Steady State Sensitivity Analysis of Continuous Time Markov Chains

TL;DR

This paper extends likelihood ratio-based Monte Carlo estimators for steady-state sensitivity analysis to continuous time Markov chains, showing their variance remains bounded over time, which is beneficial for systems with large mixing times.

Contribution

It generalizes existing results to continuous time Markov chains with countable states, demonstrating the bounded variance of estimators and practical advantages for multi-scale systems.

Findings

Variance of estimators does not grow over time.

Centered likelihood ratio estimators are preferable for large mixing times.

Numerical benchmarks confirm theoretical insights.

Abstract

In this paper we study Monte Carlo estimators based on the likelihood ratio approach for steady-state sensitivity. We first extend the result of Glynn and Olvera-Cravioto [doi:doi: 10.1287/stsy.2018.002] to the setting of continuous time Markov chains with a countable state space which include models such as stochastic reaction kinetics and kinetic Monte Carlo lattice system. We show that the variance of the centered likelihood ratio estimators does not grow in time. This result suggests that the centered likelihood ratio estimators should be favored for sensitivity analysis when the mixing time of the underlying continuous time Markov chain is large, which is typically the case when systems exhibit multi-scale behavior. We demonstrate a practical implication of this analysis on a numerical benchmark of two examples for the biochemical reaction networks.