Mixture Models and the Expectation Maximization (EM) Algorithm

A Gaussian Mixture Model (GMM), which is a generative model for data $\mathbf{x} \in \mathbb{R}^{d}$ , is defined by the following set of parameters:

K Number of mixture components
A d-dimensional Gaussian $\mathcal{N}\left(\mu^{(j)}, \sigma_{j}^{2}\right)$ for every $j=1, \ldots, K$
$p_{1}, \ldots, p_{K}$ : Mixture weights
The parameters of a K-component GMM can be represented as:
$\circ \quad \theta=\left{p_{1}, \ldots, p_{K}, \mu^{(1)}, \ldots, \mu^{(K)}, \sigma_{1}^{2}, \ldots, \sigma_{K}^{2}\right}$
The likelihood of a point $X$ in a GMM is given as
$\circ \quad p(\mathbf{x} \mid \theta)=\sum_{j=1}^{K} p_{j} \mathcal{N}\left(\mathbf{x}, \mu^{(j)}, \sigma_{j}^{2}\right)$

Observed case

Let $K=2$ and let $[-1.2-0.8]^{T},[-1-1.2]^{T},[-0.8-1]^{T}$ be three observed points in cluster 1 and $\left[\begin{array}{cc}1.2 & 0.8\end{array}\right]^{T},\left[\begin{array}{cc}1 & 1.2\end{array}\right]^{T},\left[\begin{array}{ll}0.8 & 1\end{array}\right]^{T}$ be three observed points in cluster 2 .

$\mu_{1,1}=-1$
$\mu_{1,2}=-1$
$\mu_{2,1}=1$
$\mu_{2,2}=1$

Unobserved Case: EM Algorithm

Estimates of Parameters of GMM: The Expectation Maximization (EM) Algorithm

We observe $\mathrm{n}$ data points $\mathbf{x}{1}, \ldots, \mathbf{x}{n}$ in $\mathbb{R}^{d}$ , we would like to maximize the GMM likelihood with respect to the parameter set:

$\theta=\left{p_{1}, \ldots, p_{K}, \mu^{(1)}, \ldots, \mu^{(K)}, \sigma_{1}^{2}, \ldots, \sigma_{K}^{2}\right}$

Maximizing $\log \left(\prod_{i=1}^{n} p\left(\mathbf{x}^{(i)} \mid \theta\right)\right)$ is not tractable using the settings of GMMs.
The EM algorithm is an iterative algorithm that finds a locally optimal solution $\hat{\theta}$ to the GMM likelihood maximization problem.

E-Step

Assume that the initial means and variances of two clusters in a GMM are as follows:

$\mu^{(1)}=-3, \mu^{(2)}=2, \sigma_{1}^{2}=\sigma_{2}^{2}=4$

Let $p_{1}=p_{2}=0.5$
Let $x^{(1)}=0.2, x^{(2)}=-0.9, x^{(3)}=-1, x^{(4)}=1.2, x^{(5)}=1.8$
Using the formula of E-Step:

$p(j \mid i)=\frac{p_{j} \mathcal{N}\left(x^{(i)} ; \mu^{(j)}, \sigma_{j}^{2}\right)}{p\left(x^{(i)} \mid \theta\right)}$

$p(1 \mid 1)=0.29421$
$p(1 \mid 2)=0.62246$
$p(1 \mid 3)=0.65135$
$p(1 \mid 4)=0.10669$
$p(1 \mid 5)=0.053403$

M-Step

Using the formulae corresponding to the M-step,

$\hat{n}{1}=\sum{i=1}^{5} p(1 \mid i)=1.7281$
$\hat{p}{1}=\frac{\hat{n}{1}}{n}=\frac{\hat{n}{1}}{5}=0.34562$ $\hat{\mu}{1}=\frac{1}{\hat{n}{1}} \sum{i=1}^{5} p(1 \mid i) x^{(i)}=-0.53733$
$\hat{\sigma}{1}^{2}=\frac{1}{\hat{n}{1}} \sum_{i=1}^{5} p(1 \mid i)\left(x^{(i)}-\hat{\mu}^{(1)}\right)^{2}=0.57579 .$

A Gaussian mixture model can provide information about how likely it is that a given point belongs to each cluster.