概率图模型–因子图 – 潘登同学的Machine Learning笔记

文章目录

  • 概率图模型--因子图 -- 潘登同学的Machine Learning笔记
  • 简单回顾概率图模型
    • 回顾贝叶斯网络
    • 简单回顾马尔可夫随机场(MRF)
  • 因子图
    • 将贝叶斯网络用因子图表示
    • 将马尔科夫随机场用因子图表示
  • 总结

简单回顾概率图模型

概率图就是概率论+图论;

最大的贡献就是联合概率分布可以表示为局部势函数的连乘积;

回顾贝叶斯网络

将联合概率分布可以表示为局部势函数的联乘积

P(S,C,X,B,D)=P(S)P(C∣S)P(B∣S)P(X∣C,S)P(D∣C,B)P(S,C,X,B,D) = P(S)P(C|S)P(B|S)P(X|C,S)P(D|C,B) P(S,C,X,B,D)=P(S)P(C∣S)P(B∣S)P(X∣C,S)P(D∣C,B)

简单回顾马尔可夫随机场(MRF)

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A
D
B
C

P(A,B,C,D)=1Zϕ∏i=1kϕi(Di)=1Zϕϕ1(A,B)ϕ2(B,C)ϕ3(C,D)ϕ4(D,A)\begin{aligned} P(A,B,C,D) &= \frac{1}{Z_{\phi}}\prod_{i=1}^{k}\phi_i(D_i) \\ &= \frac{1}{Z_{\phi}}\phi_1(A,B)\phi_2(B,C)\phi_3(C,D)\phi_4(D,A) \\ \end{aligned} P(A,B,C,D)​=Zϕ​1​i=1∏k​ϕi​(Di​)=Zϕ​1​ϕ1​(A,B)ϕ2​(B,C)ϕ3​(C,D)ϕ4​(D,A)​

因子图

因子图其实是上面这些概率图模型的一个统一表述;

  • 因子图是一个二部图, 一边是变量xxx, 一边是因子fff;

变量就是自变量; 因子就可以理解为势函数, 也就是参数;

  • 定义

因子图是一类无向概率图模型, 包括变量节点和因子节点。 变量节点和因子节点之间有无向边连接。 与某个因子节点相连的变量节点, 为该因子的变量。 定义在因子图上的联合概率分布可以表示为各个因子的联乘积;

看！ 又是联乘积了对叭…

  • 用各个因子的联乘积表示上图
    p(x)=1Zϕ∏AfA(xA)p(x) = \frac{1}{Z_{\phi}}\prod_{A}f_A(x_A) p(x)=Zϕ​1​A∏​fA​(xA​)
    具体来说,
    p(x1,x2,x3)=1Zϕfa(x1,x2)fb(x1,x2)fc(x2,x3)fd(x3)p(x_1, x_2, x_3) = \frac{1}{Z_{\phi}}f_{a}(x_1, x_2)f_{b}(x_1, x_2)f_{c}(x_2, x_3)f_{d}(x_3) p(x1​,x2​,x3​)=Zϕ​1​fa​(x1​,x2​)fb​(x1​,x2​)fc​(x2​,x3​)fd​(x3​)

将贝叶斯网络用因子图表示

将贝叶斯网络用因子图表示，如下:

  • 数学表示:
    P(S,C,B,D,X)=fS(S)fC(S,C)fB(S,B)fX(S,C,X)fD(C,B,D)P(S,C,B,D,X) = f_S(S)f_C(S,C)f_B(S,B)f_X(S,C,X)f_D(C,B,D) P(S,C,B,D,X)=fS​(S)fC​(S,C)fB​(S,B)fX​(S,C,X)fD​(C,B,D)

  • 再来看看原本贝叶斯网络的数学表示

P(S,C,X,B,D)=P(S)P(C∣S)P(B∣S)P(X∣C,S)P(D∣C,B)P(S,C,X,B,D) = P(S)P(C|S)P(B|S)P(X|C,S)P(D|C,B) P(S,C,X,B,D)=P(S)P(C∣S)P(B∣S)P(X∣C,S)P(D∣C,B)

其实他俩一样对吧; 但是关键点就是这个P(S)

一般的P(S)我们就单纯的把他理解发生某件事为概率,如
P明天下雨=0.6;∴P明天下雨=0.4P_{明天下雨} = 0.6; \therefore P_{明天下雨} = 0.4 P明天下雨​=0.6;∴P明天下雨​=0.4

但是因子图, 把这样的概率表示成了因子节点, 所以整个因子图就把输入变量和因子节点分隔开, 这样虽然本质不变, 但是便于目标的求解;

将马尔科夫随机场用因子图表示

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A
D
B
C
  • 表示如下:

可以看到, 因子图的一组节点是输入变量, 另一组节点是原本的边, (也可以理解为对原图的所有边都做了一个细分同构)

其实就是把原本MRF的边当做了一些新的节点, 而MRF的边的含义就是势函数, 所以因子图把势函数当做了一些新的节点, 就把输入变量与势函数分隔开了;

  • MRF的细分同构

(就是在原本边上加了一个节点)

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A
f_B
f_A
B
f_D
D
f_C
C

可以看出这个图跟上面二部图其实是一样的, 只是视觉问题而已;

  • 数学表示:
    P(A,B,C,D)=1Zf1(A,B)f2(B,C)f3(C,D)f4(D,A)P(A,B,C,D) = \frac{1}{Z}f_1(A,B)f_2(B,C)f_3(C,D)f_4(D,A) P(A,B,C,D)=Z1​f1​(A,B)f2​(B,C)f3​(C,D)f4​(D,A)

  • 再来看看原本MRF的数学表示
    P(A,B,C,D)=1Zϕ∏i=1kϕi(Di)=1Zϕϕ1(A,B)ϕ2(B,C)ϕ3(C,D)ϕ4(D,A)\begin{aligned} P(A,B,C,D) &= \frac{1}{Z_{\phi}}\prod_{i=1}^{k}\phi_i(D_i) \\ &= \frac{1}{Z_{\phi}}\phi_1(A,B)\phi_2(B,C)\phi_3(C,D)\phi_4(D,A) \\ \end{aligned} P(A,B,C,D)​=Zϕ​1​i=1∏k​ϕi​(Di​)=Zϕ​1​ϕ1​(A,B)ϕ2​(B,C)ϕ3​(C,D)ϕ4​(D,A)​

其实他俩没啥区别吧, 所以因子图就是一个大一统的模型吧, 方便求解;

但其实也能看出他的一个缺点, 就是没有贝叶斯网络和MRF那样直观, 贝叶斯网络与MRF的因果关系都很显然, 但因子图借用了二部图会难以看出因果关系;

总结

联合概率分布的因子分解是概率图模型表示的核心概念, 大大降低了模型的复杂度

因子图就是这样了, 继续下一章吧！pd的Machine Learning

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