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Chapter 12 Bayesian Networks

2022-08-05 08:09:00 【Sang Zhiwei 0208】

1 概率公式

条件概率： $P(A|B)=\frac{P(AB)}{P(B)}$

全概率公式： $P(A)=\sum_{i}P(A|B_{i})P(B_{i})$

贝叶斯公式(Bayes)： $P(B_{i}|A) =\frac{P(A|B_{i})P(B_{i})}{\sum_{j}P(A|B_{j})P(B_{j})}$

2 贝叶斯公式

2.1 Thinking brought by Bayesian formula

given some samples,Calculate a conclusion in these samples $A_{1},A_{2},...,A_{n}$ 出现的概率,即 $P(A_{i}|D)$

$maxP(A_{i}|D)=max\frac{P(D|A_{i})P(A_{i})}{P(D)}$ 贝叶斯公式

$max\frac{P(D|A_{i})P(A_{i})}{P(D)}=max(P(D|A_{i})P(A_{i}))$ sample given,则对于任何 $A_{i},P(D)$ 是常数,Normalization factor only.

$max(P(D|A_{i})P(A_{i})) -> maxP(D|A_{i}$ ：忽略 $P(A_{i})$

$maxP(A_{i}|D) -> maxP(D|A_{i})$ ：If these conclusions $A_{1},A_{2},...,A_{n}$ The prior probability is equal（或近似）,can be deduced from this.

2.2 贝叶斯公式的应用

金条问题：

Let these three boxes be B=1,B=2,B=3, Two precious metals are M=G(金条),M=S(silver bullion)

所以已知： $P(B=1)=P(B=2)=P(B=3)=\frac{1}{3}$

P(M=G|B=1)=1,P(M=S|B=1)=0

P(M=G|B=2)=0,P(M=S|B=2)=1

$P(M=G|B=3)=\frac{1}{2},P(M=S|B=3)=\frac{1}{2}$

问题就转化为求 P(B=1|M=G)=?

解答： $P(B=1|M=G)=\frac{P(B=1,M=G)}{P(M=G)}=\frac{\frac{1}{3}}{\frac{1}{3}+0+\frac{1}{3}\cdot \frac{1}{2}}=\frac{2}{3}$

2.3 贝叶斯网络

The random variables involved in a research system,根据是否条件独立绘制在一个有向图中,就形成了贝叶斯网络.
贝叶斯网络(Bayesian Network),Also known as directed acyclic graph model,It is one of the probabilistic graphical models,根据概率图的拓扑结构,考察一组随机变量 $\left \{ X_{1},X_{2},...,X_{n} \right \}$ 及其组条件概率分布.
Probabilistic graphical models are divided into Markov network models（无向图）and Bayesian network models（有向图）.
一般而言,贝叶斯网络的有向无环图中的节点表示随机变量,它们可以是可观察到的变量,或隐变量、未知参数等.连接两个节点的箭头代表此两个随机变量是具有因果关系(或非条件独立).若两个节点间以一个单箭头连接在一起,表示其中一个节点是“因(parents)”,另一个是‘果(children)”,两节点就会产生一个条件概率值.
A simple Bayesian network

2.4 Full Bayesian network

There are edge connections between each pair of nodes

$p(x_{1},...,x_{K})=p(x_{K}|x_{1},...,x_{K-1})...p(x_{2}|x_{1})p(x_{1})$

$P(X_{1}=x_{1},...,X_{n}=x_{n})=\prod_{i=1}^{n}P(X_{i}=x_{i}|X_{i+1}=x_{i+1},...,X_{n}=x_{n})$

举例说明：当K=5时 $p(x_{1},...,x_{5})=p(x_{5}|x_{1}...x_{4})p(x_{4}|x_{1}...x_{3})p(x_{3}|x_{2}x_{1})p(x_{1})$

2.5 "正常"的贝叶斯网络

Some edges are missing
如下图所示：直观上 $x_{1},x_{2}$ 独立, $x_{6},x_{7}$ 在 $x_{4}$ independent under the given conditions
$x_{1},x_{2},...,x_{7}$ 的联合分布为： $p(x_{1})p(x_{2})p(x_{3})p(x_{4}|x_{1},x_{2},x_{3})p(x_{5}|x_{1},x_{3})p(x_{6}|x_{4})p(x_{7}|x_{4},x_{5})$

举例说明：

例一：

由于呼吸困难（D）The cause is lung cancer（C）and bronchitis（B）,Hence the above table（CPD）.

例二：

The joint distribution of all random variables is ：
$P(j,m,a,\overline{b},\overline{e})=P(j|a)P(m|a)P(a|\overline{b},\overline{e})P(\overline{b})P(\overline{e})=0.9\times 0.7\times 0.001\times 0.999\times 0.998\approx 0.00063$

实际上,Joint distribution if required,Only a topology diagram needs to be given,And the probability distribution table between each random variable is enough.