The Bayesian Expectation-Maximization-Maximization for the 3PLM

The Expectation Maximization Algorithm

This note represents my attempt at explaining the EM algorithm (Hartley, 1958; Dempster et al., 1977; McLachlan and Krishnan, 1997). This is just a slight variation on TomMinka’s tutorial (Minka, 1998), perhaps a little easier (or perhaps not). It includes a graphical example to provide some intuition. 1 Intuitive Explanation of EM EM is an iterative optimizationmethod to estimate some unknown ...

Expectation Maximization

The Expectation Maximization (EM) algorithm [1, 2] is one of the most widely used algorithms in statistics. Suppose we are given some observed data X and a model family parametrized by θ, and would like to find the θ which maximizes p(X |θ), i.e. the maximum likelihood estimator. The basic idea of EM is actually quite simple: when direct maximization of p(X |θ) is complicated we can augment the...

Expectation Maximization

The Noisy Expectation-Maximization Algorithm

We present a noise-injected version of the Expectation-Maximization (EM) algorithm: the Noisy Expectation Maximization (NEM) algorithm. The NEM algorithm uses noise to speed up the convergence of the EM algorithm. The NEM theorem shows that additive noise speeds up the average convergence of the EM algorithm to a local maximum of the likelihood surface if a positivity condition holds. Corollary...

The Expectation Maximization (EM) algorithm

In the previous class we already mentioned that many of the most powerful probabilistic models contain hidden variables. We will denote these variables with y. It is usually also the case that these models are most easily written in terms of their joint density, p(d,y,θ) = p(d|y,θ) p(y|θ) p(θ) (1) Remember also that the objective function we want to maximize is the log-likelihood (possibly incl...