Bayesian nonhomogeneous Markov models via Pólya-Gamma data augmentation with applications to rainfall modeling

Abstract

Discrete-time hidden Markov models are a broadly useful class of latent-variable models with applications in areas such as speech recognition, bioinformatics, and climate data analysis. It is common in practice to introduce temporal nonhomogeneity into such models by making the transition probabilities dependent on time-varying exogenous input variables via a multinomial logistic parametrization. We extend such models to introduce additional nonhomogeneity into the emission distribution using a generalized linear model (GLM), with data augmentation for sampling-based inference. However, the presence of the logistic function in the state transition model significantly complicates parameter inference for the overall model, particularly in a Bayesian context. To address this, we extend the recently-proposed Pólya-Gamma data augmentation approach to handle nonhomogeneous hidden Markov models (NHMMs), allowing the development of an efficient Markov chain Monte Carlo (MCMC) sampling scheme. We apply our model and inference scheme to 30 years of daily rainfall in India, leading to a number of insights into rainfall-related phenomena in the region. Our proposed approach allows for fully Bayesian analysis of relatively complex NHMMs on a scale that was not possible with previous methods. Software implementing the methods described in the paper is available via the R package NHMM.