# Mixture model

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

### One 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.

### Another definition

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

[itex]p_{X}(x) = \sum_{k = 1}^{K} a_{k} h(x|\lambda _k)[itex]

where [itex]p_{X}(x)[itex] is the modeled probability distribution function, [itex]K[itex] is the number of components in the mixture model, and [itex]a_{k}[itex] is mixture proportion of component [itex]k[itex]. By definition, [itex]0 < a_{k} < 1[itex] for all [itex]k = 1 ... K [itex] and [itex]a_{1} + ... + a_{K} = 1[itex].

[itex]h(x|\lambda _k)[itex] is a probability distribution parameterized by [itex]\lambda _k[itex].

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

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