# Learning Hidden Markov Models from Aggregate Observations

arXiv: https://arxiv.org/abs/2011.11236 · HTML: https://mlsanity.com/paper/2011.11236/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Nov 2020.

## Built on by (most cited first, 3 of 3)

- [Trajectory Inference with Smooth Schr"odinger Bridges](https://mlsanity.com/paper/2503.00530/) (2025)
- [Estimating Latent Population Flows from Aggregated Data via Inversing Multi-Marginal Optimal Transport](https://mlsanity.com/paper/2212.14527/) (2022)
- [Learning Collective Dynamics with Differentiable Gaussian Representations](https://mlsanity.com/paper/2609.28405/) (2026)

## Abstract (arXiv)

In this paper, we propose an algorithm for estimating the parameters of a time-homogeneous hidden Markov model from aggregate observations. This problem arises when only the population level counts of the number of individuals at each time step are available, from which one seeks to learn the individual hidden Markov model. Our algorithm is built upon expectation-maximization and the recently proposed aggregate inference algorithm, the Sinkhorn belief propagation. As compared with existing methods such as expectation-maximization with non-linear belief propagation, our algorithm exhibits convergence guarantees. Moreover, our learning framework naturally reduces to the standard Baum-Welch learning algorithm when observations corresponding to a single individual are recorded. We further extend our learning algorithm to handle HMMs with continuous observations. The efficacy of our algorithm is demonstrated on a variety of datasets.