【ML从入门到入土系列08】EM

1 理论

EM算法通过迭代求解观测数据的对数似然函数$L(\theta )=\log P(\mathrm{Y}|\theta )$的极大化，实现极大似然估计。每次迭代包括两步：

• $E$步：求期望
  $$Q\left(\theta ,{\theta }^{(i)}\right)=\sum _z\log P(Y,Z\mid \theta )P\left(Z\mid Y,{\theta }^{(i)}\right)$$
• $M$步：求极大
  $${\theta }^{(i+1)}=\arg \underset{\theta }{\max }Q\left(\theta ,{\theta }^{(i)}\right)$$

2 代码

class EM:
    def __init__(self, prob):
        self.pro_A, self.pro_B, self.pro_C = prob

    # E步
    def pmf(self, i):
        pro_1 = self.pro_A * math.pow(self.pro_B, data[i]) * math.pow(
            (1 - self.pro_B), 1 - data[i])
        pro_2 = (1 - self.pro_A) * math.pow(self.pro_C, data[i]) * math.pow(
            (1 - self.pro_C), 1 - data[i])
        return pro_1 / (pro_1 + pro_2)

    # M步
    def fit(self, data):
        count = len(data)
        for d in range(count):
            _ = yield
            _pmf = [self.pmf(k) for k in range(count)]
            pro_A = 1 / count * sum(_pmf)
            pro_B = sum([_pmf[k] * data[k] for k in range(count)]) / sum(
                [_pmf[k] for k in range(count)])
            pro_C = sum([(1 - _pmf[k]) * data[k]
                         for k in range(count)]) / sum([(1 - _pmf[k])
                                                        for k in range(count)])
            self.pro_A = pro_A
            self.pro_B = pro_B
            self.pro_C = pro_C

3 参考

理论：周志华《机器学习》，李航《统计学习方法》
代码：https://github.com/fengdu78/lihang-code