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

TL;DR

The paper tackles Bayesian inference for quasi-periodic data where the period parameter yields a highly concentrated posterior. It introduces a data-driven prior for the period by fitting a Gaussian process with a periodic kernel, then uses the GP hyperparameter posterior as a prior for the parametric model in an empirical Bayes, modular framework via adaptive importance sampling. The method is demonstrated on synthetic sine data and real astrophysical time series (radial velocity and light curves), showing tighter, more informative posteriors for the period and improved inference relative to uninformative priors and purely frequency-based methods. This approach reduces search space dimensionality and enhances robustness in exoplanet and binary-star analyses, with broad applicability to quasi-periodic time series.

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.

Figures (11)

• Figure 1: Simulated data (red dots). The green continuous line is the theoretical periodic curve. The result of the GP regression is shown for completeness (black dashed-line).
• Figure 2: Marginal posterior distribution for the hyperparametres $A$, $L$ and $P$. The value over each histogram is the mean and the 90% credibility level (continuous black line and dashed lines respectively). The red continuous line represents the estimated MAP.
• Figure 3: Kernel density estimation (continuous line) for the marginal posterior of the hyperparameter $P$ (histogram).
• Figure 4: Marginal posterior distribution for the model parameters $A$, $\phi$, $b$ and $P$. The value over each histogram is the mean and the 90% credibility level (continuous black line and dashed lines, respectively). The continuous red line represents the estimated MAP.
• Figure 5: Box chart with the MMSE distribution for the hyperparameter $P$, for simulations with different number of points.