Dirichlet or Potts ?

TL;DR

This paper compares Dirichlet and Potts models for Gaussian mixture data, focusing on their application in image segmentation and discussing algorithms for simulation and parameter estimation.

Contribution

It introduces the use of Potts distribution as a Markovian alternative to Dirichlet priors in mixture models, especially for image segmentation tasks.

Findings

Potts model better captures spatial dependencies in image segmentation.

Algorithms for simulation and parameter estimation are detailed.

Equivalence of i.i.d. hidden variables with Dirichlet prior is demonstrated.

Abstract

When modeling the distribution of a set of data by a mixture of Gaussians, there are two possibilities: i) the classical one is using a set of parameters which are the proportions, the means and the variances; ii) the second is to consider the proportions as the probabilities of a discrete valued hidden variable. In the first case a usual prior distribution for the proportions is the Dirichlet which accounts for the fact that they have to sum up to one. In the second case, to each data is associated a hidden variable for which we consider two possibilities: a) assuming those variables to be i.i.d. We show then that this scheme is equivalent to the classical mixture model with Dirichlet prior; b) assuming a Markovian structure. Then we choose the simplest markovian model which is the Potts distribution. As we will see this model is more appropriate for the case where the data represents the pixels of an image for which the hidden variables represent a segmentation of that image. The main object of this paper is to give some details on these models and different algorithms used for their simulation and the estimation of their parameters. Key Words: Mixture of Gaussians, Dirichlet, Potts, Classification, Segmentation.