Discretization approximation: An alternative to Monte Carlo in Bayesian computation

TL;DR

This paper introduces a simple, deterministic discretization approximation method for Bayesian computation, offering faster convergence than traditional Monte Carlo methods, especially when using quasi-Monte Carlo sequences.

Contribution

The paper proposes a novel discretization approximation technique for Bayesian computation that is easy to implement and can outperform MCMC in convergence speed.

Findings

Discretization approximation converges faster than MCMC.

Supports high-dimensional Bayesian problems effectively.

Numerical examples show strong performance across various cases.

Abstract

In this paper we propose a new deterministic approximation method, called discretization approximation, for Bayesian computation. Discretization approximation is very simple to understand and to implement, It only requires calculating posterior density values as probability masses at pre-specified support points. The resulted discrete distribution can be a good approximation to the target posterior distribution. All posterior quantities, including means, standard deviations, and quantiles, can be approximated by those of this completely known discrete distribution. We establish the convergence rate of discretization approximation as the number of support points goes to infinity. If the support points are generated from quasi-Monte Carlo sequences, then the rate is actually the same as that in integration approximation, generally faster than the optimal statistical rate. In this sense, discretization approximation is superior to the popular Markov chain Monte Carlo method. We also provide random sampling and representation point construction methods from discretization approximation. Numerical examples including some benchmarks demonstrate that the proposed method performs quite well for both low-dimensional and high-dimensional cases.