In mathematical statistics, the term mixture model has two different meanings.

First definition

A mixture model is a model in which the independent variables are measured as fractions of a total. For example, suppose researchers are trying to find the optimal mixture of ingredients for a fruit punch consisting of grape juice, mango juice, and pineapple juice. A mixture model is suitable here because the results of the taste tests will not depend on the amount of ingredients used to make the batch but rather on the fraction of each ingredient present in the punch. The sum of the mixture components is always 100%, and a mixture model takes this restriction into account.

Second definition

A mixture model can also be a formalism for modeling a probability density function as a sum of parameterized functions. In mathematical terms,

$p\_\{X\}(x)\; =\; \backslash sum\_\{k\; =\; 1\}^\{K\}\; a\_\{k\}\; h(x|\backslash lambda\; \_k)$

where $p\_\{X\}(x)$ is the modeled probability distribution function, $K$ is the number of components in the mixture model, and $a\_\{k\}$ is mixture proportion of component $k$. By definition, for all $k\; =\; 1,\backslash dots,K$ and $a\_\{1\}\; +\backslash cdots+\; a\_\{K\}\; =\; 1$.

$h(x|\backslash lambda\; \_k)$ is a probability distribution parameterized by $\backslash lambda\; \_k$.

Mixture models are often used when we know $h(x)$ and we can sample from $p\_\{X\}(x)$, but we would like to determine the $a\_\{k\}$ and $\backslash lambda\; \_k$ values. Such situations can arise in studies in which we sample from a population that is composed of several distinct subpopulations.

Common approaches for estimation in mixture models

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It's common to think of mixture modeling (under the second definition) as a missing data problem. One way to understand this is to assume that the data points under consideration have "membership" in one of the distributions we are using to model the data. When we start, this membership is unknown, or missing. The job of estimation is to devise appropriate parameters for the model functions we choose, with the connection to the data points being represented as their membership in the individual model distributions.

Expectation maximization

The Expectation-maximization algorithm is one way to compute the missing memberships of data points in our chosen distribution model. It is an iterative procedure, where we start with initial parameters for our model distribution (the $a\_\{k\}$'s and $\backslash lambda\_\{k\}$'s of the model listed above). The estimation process proceeds iteratively in two steps, the Expectation Step, and the Maximization Step.

The expectation step

With initial guesses for the parameters in our mixture model, we compute "partial membership" of each data point in each constituent distribution. This is done by calculating expectation values for the membership variables of each data point. An example will provide some clarity. Let's consider a simple example. We have a collection of data points $x\_\{i\}$ that can be modeled as coming from a sum of two Gaussian distributions. The probability expression for our model is

$P(x\_\{i\})\; =\; (1\; -\; f)\; \backslash mathcal\{N\}(x\_\{i\};\backslash mu\_\{1\},\backslash sigma)\; +\; f\; \backslash mathcal\{N\}(x\_\{i\};\backslash mu\_\{2\},\backslash sigma)$

Where $f$ is the mixing coefficient in $*$ ($a\_\{k\; =\; 2\}\; =\; f$ and $a\_\{k\; =\; 1\}\; =\; 1\; -\; f$), and we assume $\backslash sigma$ is known and constant. For each of our data points $x\_\{i\}$, we can compute a membership value for each of the two Gaussians as follows

$y\_\{1,i\}\; =\; \backslash frac\{(1\; -\; f)\; \backslash mathcal\{N\}(x\_\{i\};\backslash mu\_\{1\},\backslash sigma)\}\{(1\; -\; f)\; \backslash mathcal\{N\}(x\_\{i\};\backslash mu\_\{1\},\backslash sigma)\; +\; f\; \backslash mathcal\{N\}(x\_\{i\};\backslash mu\_\{2\},\backslash sigma)\}$

and similarly for $y\_\{2,i\}$.

In the case of a Gaussian mixture model,

$\backslash mathcal\{N\}(x;\backslash mu,\backslash sigma)\; =\; \backslash frac\{1\}\{\backslash sigma\; \backslash sqrt\{2\backslash pi\}\}\; e^\{-\backslash frac\{(x\; -\; \backslash mu)^2\}\{2\; \backslash sigma^2\}\}$

The maximization step

With our expectation values in hand for group membership, we can recompute plug-in estimates of our distribution parameters. For the mixing coefficient $f$ this is simply the fractional membership of all data points in the second Gaussian.

$f\; =\; \backslash frac\{\backslash sum\_\{i\}\; y\_\{2,i\}\}\{N\}$

where $N$ is the total number of data points. For μ₁,

$\backslash mu\_\{1\}\; =\; \backslash frac\{\backslash sum\_\{i\}\; y\_\{1,i\}x\_\{i\}\}\{\backslash sum\_\{i\}\; y\_\{1,i\}\}$

With new estimates for $f$ and the $\backslash mu$'s, we proceed back to the Expectation step to recompute new membership values. The procedure is repeated until there is no further change in the mixture model parameters.

Markov chain Monte Carlo

As an alternative to the EM algorithm, we can use posterior sampling as indicated by Bayes' theorem to deduce parameters in our mixture model. Once again we regard this as an incomplete data problem where membership of data points is our missing data. We resort to a method called Gibbs sampling which is once again a two step iterative procedure.

We'll use the example from the previous section to demonstrate how the method works. We start again with initial guessed parameters for our mixture model. Instead of computing partial memberships for each elemental distribution, we draw a membership value for each data point from a binomial distribution (it will be assigned to either the first or the second Gaussian). The binomial parameter $\backslash theta$ is determined for each data point on the basis of one of the constituent distributions. Draws from the distribution generate membership associations for each data point. We can then use plug-in estimators as in the M step of EM to generate a new set of mixture model parameters, and return to the binomial draw step.

Spectral method

Some problems in mixture model estimation can be solved using Spectral Techniques. In particular it becomes useful if data points $x\_i$ are points in high-dimensional Euclidean space, and the hidden distributions are known to be log-concave (such as Gaussian distribution or Exponential distribution).

Spectral methods of learning mixture models are based on the use of Singular Value Decomposition of a matrix which contains data points. The idea is to consider the top $k$ singular vectors, where $k$ is the number of distributions we are trying to learn. The projection of each data point to a linear subspace spanned by those vectors, groups points originating from the same distribution very close together, and points from different distributions stay far apart.

One distinctive feature of the spectral method is that it allows to prove that if distributions satisfy certain separation condition (e.g. not too close), then the estimated mixture will be very close to the true one with high probability.

Other methods

Other methods which guarantee accurate estimation have emerged in the last few years. Some of them can even provably learn mixtures of heavy-tailed distributions including those with infinite moments (see links to papers below). In this setting, EM based methods would not work, since Expectation step would diverge due to presence of outliers.

Further reading

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Books on Mixture Models

1. G. McLachlan, D. Peel Finite Mixture Models, , Wiley (2000)
2. Marin, J.M., Mengersen, K. and Robert, C.P. "Bayesian modelling and inference on mixtures of distributions". Handbook of Statistics 25, D. Dey and C.R. Rao (eds). Elsevier-Sciences (to appear). available as PDF

Recent papers

External links

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• Mixture modelling page (and the Snob program for Minimum Message Length (MML) applied to finite mixture models), maintained by D.L. Dowe.

Statistics

Modèle de mélanges gaussiens