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pgm_Bayesian_Network_fundamentals

时间:2015-12-02 20:26:48      阅读:177      评论:0      收藏:0      [点我收藏+]

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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。

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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.

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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

 

pgm_Bayesian_Network_fundamentals

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原文地址:http://www.cnblogs.com/kkangwing/p/5014002.html

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