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An important feature of Bayesian statistics is the opportunity to dosequential inference: the posterior distribution obtained after seeing adataset can be used as prior for a second inference. However, when Monte Carlosampling methods are used for inference, we only have a set of samples from theposterior distribution. To do sequential inference, we then either have toevaluate the second posterior at only these locations and reweight the samplesaccordingly, or we can estimate a functional description of the posteriorprobability distribution from the samples and use that as prior for the secondinference. Here, we investigated to what extent we can obtain an accurate jointposterior from two datasets if the inference is done sequentially rather thanjointly, under the condition that each inference step is done using Monte Carlosampling. To test this, we evaluated the accuracy of kernel density estimates,Gaussian mixtures, vine copulas and Gaussian processes in approximatingposterior distributions, and then tested whether these approximations can beused in sequential inference. In low dimensionality, Gaussian processes aremore accurate, whereas in higher dimensionality Gaussian mixtures or vinecopulas perform better. In our test cases, posterior approximations arepreferable over direct sample reweighting, although joint inference is stillpreferable over sequential inference. Since the performance is case-specific,we provide an R package mvdens with a unified interface for the densityapproximation methods.

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