An Imprecise Probabilistic Estimator for the Transition Rate Matrix of a Continuous-Time Markov Chain

TL;DR

This paper introduces an imprecise probabilistic estimator for the transition rate matrix of continuous-time Markov chains, leveraging prior sets and the Imprecise Dirichlet Model to derive a simple, closed-form estimator.

Contribution

It develops a novel imprecise probabilistic framework for estimating continuous-time Markov chain transition rates, connecting discrete and continuous estimators with a simple closed-form expression.

Findings

The estimator is easy to compute due to conjugacy.

The limit of discrete-time estimators yields a continuous-time estimator.

The estimator has a simple closed-form expression.

Abstract

We consider the problem of estimating the transition rate matrix of a continuous-time Markov chain from a finite-duration realisation of this process. We approach this problem in an imprecise probabilistic framework, using a set of prior distributions on the unknown transition rate matrix. The resulting estimator is a set of transition rate matrices that, for reasons of conjugacy, is easy to find. To determine the hyperparameters for our set of priors, we reconsider the problem in discrete time, where we can use the well-known Imprecise Dirichlet Model. In particular, we show how the limit of the resulting discrete-time estimators is a continuous-time estimator. It corresponds to a specific choice of hyperparameters and has an exceptionally simple closed-form expression.