Data-driven informative priors for Bayesian inference with quasi-periodic data

TL;DR

This paper introduces a data-driven approach to construct informative priors for Bayesian inference in models with periodic data, using Gaussian processes to approximate the period's posterior distribution and improve inference efficiency.

Contribution

It proposes a novel empirical Bayes workflow that leverages Gaussian process hyperparameter posteriors as priors for the period in parametric models, enhancing Bayesian inference in periodic settings.

Findings

The method accurately approximates the period's posterior distribution.

Using the data-driven prior improves inference efficiency.

The approach is validated on synthetic and real datasets.

Abstract

Bayesian computational strategies for inference can be inefficient in approximating the posterior distribution in models that exhibit some form of periodicity. This is because the probability mass of the marginal posterior distribution of the parameter representing the period is usually highly concentrated in a very small region of the parameter space. Therefore, it is necessary to provide as much information as possible to the inference method through the parameter prior distribution. We intend to show that it is possible to construct a prior distribution from the data by fitting a Gaussian process (GP) with a periodic kernel. More specifically, we want to show that it is possible to approximate the marginal posterior distribution of the hyperparameter corresponding to the period in the kernel. Subsequently, this distribution can be used as a prior distribution for the inference method. We use an adaptive importance sampling method to approximate the posterior distribution of the hyperparameters of the GP. Then, we use the marginal posterior distribution of the hyperparameter related to the periodicity in order to construct a prior distribution for the period of the parametric model. This workflow is empirical Bayes, implemented as a modular (cut) transfer of a GP posterior for the period to the parametric model. We applied the proposed methodology to both synthetic and real data. We approximated the posterior distribution of the period of the GP kernel and then passed it forward as a posterior-as-prior with no feedback. Finally, we analyzed its impact on the marginal posterior distribution.