Mixture distribution: Difference between revisions

Revision as of 14:39, 13 September 2008

Revision as of 22:11, 21 January 2009

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:<math> q(x) = \sum_{i=1}^n \, w_i \, p(x,a_i) </math>

Otherwise, the mixture is called an ''infinite mixture''. Note that it is possible to have discrete but infinite mixtures.

In applications, finite mixtures are more common than infinite mixtures, and an unqualified reference to a mixture density usually means a finite mixture. Continuous infinite mixtures often have other names, e.g. hierarchical models.

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'').