Adaptive ABC model choice and geometric summary statistics for hidden Gibbs random fields

Abstract

Selecting between different dependency structures of hidden Markov random field can be very challenging, due to the intractable normalizing constant in the likelihood. We answer this question with approximate Bayesian computation (ABC) which provides a model choice method in the Bayesian paradigm. This comes after the work of Grelaud et al. (2009) who exhibited sufficient statistics on directly observed Gibbs random fields. But when the random field is latent, the sufficiency falls and we complement the set with geometric summary statistics. The general approach to construct these intuitive statistics relies on a clustering analysis of the sites based on the observed colors and plausible latent graphs. The efficiency of ABC model choice based on these statistics is evaluated via a local error rate which may be of independent interest. As a byproduct we derived an ABC algorithm that adapts the dimension of the summary statistics to the dataset without distorting the model selection.

Approximate Bayesian Computation: A Survey on Recent Results

“…In [59], the authors consider summary statistics for ABC model choice in hidden Gibbs random fields. The move to a hidden Markov random field means that the original approach of [28] does not apply: there is no dimension-reduction sufficient statistics in that case.…”

Conditional Random Fields for Pattern Recognition Applied to Structured Data

“…In the context of hidden Gibbs random fields, [64] recently developed a method to choose the dependency structure in Gibbs fields that relies on approximate Bayesian computation. Another way to assess candidate model structure (such as function forms or number of connections used in Equation (1)) is empirical classification accuracy.…”

Hidden Gibbs random fields model selection using Block Likelihood Information Criterion

“…Celeux et al, ; Friel et al, ; McGrory et al, ; Everitt, ) and model selection (e.g. Grelaud et al, ; Friel, ; Cucala & Marin, ; Stoehr et al, ). Remark the exception of small latices on which we can apply the recursive algorithm of Reeves and Pettitt () and Friel and Rue () and obtain an exact computation of the normalizing constant.…”

“…If the problem of recovering the number of hidden states is common in image segmentation, the problem of selecting a dependence structure has received little attention in the literature. Stoehr et al () have proposed to use approximate Bayesian computation (ABC) model choice (e.g. Marin et al, ) based on geometric summary statistics to tackle the choice of an underlying graph, but their approach is restricted to the latter.…”

“…Such ideas have occurred in the context of composite likelihood, but the use of non‐genuine probability distribution results in misspecified model (e.g. Okabayashi et al, ; Friel, ; Stoehr & Friel, ) that we have decided to avoid.…”