Finite Mixture Models

Today, I went on searching for mathematical solution. Finite Mixture Models may be helpful. They are combined with Maximum Likelihood Estimate.

Summary:

Finite Mixture Models

Pdf: probability density function

For a given data X with N observations, the likelihood of the data assuming that xi are independently distributed is given by

The problem of mixture estimation from data X can be formulated as to find the set of parameters Θ that gives the maximum likelihood estimate (MLE) solution

One way is to maximize the complete likelihood in an expectation-maximization (EM) approach.

Expectation-maximization Algorithm

The likelihood of the complete data (X; Y) takes the following multinomial form

where 1 is the indicator function, i.e. 1(yi = k) = 1 if yi = k holds, and 1(yi = k) = 0 otherwise.

There are some examples that I have not read. Today, I focused on the theoretical part.

Tomorrow, I will read more examples and try to associate them with my research.