Markov Observation Models

TL;DR

This paper introduces a generalized Hidden Markov Model where observations form a Markov chain influenced by hidden states, along with algorithms for parameter estimation and state sequence decoding.

Contribution

It extends the Hidden Markov Model to include Markov chain observations and develops EM, filtering, and Viterbi algorithms for this new framework.

Findings

Developed an EM algorithm for parameter estimation in the new model.

Created a filtering recursion to track hidden states.

Designed a Viterbi-like algorithm for most likely state sequence estimation.

Abstract

Herein, the Hidden Markov Model is expanded to allow for Markov chain observations. In particular, the observations are assumed to be a Markov chain whose one step transition probabilities depend upon the hidden Markov chain. An Expectation-Maximization analog to the Baum-Welch algorithm is developed for this more general model to estimate the transition probabilities for both the hidden state and for the observations as well as to estimate the probabilities for the initial joint hidden-state-observation distribution. A believe state or filter recursion to track the hidden state then arises from the calculations of this Expectation-Maximization algorithm. A dynamic programming analog to the Viterbi algorithm is also developed to estimate the most likely sequence of hidden states given the sequence of observations.