Bayesian Inference on a Mixture Model With Spatial Dependence

Abstract

We introduce a new technique to select the number of components of a mixture model with spatial dependence. The method consists of an estimation of the integrated completed likelihood based on a Laplace’s approximation and a new technique to deal with the normalizing constant intractability of the hidden Potts model. Our proposal is applied to a real satellite image. Supplementary materials are available online.

Key Words:

• Auxiliary Markov chain Monte Carlo schemes
• Bayesian model choice
• Hidden Gibbs random fields
• Integrated completed likelihood

SUPPLEMENTARY MATERIALS

Here are some brief descriptions of the supplementary files.

file menteith.txt: This text file contains the Menteith dataset. Each line corresponds to one pixel: the first column gives the pixel’s index, the second one gives the grey level, and the following numbers give the indexes of the neighbor pixels.

file gibbs_sampler.c: This C program contains the global Gibbs sampler associated to the mixture model with spatial dependence. It requires a data file such as menteith.txt.

file pottsbeta.c: This C program is used to generate some independent realizations of S(z _k ) for given values of k and β _k and the same structure as menteith.txt.

file linkfield.c: This C program is used to generate some independent realizations of S(z _k ) from its conditional distribution given N ₀ and for β _k = 0.

file post_treatment.R: This R program uses the results produced by the C codes to compute the ICL criterion.

ACKNOWLEDGMENTS

We are grateful to two reviewers for their valuable comments and suggestions on a previous version of this article. They greatly contributed in improving the quality and presentation of this study.