# Maximum Likelihood Estimation of Mixture Densities

Burke, James

### Abstract

Suppose you are given a parameterized family of densities and a data set and asked to find the mixture density over the family of densities that most likely gave rise to this data. More formally, you wish to find the regular Borel probability measure over the parameter space that yields the maximum likelihood mixture density for the given data set. This is a classical problem in probability and statistics which can be posed as convex programming problem on the space of regular Borel measures. We consider extensions of this problem that include constraints and develop their duality theory. We also discuss embeddings of the problem in finite dimensions as well as solution methods.

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