)

b) for each node Xi, a CPD P(Xi|ParG(Xi)) is assigned. ParG(Xi) 表示 图中Xi 的双亲结点. CPD： Conditional Probability Distribution；但是某些无入边的结点（即没有父结点）的CPD可能就是其概率分布。

一个Bayesian Network 表示一个 joint distribution ,via chain rule for the BN:

　　P(X1,...,Xn) = ∏iP(Xi|ParG(Xi)) ； note that BN is a legal Distribution : i.e., ΣP = 1.

定义 P factorizes over G:

Let G be a graph over X1,...Xn, P factorizes over G if P(X1,...,Xn) = ∏iP(Xi|ParG(Xi))

3 types of Reasoning:

a) Causal Reasoning : reasoning from up to bottom. (沿着有向边的指向进行的reasoning)

b) Evidential Reasoning: bottom-up.

c) InterCausal reasoning: reasoning between two causes of a single effects. (V-structure)

Influence 的概念（理解的不好，还要重新再看一下）:Y influence X for that: condition on Y change the Beliefs about X. 即基于Y的观测会改变我们对X的分布的认识.

V-structure 的概念： X-->Z<--Y 在没有evidence 的情况下，X无法影响到Y。即X，Y之间无法形成一条active trail。

Active Trail的概念： A trail X1...Xn is active given Z if:

　　　　　　　　　　1) for any Xi-1-->Xi<--Xi+1 ; we have that Xi or any of its descendents belong to Z ;

　　　　　　　　　　2) no other Xi in Z

　　　　　　　　　　也就是说在一个trail里面，对任意的V-structure Xi-1-->Xi<--Xi+1， Xi必须可以被观察到，且Xi 及其descendents之多只有一个被观测到。

***************************************************************************************************************************

Independency & Factorization

a Joint distribution P(X,Y,Z) is proportion to Φ1(X,Z)*Φ2(Y,Z) ,Then (X is independent on Y |Z)

也就是说对一个分布P 进行factorize 的过程 揭示了分布P中的变量之间的independencies。

d-separated 的定义：

X and Y are d-separated in G given Z if : there is no active trail in G between X and Y given Z : d-sepG(X,Y|Z)

定理： If P factorizes over G, and d-sepG(X,Y|Z), then P sattisfies (X is independent on Y|Z).

推论： Any node is d-separated from its non-descendents given its parents.

(*) If P factorizes over G, then in P, any variables is independent of its non-descendents given its parents.

***************************************************************************************************************************

I-maps (Independency Map)

I(G) = {(X is independent of Y|Z):d-sepG(X,Y|Z)} 如果P 满足上式， 那么G 就是P的一个I—map。

而 P factorizes over G <==> G is an I-map for P