Learning hidden Markov models from aggregate observations

In this paper, we propose an algorithm for estimating the parameters of a time-homogeneous hidden Markov model (HMM) from aggregate observations. This problem arises when only population level counts number individuals at each time step are available, and one seeks to learn individual HMM these Our is built upon classical expectation–maximization recently proposed inference (Sinkhorn belief propagation). We present parameter learning two different settings HMMs: with discrete observations continuous observations, exhibits convergence guarantees in both cases. Moreover, our framework naturally reduces standard Baum–Welch HMMs size 1. The efficacy demonstrated through several numerical experiments.