# Baum-Welch Algorithm

The hidden Markov Ising model makes an Ising model, a binary graphical model from statistical physics, the emission distribution of a hidden Markov chain, so that a binary network's dependence structure is modeled directly as its state changes over time. Because the Ising normalizing constant grows exponentially with the number of nodes, fitting uses pseudolikelihood inside Baum-Welch reestimation, with results on the algorithm's validity, model identifiability and large-network consistency. Exogenous covariates extend it to a hidden Markov autologistic model, applied to a five-year sequence of weekly gun-death events across 3109 continental U.S. counties from 2021 to 2025, where it identifies state changes in gun violence activity.

The hidden Markov Ising model makes an Ising model, a binary graphical model from statistical physics, the emission distribution of a hidden Markov chain, so that a binary network's dependence structure is modeled directly as its state changes over time. Because the Ising normalizing constant grows exponentially with the number of nodes, fitting uses pseudolikelihood inside Baum-Welch reestimation, with results on the algorithm's validity, model identifiability and large-network consistency. Exogenous covariates extend it to a hidden Markov autologistic model, applied to a five-year sequence of weekly gun-death events across 3109 continental U.S. counties from 2021 to 2025, where it identifies state changes in gun violence activity.