Leveraging single-case results to Bayesian hierarchical modelling

In scientific research, we often aim to learn one or more parameters of instances(objects) from a population—such as the batting averages of a group of baseball players and characteristics of white dwarfs from the Galactic Halo-and the distribution of fitted parameters across the population. Bayesian hierarchical models are well suited to this kind of situation. Despite there are many general-purpose and specialized Bayesian inference packages, many of them are designed for the single-case analysis, i.e., fitting a single unit of data at a time, rather than simultaneously fitting the hierarchical model for multiple datasets. This is especially true when the likelihood function is complicated and has no analytical form. In this paper, we fill this gap by proposing general algorithms to efficiently compute the exact hierarchical models by utilizing available packages that can perform Bayesian inference for single-case analysis. Our algorithms are efficient and easy-to-implement, thus significantly saving time and effort. We illustrate the application of our methods on three datasets, to verify the effectiveness, efficiency and benefits of our methods.