EM算法

本文主要是介绍EM算法，对大家解决编程问题具有一定的参考价值，需要的程序猿们随着小编来一起学习吧！

EM Algorithm

convex function: \(f''(x) \ge 0\) or \(H \ge 0\) (Hessian matrix when x is a vector)

\[E[f(x)] \ge f(EX) \]

EM can be proved that it make the likelihood function increase monotonically.

maximize the lower-bound on the likelihood \(\ell\), \(\log\) is a concave function. the process is

\[\theta_{t-1} \rightarrow^{assign} Q_{z,t-1} \rightarrow^{\arg\max} \theta_{t} \rightarrow Q_{z,t} \]

define:

\[J(Q,\theta)=\sum_{i} \sum_{z^{(i)}}Q_i(z^{(i)})\log \frac{p(x^{(i)},z^{(i)};\theta)}{Q_i(z^{(i)})} \]

the EM algorithm can also be defined as coordinate ascent on \(J\).

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