Nonparametric Bayesian assessment of the order of dependence for binary sequences

Abstract

This article discusses inference on the order of dependence in binary sequences. The proposed approach is based on the notion of partial exchangeability of order k. A partially exchangeable binary sequence of order k can be represented as a mixture of Markov chains. The mixture is with respect to the unknown transition probability matrix theta. We use this defining property to construct a semiparametric model for binary sequences by assuming a nonparametric prior on the transition matrix theta. This enables us to consider inference on the order of dependence without constraint to a particular parametric model. Implementing posterior simulation in the proposed model is complicated by the fact that the dimension of theta changes with the order of dependence k. We discuss appropriate posterior simulation schemes based on a pseudo prior approach. We extend the model to include covariates by considering an alternative parameterization as an autologistic regression which allows for a straightforward introduction of covariates. The regression on covariates raises the additional inference problem of variable selection. We discuss appropriate posterior simulation schemes, focusing on inference about the order of dependence. We discuss and develop the model with covariates only to the extent needed for such inference.