Putting an Ising model inside a hidden Markov chain: pseudolikelihood sidesteps the partition function, and U.S. county gun-death weeks split into four states

Each observation in the hidden Markov chain is no longer drawn from a freely specified discrete distribution but from an Ising model on a binary network, so dependence between neighboring nodes enters the emission distribution instead of being left to the observation vector. Previously, if the observation space of a hidden Markov model consisted of binary vectors of length n, there were 2 to the n possible values, and characterizing an arbitrary discrete distribution required a number of parameters growing exponentially; node and edge weights in the Ising model reduce that parameter count. In a simulation with a 3×3 grid and 2 states, direct likelihood maximization with exact log-partition computation and pseudolikelihood maximization gave very similar fits: the fitted transition matrices and Ising parameters were close, and the adjusted Rand index of the Viterbi path against the true state sequence was similar.

Fitting no longer depends on the Ising normalizing constant; the emission probability is replaced by a pseudolikelihood, and the forward, backward and reestimation steps of Baum-Welch proceed as usual. Pseudolikelihood conditions each node's value on the remaining nodes and replaces the sum over all configurations with a local field, avoiding a partition function that grows exponentially with the number of nodes. In the same 3×3 grid simulation, likelihood-based Baum-Welch took 16.1 seconds to converge while the pseudolikelihood fit completed instantaneously; in a 64×64 happy-sad simulation, fitting 20 models took 19 minutes 37 seconds in the low-correlation setting and 54 minutes 32 seconds in the high-correlation setting.

The demonstrated scope is simulated square-grid Ising data and a binary sequence of weekly gun-death events across continental U.S. counties from 2021 to 2025; the real-data analysis is framed as a proof of concept. Covariates in the real data are time-invariant because of data limitations.

Identifiability results rely on linearly independent or distinct emission distributions, a full-rank transition matrix and a stationary Markov chain; relaxing stationarity or linear independence requires a longer chain of observations. Large-network consistency relies on bounded maximum degree, a lower bound on KL divergence and separation between states. On the real data, freeing all edge parameters made the pseudolikelihood estimate perform clearly worse under the classical likelihood, so the analysis uses shared edge parameters; the authors also caution that parameter signs in the autologistic model need careful interpretation.