A reaction network scheme for hidden Markov model parameter learning

With a view towards artificial cells, molecular communication systems, molecular multiagent systems and federated learning, we propose a novel reaction network scheme (termed the Baum-Welch (BW) reaction network) that learns parameters for hidden Markov models (HMMs). All variables including inputs and outputs are encoded by separate species. Each reaction in the 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 positive fixed point of the BW algorithm for HMMs is a fixed point of the reaction network scheme, and vice versa. Furthermore, we prove that the 'expectation' step and the 'maximization' step of the reaction network separately converge exponentially fast and compute the same values as the E-step and the M-step of the BW algorithm. We simulate example sequences, and show that our reaction network learns the same parameters for the HMM as the BW algorithm, and that the log-likelihood increases continuously along the trajectory of the reaction network.

Automated summary

In this study, we propose a novel reaction network scheme called the Baum-Welch (BW) reaction network for learning parameters of hidden Markov models (HMMs). The scheme encodes all variables using separate species and each reaction in the scheme changes only one molecule from one species to another. We show that the BW algorithm and the reaction network scheme have the same fixed points and that the expectation step and the maximization step of the reaction network converge fast and compute the same values as the E-step and M-step of the BW algorithm. Simulation results demonstrate that our reaction network learns the same parameters as the BW algorithm and the log-likelihood increases continuously.