信息检索与数据挖掘 2019/5/513 M算法示例：求两个硬币随机抛出后正反面的慨率 不完全信息：不知道每次抛的是A还是B 9H1T→0.80*9H1T+0.20*9H1T Expectation maximization 如果是A硬币，抛出 →7.2H,0.8T+1.8H,0.2T E-step →0A= 7.2 1.8 HHHHTHHHHH的概率是，(1- 04)=0.69×0.41=0.010077696×0.4 7.2+0.8 1.8+0.2 Coin A Coin B ≈0.004 T下THTNTN 0.45X 0.55X =2.2H.2.2T ≈2.8H,2.8T 如果是B硬币，抛出 HHHHTHHHHH ■日s8gg▣■88gggg88无8日88g。9”。目””” ,0.80X 0.20X ≈7.2H,0.8T ≈1.8H,02T HHHHTHHHHH的概率是0，(1- HTHTTTHHTT 884g”1t88g88gg 0.73x 0.27X =5.9H,1.5T =2.1H,0.5T 0g)1=0.510≈0.001 035X 0.65X =1.4H,2.1T =2.6H,3.9T =0.60 0.65× 0.35X =4.5H,1.9T =2.5H,11T 日850 =21.3H,8.6T =11.7H,8.4T 故这次试验是硬币A的期望为 21.3 0.004/(0.004+0.001)=0.80,是硬币B 0213+8.60.71 M-step 的期望为0.004/(0.004+0.001)=0.20 11.7 60-m+840.58 60=0.80 6=0.52 1.EM starts with an initial guess of the parameters. 2.In the E-step,a probability distribution over possible completions is computed using the current parameters.The counts shown in the table are the expected numbers of heads and tails according to this distribution. 3.In the M-step,new parameters are determined using the current completions. 4.After several repetitions of the E-step and M-step,the algorithm converges. 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 13 EM算法示例：求两个硬币随机抛出后正反面的概率 不完全信息：不知道每次抛的是A还是B 1. EM starts with an initial guess of the parameters. 2. In the E-step, a probability distribution over possible completions is computed using the current parameters. The counts shown in the table are the expected numbers of heads and tails according to this distribution. 3. In the M-step, new parameters are determined using the current completions. 4. After several repetitions of the E-step and M-step, the algorithm converges. 故这次试验是硬币A的期望为 0.004/(0.004+0.001)=0.80，是硬币B 的期望为0.004/(0.004+0.001)=0.20 9H1T0.80*9H1T + 0.20*9H1T 7.2H,0.8T + 1.8H,0.2T 𝜃መ 𝐴 = 7.2 7.2+0.8 , 𝜃መ 𝐵 = 1.8 1.8+0.2 如果是A硬币，抛出 HHHHTHHHHH的概率是𝜃෠ 𝐴 9 (1- 𝜃෠ 𝐴 ) 1=0.69×0.41=0.010077696×0.4 ≈ 0.004 如果是B硬币，抛出 HHHHTHHHHH的概率是𝜃෠ 𝐵 9 (1- 𝜃෠ 𝐵 ) 1=0.510≈ 0.001 C. B. Do and S. Batzoglou, "What is the expectation maximization algorithm?," Nature Biotechnology, vol. 26, p. 897, 08/01/online 2008