Last updated: 2021-04-20
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Knit directory: fiveMinuteStats/analysis/
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This vignette describes the EM algorithm, using the very useful view due to Neal and Hinton. [Note on notation: Neal and Hinton use \(Z\) to denote observed data, but I use it to denote latent unobserved values. So be sure to keep this in mind if you are trying to relate their paper to my summary here.]
Suppose we have data \(X\) from a distribution \(p(X|\theta)\) where \(\theta\) denotes parameters whose values are unknown and live in some space \(\Theta\). We aim to find the maximum likelihood estimate for \(\theta\) \[\hat\theta:= \arg \max_{\theta \in \Theta} p(X | \theta).\] Suppose further that we can write \(p(X | \theta)\) as an integration (or sum) over latent variables \(Z\). That is \[p(X | \theta) = \int p(X, Z| \theta)\] for some joint distribution \(p(X,Z| \theta)\). (When considered as a function of \(\theta\) this distribution, \(p(X,Z | \theta)\), is sometimes referred to as ``the complete data likelihood".)
The EM algorithm provides a convenient way to find \(\hat\theta\) when the complete data log-likelihood has certain “friendly” features, which will become apparent below.
Note: throughout this note I treat \(Z\) as a continuous random variable, using integrals and density functions. However, the same ideas apply if \(Z\) is discrete: you can just replace integrals with sums and replace density functions with probability mass functions.
Note: \(X\) and \(Z\) willl typically both be multi-dimensional, not just scalars.
Let \(Q\) denote the space of all “distributions on \(Z\)”. So \(q \in Q\) means that \(q\) is a distribution on \(Z\), with density \(q(Z)\) say.
Define the following function \(F\) that maps \((\Theta, Q)\) to the real line: \[F(\theta,q):= E_q \log p(X, Z | \theta) + H(q)\] where \(H(q)\) denotes the entropy of \(q\): \[H(q) = - E_q \log(q(Z))\] Note that \(F\) can be rewritten as: \[F(q,\theta) = E_q(\log p(Z|X,\theta) p(X| \theta)/ q(Z)) = KL[ q || p(Z|X,\theta) ] + \log p(X| \theta).\]
where \(KL(q,p)\) denotes the Kullback–Leibler divergence from \(q\) to \(p\).
There are two key results.
For any given \(\theta\) let \(\hat{q}_\theta\) denote the value of \(q\) that maximizes \(F(q,\theta)\). That is, \[\hat{q}_\theta:= \arg \max_q F(\theta,q).\] This optimal \(q\) is given by the conditional distribution of the latent variables \(Z\) given the data \(X\) and \(\theta\): \[\hat{q}_\theta(z) = p(Z | X, \theta)\]
The likelihood \(l(\theta):=\log p(X| \theta)\) is related to \(F\) by: \[l(\theta) = \max_q F(\theta,q)\] Both results follow directly from \[F(q,\theta) = KL[ q || p(Z|X,\theta) ] + \log p(X| \theta).\] and the fact that the minimum of \[KL(q || p)\] over \(q\) is 0, attained at \(q=p\).
It follow from Result 2 above that the maximum likelihood estimate \(\hat\theta\) can be obtained by jointly maximizing \(F\) over \(q\) and \(\theta\).
The EM algorithm simply does this by iterating the following two steps:
Step 1: \(q \leftarrow \arg \max_q F(q,\theta)\)
Step 2: \(\theta \leftarrow \arg \max_\theta F(q,\theta)\)
From result 1, step 1 is achieved by \(q = p(Z | X,\theta)\). Step 2 involves maximizing the expected complete-data log-likelihood, \[\theta \leftarrow \arg \max_\theta E_q \log p(X,Z|\theta).\] Of course the EM algorithm is only useful if both steps are straightforward.
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