An alternative to the Baum-Welch recursions for hidden Markov models

TL;DR

This paper introduces a new recursion method for hidden Markov models of any order, simplifying calculations and improving numerical stability compared to traditional Baum-Welch recursions, with applications in financial volatility analysis.

Contribution

It presents a direct, numerically stable recursion for HMMs of any order, enhancing efficiency and ease of implementation over existing Baum-Welch algorithms.

Findings

Recursion is more straightforward to implement.

It avoids numerical issues common in Baum-Welch.

Application to financial data demonstrates practical utility.

Abstract

We develop a recursion for hidden Markov model of any order h, which allows us to obtain the posterior distribution of the latent state at every occasion, given the previous h states and the observed data. With respect to the well-known Baum-Welch recursions, the proposed recursion has the advantage of being more direct to use and, in particular, of not requiring dummy renormalizations to avoid numerical problems. We also show how this recursion may be expressed in matrix notation, so as to allow for an efficient implementation, and how it may be used to obtain the manifest distribution of the observed data and for parameter estimation within the Expectation-Maximization algorithm. The approach is illustrated by an application to financial data which is focused on the study of the dynamics of the volatility level of log-returns.