One can naturally model the unobserved population heterogeneity or extra-population heterogeneity via mixture models. The simplest and most natural occurrence of the mixture model arises when one samples from a super population g which is a mixture of a finite number, say m, of populations (g₁,…, g_m), called the components of the population. Suppose a sample from a super population g is recorded as data (Y_i, J _i) for i = 1, …, n, where Y_i = y_i is a measurement on the i^th sampled unit and J_i = j_i indicates the index number of the component to which the unit belongs. If sampling is done from the j^th component, an appropriate probability model for the sampling distribution is given by

The function f_j(y_i; θ_j) represents a density function, called the component density for the i^th observation y_i and the parameter θ_j Θ_j ^d_j called the component parameter, that describes the specific attributes of the j^th component population, g_j.

We treat the J_i as missing data and define the latent random variable Φ= (Ф₁,…, Ф_n) to be the values of the parameter θ {θ₁, …,θ_m} corresponding to the sampled components J₁, …, J_n; i.e., if the i^th observation came from the j^th component, then define Ф_i = θ_j. Then the Ф_i are a random sample from the discrete probability measure Q that assigns a positive probability π_j to the support point θ_j for j = 1, …, m. That is, the latent