Delayed Acceptance Markov Chain Monte Carlo for Robust Bayesian Analysis

TL;DR

The results demonstrate that, although DA-MCMC slightly reduces the effective sample size per iteration compared with the standard MCMC, it achieves substantial improvement in terms of effective sample size per second, approximately doubling the efficiency.

Abstract

This study introduces a computationally efficient algorithm, delayed acceptance Markov chain Monte Carlo (DA-MCMC), designed to improve posterior simulation in quasi-Bayesian inference. Quasi-Bayesian methods, which do not require fully specifying a probabilistic model, are often computationally expensive owing to the need to evaluate the inverse and determinant of large covariance matrices. DA-MCMC addresses this challenge by employing a two-stage process: In the first stage, proposals are screened using an approximate posterior, whereas a final acceptance or rejection decision is made in the second stage based on the exact target posterior. This reduces the need for costly matrix computations, thereby improving efficiency without sacrificing accuracy. We demonstrate the effectiveness of DA-MCMC through applications to both synthetic and real data. The results demonstrate that, although DA-MCMC slightly reduces the effective sample size per iteration compared with the standard MCMC, it achieves substantial improvement in terms of effective sample size per second, approximately doubling the efficiency. This makes DA-MCMC particularly useful for cases where posterior simulation is computationally intensive. Thus, the DA-MCMC algorithm offers a significant advancement in computational efficiency for quasi-Bayesian inference, making it a valuable tool for robust Bayesian analysis.

Figures (3)

• Figure 1: MultiESS/iter for different target acceptance rate
• Figure 2: Relative frequency of second-stage acceptance probability
• Figure 3: Posterior distribution of $\beta$