Mixture Model Distribution and Single Model Distribution Comparisons

Question

I have two groups of subjects (healthy & patients). Each subject's data (>70000 samples) were displayed using a histogram. For some, it appeared to be a mixture of two Gamma function, whereas for others it appears to be that a Gamma function would be sufficient for modelling.

I am working on a Matlab script to apply mixture Gamma modelling. I have a question on how to deal with the fact that some of the subjects' data may be modelled using mixture modelling, but not others.

My plan is to apply the mixture modelling to all datasets, then discard the model with a small probability. Would that suffice?

For comparing the two groups, what would you recommend for comparing the alpha & gamma values from the modelling results?

Say, group A has 10 subjects: 4 of them were modelled successfully by mixture modelling, i.e., the probabilities corresponding to two models are similar. For the rest, the histogram can be described by a single Gamma function, i.e., the probability corresponds to one of the distribution is small. Group B has something similar too. In that case, how could I compare the coefficients between two groups?

Answer 1

This may not answer all your concerns but at least you can build upon it.

I would say that yes, use mixture modeling for everything. You can see a single Gamma function as a mixture of two Gamma functions, with one having a weight of 0 and the other one a weight of 1 for example.

Distances between mixture models exist, like the KL-divergence. This paper: Approximating the Kullback Leibler Divergence between Gaussian Mixture Models, by John R. Hershey and Peder A. Olsen, presents many different ways of computing it for the Gaussian case. However, most of the presented approximations are also valid for the Gamma distribution and are rather easy to compute (the one presented in section 7 especially).

As I understand you problem, I would represent each group by a mixture (with more components than 2 maybe). However, not knowing the output you are looking for I cannot say it for sure.