Rigorous confidence bounds for MCMC under a geometric drift condition

TL;DR

This paper develops rigorous, nonasymptotic confidence bounds for MCMC estimators under a geometric drift condition, enabling precise determination of simulation length and burn-in time for reliable Bayesian inference.

Contribution

It introduces explicit bounds on MCMC error and confidence intervals based on drift conditions, improving practical assessment of estimator accuracy.

Findings

Derived lower bounds for trajectory length and burn-in time.

Constructed fixed-width confidence intervals with explicit error probabilities.

Analyzed median-of-multiple-run estimators for sharper bounds.

Abstract

We assume a drift condition towards a small set and bound the mean square error of estimators obtained by taking averages along a single trajectory of a Markov chain Monte Carlo algorithm. We use these bounds to construct fixed-width nonasymptotic confidence intervals. For a possibly unbounded function $f:\stany \to R,$ let $I=\int_{\stany} f(x) \pi(x) dx$ be the value of interest and $\hat{I}_{t,n}=(1/n)\sum_{i=t}^{t+n-1}f(X_i)$ its MCMC estimate. Precisely, we derive lower bounds for the length of the trajectory $n$ and burn-in time $t$ which ensure that $$P(|\hat{I}_{t,n}-I|\leq \varepsilon)\geq 1-\alpha.$$ The bounds depend only and explicitly on drift parameters, on the $V-$norm of $f,$ where $V$ is the drift function and on precision and confidence parameters $\varepsilon, \alpha.$ Next we analyse an MCMC estimator based on the median of multiple shorter runs that allows for sharper bounds for the required total simulation cost. In particular the methodology can be applied for computing Bayesian estimators in practically relevant models. We illustrate our bounds numerically in a simple example.