Stochastic Chemical Reaction Networks for Robustly Approximating Arbitrary Probability Distributions

TL;DR

This paper demonstrates that stochastic chemical reaction networks can approximate any discrete probability distribution on non-negative integers, with constructions that are both precise and robust to initial perturbations.

Contribution

It introduces new classes of stochastic chemical reaction networks capable of arbitrarily approximating any discrete distribution, including robust ergodic models.

Findings

Detailed balanced networks can approximate any discrete distribution.

Robust ergodic models approximate distributions regardless of initial conditions.

Models are resilient to perturbations over time.

Abstract

We show that discrete distributions on the $d$-dimensional non-negative integer lattice can be approximated arbitrarily well via the marginals of stationary distributions for various classes of stochastic chemical reaction networks. We begin by providing a class of detailed balanced networks and prove that they can approximate any discrete distribution to any desired accuracy. However, these detailed balanced constructions rely on the ability to initialize a system precisely, and are therefore susceptible to perturbations in the initial conditions. We therefore provide another construction based on the ability to approximate point mass distributions and prove that this construction is capable of approximating arbitrary discrete distributions for any choice of initial condition. In particular, the developed models are ergodic, so their limit distributions are robust to a finite number of perturbations over time in the counts of molecules.