Mixture density
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In statistics, a mixture density is a probability density function which is a convex linear combination of other probability density functions.
Consider a probability density function p(x,a) for a variable x, parameterized by a. That is, for each value of a in some set A, p(x,a) is a probability density function with respect to x. Given a nonnegative function w such that w(a) integrates to 1, the function
- <math> q(x) = \int_A \, w(a) \, p(x,a) \, da <math>
is again a probability density function for x, called the mixture density defined by the mixture components p(x,a) and the weighting function w.
If the possible values of the parameter a are finite, the mixture density is called a finite mixture, and the integration is replaced by a summation in the definition.
- <math> q(x) = \sum_{i=1}^n \, w_i \, p(x,a_i) <math>
Otherwise, the mixture is called an infinite mixture. In applications, finite mixtures are more common than infinite mixtures, and an unqualified reference to a mixture density usually means a finite mixture.
A general linear combination of probability density functions is not necessarily a probability density, since it may be negative or it may integrate to something other than 1. It can be shown that if w is nonnegative and integrates to 1, then the function q as defined above is indeed a probability density. The combination is called "convex" because q is in the convex hull of the set of functions p(x,a).