A convex approach for Markov chain estimation from aggregate data via inverse optimal transport

TL;DR

This paper introduces a convex optimization method using inverse optimal transport to estimate Markov transition matrices from aggregate population data, enabling efficient and accurate dynamical law identification.

Contribution

It formulates the inverse Markov estimation as a convex entropic optimal transport problem and proposes an efficient iterative algorithm for solving it.

Findings

Accurate estimation demonstrated in numerical experiments.

Convergence of the proposed algorithm confirmed.

Effective from both snapshot and time series data.

Abstract

We address the problem of identifying the dynamical law governing the evolution of a population of indistinguishable particles, when only aggregate distributions at successive times are observed. Assuming a Markovian evolution on a discrete state space, the task reduces to estimating the underlying transition probability matrix from distributional data. We formulate this inverse problem within the framework of entropic optimal transport, as a joint optimization over the transition matrix and the transport plans connecting successive distributions. This formulation results in a convex optimization problem, and we propose an efficient iterative algorithm based on the entropic proximal method. We illustrate the accuracy and convergence of the method in two numerical setups, considering estimation from independent snapshots and estimation from a time series of aggregate observations, respectively.