# Expectation-Maximization (EM) Algorithm

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An Expectation-Maximization (EM) Algorithm is a deterministic nonparametric maximum marginal likelihood estimation algorithm that alternates between performing an expectation (E) step (which computes an expectation of the likelihood by including the latent variables as if they were observed) and a maximization (M) step (which computes the maximum likelihood estimates of the parameters by maximizing the expected likelihood found on the E step).

## References

### 2009

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Expectation-maximization_algorithm
• … Given a statistical model consisting of a set $\mathbf{X}$ of observed data, a set of unobserved latent data or missing values $\mathbf{Z}$, and a vector of unknown parameters $\boldsymbol\theta$, along with a likelihood function $L(\boldsymbol\theta; \mathbf{X}, \mathbf{Z}) = p(\mathbf{X}, \mathbf{Z}|\boldsymbol\theta)$, the maximum likelihood estimate (MLE) of the unknown parameters is determined by the marginal likelihood of the observed data $L(\boldsymbol\theta; \mathbf{X}) = p(\mathbf{X}|\boldsymbol\theta) = \sum_{\mathbf{Z}} p(\mathbf{X}, \mathbf{Z}|\boldsymbol\theta)$ However, this quantity is often intractable.

The EM algorithm seeks to find the MLE of the marginal likelihood by iteratively applying the following two steps:

1. Expectation step (E-step): Calculate the expected value of the log likelihood function, with respect to the conditional distribution of $\mathbf{Z}$ given $\mathbf{X}$ under the current estimate of the parameters

$\boldsymbol\theta^{(t)}$: $Q(\boldsymbol\theta|\boldsymbol\theta^{(t)}) = \operatorname{E}_{\mathbf{Z}|\mathbf{X},\boldsymbol\theta^{(t)}}\left[ \log L (\boldsymbol\theta;\mathbf{X},\mathbf{Z}) \right] \,$

2. Maximization step (M-step): Find the parameter that maximizes this quantity:

$\boldsymbol\theta^{(t+1)} = \underset{\boldsymbol\theta}{\operatorname{arg\,max}} \ Q(\boldsymbol\theta|\boldsymbol\theta^{(t)}) \,$

• Note that in typical models to which EM is applied:
1. The observed data points $\mathbf{X}$ may be discrete (taking values in a finite or countably infinite set) or continuous (taking values in an uncountably infinite set). There may in fact be a vector of observations associated with each data point.
2. The missing values (aka latent variables) $\mathbf{Z}$ are discrete, drawn from a fixed number of values, and there is one latent variable per observed data point.
3. The parameters are continuous, and are of two kinds: Parameters that are associated with all data points, and parameters associated with a particular value of a latent variable (i.e. associated with all data points whose corresponding latent variable has a particular value).
• (Wu & Kumar, 2009) ⇒ Xindong Wu, and Vipin Kumar, editors. (2009). “The Top Ten Algorithms in Data Mining.” Chapman & Hall. ISBN:1420089641
• http://www.ics.uci.edu/~smyth/courses/ics274/notes4.pdf