  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.
