MCMC Confidence Intervals and Biases

TL;DR

This paper examines the assumptions behind MCMC confidence intervals derived without CLTs, demonstrating that the bias assumption may not always hold and proposing to weaken these assumptions for broader applicability.

Contribution

It weakens and simplifies the assumptions needed for MCMC confidence intervals without CLTs, addressing potential limitations of previous methods.

Findings

The bias assumption $o(1/\sqrt{n})$ may not always hold in practice.

Previous confidence interval methods rely on assumptions that can be invalid.

Proposes more general conditions to make MCMC confidence intervals more widely applicable.

Abstract

The recent paper "Simple confidence intervals for MCMC without CLTs" by J.S. Rosenthal, showed the derivation of a simple MCMC confidence interval using only Chebyshev's inequality, not CLT. That result required certain assumptions about how the estimator bias and variance grow with the number of iterations $n$. In particular, the bias is $o(1/\sqrt{n})$. This assumption seemed mild. It is generally believed that the estimator bias will be $O(1/n)$ and hence $o(1/\sqrt{n})$. However, questions were raised by researchers about how to verify this assumption. Indeed, we show that this assumption might not always hold. In this paper, we seek to simplify and weaken the assumptions in the previously mentioned paper, to make MCMC confidence intervals without CLTs more widely applicable.