Last updated: 2021-04-20
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This vignette describes a very useful view of the EM algorithm 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 log-likelilhood".)
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 \(q\): \[H(q) = - E_q \log(q(Z))\]
sessionInfo()R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
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[29] compiler_3.6.0 pillar_1.4.6 backports_1.1.10 httpuv_1.5.4
[33] pkgconfig_2.0.3 This site was created with R Markdown