# Bayesian Non-Homogeneous Markov Models via Polya-Gamma Data Augmentation with Applications to Rainfall Modeling

## TL;DR

This paper introduces a Bayesian non-homogeneous hidden Markov model for rainfall data, utilizing Polya-Gamma data augmentation to enable efficient inference in complex models with time-varying transition and emission probabilities.

## Contribution

It extends Polya-Gamma data augmentation to non-homogeneous HMMs, facilitating scalable Bayesian inference for complex models with time-dependent parameters.

## Key findings

- Applied to 30 years of Indian rainfall data, revealing new climate insights.
- Demonstrated efficient MCMC sampling for complex NHMMs.
- Enabled fully Bayesian analysis of non-homogeneous models at large scale.

## 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 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.