Approximating conditional distributions

TL;DR

This paper extends the Stein-Chen method to approximate and bound the distance between conditional distributions, providing explicit bounds and samplers for random walk bridges and related schemes.

Contribution

It introduces a new approach using an integration-by-parts formula to derive Stein operators for conditional distributions, enabling quantitative bounds and sampler constructions.

Findings

Derived bounds for filtering equations and bridge distances

Constructed samplers with explicit convergence rates

Applied method to discrete schemes approximating continuous processes

Abstract

In this article, we discuss the basic ideas of a general procedure to adapt the Stein-Chen method to bound the distance between conditional distributions. From an integration-by-parts formula (IBPF), we derive a Stein operator whose solution can be bounded, for example, via ad hoc couplings. This method provides quantitative bounds in several examples: the filtering equation, the distance between bridges of random walks and the distance between bridges and discrete schemes approximating them. Moreover, through the coupling construction for a certain class of random walk bridges we determine samplers, whose convergence to equilibrium is computed explicitly.