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Bayesian nonparametric priors for hidden Markov random fields

Hongliang Lu 1, 2 Julyan Arbel 1, 2 Florence Forbes 1, 2
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology , Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann
Abstract : One of the central issues in statistics and machine learning is how to select an adequate model that can automatically adapt its complexity to the observed data. In the present paper, we focus on the issue of determining the structure of clustered data, both in terms of finding the appropriate number of clusters and of modelling the right dependence structure between the observations. Bayesian nonparametric (BNP) models, which do not impose an upper limit on the number of clusters, are appropriate to avoid the required guess on the number of clusters but have been mainly developed for independent data. In contrast, Markov random fields (MRF) have been extensively used to model dependencies in a tractable manner but usually reduce to finite cluster numbers when clustering tasks are addressed. Our main contribution is to propose a general scheme to design tractable BNP-MRF priors that combine both features: no commitment to an arbitrary number of clusters and a dependence modelling. A key ingredient in this construction is the availability of a stick-breaking representation which has the threefold advantage to allowing us to extend standard discrete MRFs to infinite state space, to design a tractable estimation algorithm using variational approximation and to derive theoretical properties on the predictive distribution and the number of clusters of the proposed model. This approach is illustrated on a challenging natural image segmentation task for which it shows good performance with respect to the literature.
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Submitted on : Wednesday, July 29, 2020 - 2:37:09 PM
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Hongliang Lu, Julyan Arbel, Florence Forbes. Bayesian nonparametric priors for hidden Markov random fields. Statistics and Computing, Springer Verlag (Germany), 2020, 30, pp.1015-1035. ⟨10.1007/s11222-020-09935-9⟩. ⟨hal-02163046v3⟩

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