Approximating multivariate posterior distribution functions from Monte Carlo samples for sequential Bayesian inference

TL;DR

This paper evaluates methods for approximating multivariate posterior distributions from Monte Carlo samples to enable effective sequential Bayesian inference, comparing accuracy of different density estimators across dimensions.

Contribution

It systematically compares kernel density estimates, Gaussian mixtures, vine copulas, and Gaussian processes for posterior approximation in sequential inference, providing practical guidance.

Findings

Gaussian processes excel in low dimensions

Gaussian mixtures and vine copulas perform better in high dimensions

Posterior approximations outperform direct sample reweighting

Abstract

An important feature of Bayesian statistics is the opportunity to do sequential inference: the posterior distribution obtained after seeing a dataset can be used as prior for a second inference. However, when Monte Carlo sampling methods are used for inference, we only have a set of samples from the posterior distribution. To do sequential inference, we then either have to evaluate the second posterior at only these locations and reweight the samples accordingly, or we can estimate a functional description of the posterior probability distribution from the samples and use that as prior for the second inference. Here, we investigated to what extent we can obtain an accurate joint posterior from two datasets if the inference is done sequentially rather than jointly, under the condition that each inference step is done using Monte Carlo sampling. To test this, we evaluated the accuracy of kernel density estimates, Gaussian mixtures, vine copulas and Gaussian processes in approximating posterior distributions, and then tested whether these approximations can be used in sequential inference. In low dimensionality, Gaussian processes are more accurate, whereas in higher dimensionality Gaussian mixtures or vine copulas perform better. In our test cases, posterior approximations are preferable over direct sample reweighting, although joint inference is still preferable over sequential inference. Since the performance is case-specific, we provide an R package mvdens with a unified interface for the density approximation methods.