# Variance bounding of delayed-acceptance kernels

## TL;DR

This paper investigates conditions under which delayed-acceptance Metropolis-Hastings algorithms inherit variance bounding properties from their parent kernels, improving computational efficiency in Bayesian inference.

## Contribution

It provides sufficient conditions for delayed-acceptance kernels to inherit variance bounding, enhancing understanding of their efficiency in computationally expensive Bayesian inference.

## Key findings

- Delayed-acceptance kernels can be variance bounding under certain conditions.
- Bounded discrepancy between approximate and true log densities ensures inheritance.
- Sufficient conditions for proposal pairs to preserve variance bounding property.

## Abstract

A delayed-acceptance version of a Metropolis--Hastings algorithm can be useful for Bayesian inference when it is computationally expensive to calculate the true posterior, but a computationally cheap approximation is available; the delayed-acceptance kernel targets the same posterior as its associated "parent" Metropolis-Hastings kernel. Although the asymptotic variance of the ergodic average of any functional of the chain cannot be less than that obtained using its parent, the average computational time per iteration can be much smaller and so for a given computational budget the delayed-acceptance kernel can be more efficient. When the asymptotic variance of the ergodic averages of all $L^2$ functionals of the chain is finite, the kernel is said to be variance bounding. It has recently been noted that a delayed-acceptance kernel need not be variance bounding even when its parent is. We provide sufficient conditions for inheritance: for non-local algorithms, such as the independence sampler, the discrepancy between the log density of the approximation and that of the truth should be bounded; for local algorithms, two alternative sets of conditions are provided. As a by-product of our initial, general result we also supply sufficient conditions on any pair of proposals such that, for any shared target distribution, if a Metropolis-Hastings kernel using one of the proposals is variance bounding then so is the Metropolis-Hastings kernel using the other proposal.