Learning Gaussian Hidden Markov Models From Aggregate Data

We consider system identification (learning) problems for Gaussian hidden Markov models (GHMMs). We propose an algorithm to tackle the cases where the data is recorded in aggregate (collective) form generated by a large population of individuals following a certain dynamics. Our parameter learning algorithm is built upon the expectation-maximization algorithm with a novel expectation step proposed recently known as the collective Gaussian forward-backward algorithm. The proposed learning algorithm generalizes the traditional Baum-Welch learning algorithm for GHMMs as it naturally reduces to the latter in case of individual observations.