# A reaction network scheme which implements inference and learning for Hidden Markov Models

## TL;DR

This paper introduces the Chemical Baum-Welch Algorithm, a novel reaction network scheme that enables molecular systems to learn Hidden Markov Model parameters through biochemical reactions, mimicking traditional algorithms.

## Contribution

It presents a biochemical reaction network implementation of the Baum-Welch algorithm for HMMs, with proven convergence properties and simulation validation.

## Key findings

- Reaction network converges exponentially fast in expectation and maximization steps.
- Fixed points of the network correspond to those of the Baum-Welch algorithm.
- Simulations show the network learns parameters identical to classical HMM algorithms.

## Abstract

With a view towards molecular communication systems and molecular multi-agent systems, we propose the Chemical Baum-Welch Algorithm, a novel reaction network scheme that learns parameters for Hidden Markov Models (HMMs). Each reaction in our scheme changes only one molecule of one species to one molecule of another. The reverse change is also accessible but via a different set of enzymes, in a design reminiscent of futile cycles in biochemical pathways. We show that every fixed point of the Baum-Welch algorithm for HMMs is a fixed point of our reaction network scheme, and every positive fixed point of our scheme is a fixed point of the Baum-Welch algorithm. We prove that the "Expectation" step and the "Maximization" step of our reaction network separately converge exponentially fast. We simulate mass-action kinetics for our network on an example sequence, and show that it learns the same parameters for the HMM as the Baum-Welch algorithm.