Probabilistic Systems with Hidden State and Unobservable Transitions

TL;DR

This paper extends Hidden Markov Models to include unobservable epsilon-transitions, providing algorithms for inference and parameter learning that handle the complexities introduced by these null transitions.

Contribution

It introduces a generalized framework for probabilistic systems with hidden states and epsilon-transitions, along with algorithms for decoding and learning in this setting.

Findings

Algorithm for most probable explanation with epsilon-transitions

Parameter learning method ensuring increased observation probability

Theoretical foundation based on EM algorithm guarantees correctness

Abstract

We consider probabilistic systems with hidden state and unobservable transitions, an extension of Hidden Markov Models (HMMs) that in particular admits unobservable {\epsilon}-transitions (also called null transitions), allowing state changes of which the observer is unaware. Due to the presence of {\epsilon}-loops this additional feature complicates the theory and requires to carefully set up the corresponding probability space and random variables. In particular we present an algorithm for determining the most probable explanation given an observation (a generalization of the Viterbi algorithm for HMMs) and a method for parameter learning that adapts the probabilities of a given model based on an observation (a generalization of the Baum-Welch algorithm). The latter algorithm guarantees that the given observation has a higher (or equal) probability after adjustment of the parameters and its correctness can be derived directly from the so-called EM algorithm.