# Estimating ensemble flows on a hidden Markov chain

## TL;DR

This paper introduces a new method to estimate the evolution of an ensemble of indistinguishable agents on a hidden Markov chain using aggregate data, extending optimal transport concepts to finite state spaces.

## Contribution

It develops a convex maximum likelihood framework and a fast algorithm for estimating ensemble flows on hidden Markov chains from aggregate data.

## Key findings

- Effective tracking of ensemble flows demonstrated in numerical examples.
- Framework extends optimal transport to finite state hidden Markov models.
- Convex formulation enables efficient computation.

## Abstract

We propose a new framework to estimate the evolution of an ensemble of indistinguishable agents on a hidden Markov chain using only aggregate output data. This work can be viewed as an extension of the recent developments in optimal mass transport and Schr\"odinger bridges to the finite state space hidden Markov chain setting. The flow of the ensemble is estimated by solving a maximum likelihood problem, which has a convex formulation at the infinite-particle limit, and we develop a fast numerical algorithm for it. We illustrate in two numerical examples how this framework can be used to track the flow of identical and indistinguishable dynamical systems.