Improved Method to Estimating Parameters of a Poisson Hidden Markov Model Using Bayesian Approach

Abstract

Expectation-Maximization (EM) is a popular method for estimating parameters of the Poisson-Hidden Markov Model (P-HMM). However, over-dispersion in comparison to the Poisson distribution remains a concern. This study developed a Bayesian method to Poisson count models. The study compares the Mean Square Errors and sufficiency of the EM to the Gibbs sampler technique using Akaike Information Criterion, Bayesian Information Criterion, and Deviance Information Criterion as model selectors. The Maximum Likelihood Estimates and Maximum a \textit{posteriori} were calculated using both simulated and real data in this case. The study's findings indicate that using any of the two approaches depends on the data type and the sample size. Whereas the Poisson Hidden Markov model, which uses the EM algorithm is preferred when using Zero-inflated data with a sample size  n ≤ 20 , the Bayesian Poisson-Hidden Markov Model, which uses a Gibbs sampler is better used for Heavy and Mixture data types irrespective of the sample size. The model predicted parameters of simulated data with remarkable accuracy and produced some unique statistical property results. This method is applicable to Poisson-hidden Markov models with homogeneous time series.