Mixture density

In statistics, a mixture density is a probability density function which is a convex 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

:$q(x)\; =\; int\_A\; ,\; w(a)\; ,\; p(x,a)\; ,\; da$

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 in number, the mixture density is called a "finite mixture", and the integration is replaced by a summation in the definition.

:$q(x)\; =\; sum\_\{i=1\}^n\; ,\; w\_i\; ,\; p(x,a\_i)$

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").

See also

* Mixture (probability)
* Mixture model
* Expectation-maximization algorithm