信息检索与数据挖掘 2019/5/514 EM算法示例：求两个硬币随机抛出后正反面的概率 迭代过程分析 x=(x,x2,,x),x,∈{0,1，，10}第i次抛硬币试验中正面(head)朝上的次数 2=(31,22,,5),,∈{4，B}第i次抛硬币试验 ②求隐变量Z的后验概率 X和Z的联合概率 X=(K1=5,x2=9,x3=8,x4=4,X=7) P(z1=Ax;00) Z是隐变量 P(x,Z1=Ax1;0) P(21=B1x:00) E-step P(xz=B:) Coin A Coin B THHTHTH: 0.45 A =2.2H,2.2T ≈2.8H,2.8T 0.80 0.20 ≈72H,0.8T ≈1.8H,0.2T HTTTHHTT HHHTHHHTH 0.73 0.21 =5.9H,1.5T ≈2.1H,0.5T 0.35 0.65x 1.4H,2.1T ③从联合概率求边缘概率 6=0.60 0.65 ≈4.5H.1.9T 在88■gg中 6=0.50 ≈21.3H,8.6T a=Pa=Ax:00 g00ngg▣ggg▣0gge0g00ege 21.3 =P(z:=Bx:0) 21.3+8.6=0.71 M-step 11.7+8.4≈0.58 6o=0.80 60=0.52 C.B.Do and S.Batzoglou,"What is the expectation maximization algorithm?,"Nature Biotechnology,vol.26,p.897,08/01/online 2008信息检索与数据挖掘 2019/5/5 14 EM算法示例：求两个硬币随机抛出后正反面的概率 迭代过程分析 x = (x1, x2, …, x5), xi ∈ {0,1,…,10} 第i次抛硬币试验中正面(head)朝上的次数 z = (z1 , z2 ,…, z5 ), zi ∈ {A,B} 第i次抛硬币试验中被抛掷的硬币是A还是B ②求隐变量Z的后验概率 𝑷(𝒛𝒊 = 𝑨|𝒙𝒊 ; 𝜽෡ 𝑨 (𝟎) ) 𝑷(𝒛𝒊 = 𝑩|𝒙𝒊 ; 𝜽෡ 𝑩 (𝟎) ) X和Z的联合概率 𝑷(𝒙𝒊 , 𝒛𝒊 = 𝑨|𝒙𝒊 ; 𝜽෡ 𝑨 (𝟎) ) 𝑷(𝒙𝒊 , 𝒛𝒊 = 𝑩|𝒙𝒊 ;𝜽෡ 𝑩 (𝟎) ) x=(x1=5,x2=9,x3=8,x4=4,x5=7) Z是隐变量 ③从联合概率求边缘概率 𝜽෡ 𝑨 (𝟏) = 𝑷(𝒛𝒊 = 𝑨|𝒙𝒊 ; 𝜽෡ 𝑨 (𝟎) ) 𝜽෡ B (𝟏) = 𝑷(𝒛𝒊 = 𝑩|𝒙𝒊 ;𝜽෡ 𝑩 (𝟎) ) C. B. Do and S. Batzoglou, "What is the expectation maximization algorithm?," Nature Biotechnology, vol. 26, p. 897, 08/01/online 2008