  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 non-homogeneity 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 non-homogeneity 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 Polya-Gamma data augmentation approach to handle
non-homogeneous 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.
