### A Note on Non-Identifibiality Problem of Finite Mixture Distribution Models in Model-Based Classification

#### Abstract

The probability density functions (pdfs) of the mixture distribution models (mdms) for two different populations can be compared by using a distance function (metric) between them in model-based classification applications. The result of the comparison may not be true if the component densities of the mdms are permutation functions. Thus, non-identifibiality problem of finite mixture distribution models. In other words, the order of the component densities of the mdms should be taken into account. If the component densities of the mdms are permutation functions then the pdfs of the mdms for two different population looks like similar but in fact they are completely different. Such a case may cause wrong inference in the applications in which the mdms used, for example in classification applications. The componentwise distance function is proposed for the comparison of the pdfs of the mdms for two different populations if the component densities are permutation functions. The condition under which the value of the distance function between the pdfs of the mdms for two different populations is equal to the value of the componentwise distance function between the pdfs of the mdms for two different populations is given.

#### Keywords

Finite mixture distribution model, Hellinger distance, model-based classification, non-identifibiality, permutation functions