Matching Bayesian and frequentist coverage probabilities when using an approximate data covariance matrix

TL;DR

This paper introduces a Bayesian prior that ensures credible intervals align with frequentist confidence intervals when the data covariance matrix is estimated from simulations, providing a more consistent interpretation of parameter uncertainties in astrophysics.

Contribution

The authors propose a new prior that achieves approximate frequentist coverage in Bayesian credible intervals, addressing the challenge of covariance matrix estimation from simulations.

Findings

The new prior leads to credible intervals with coverage probabilities close to nominal levels.

The approach links Bayesian and frequentist methods for covariance matrix uncertainties.

Simulations demonstrate improved interval coverage in astrophysical data analysis.

Abstract

Observational astrophysics consists of making inferences about the Universe by comparing data and models. The credible intervals placed on model parameters are often as important as the maximum a posteriori probability values, as the intervals indicate concordance or discordance between models and with measurements from other data. Intermediate statistics (e.g. the power spectrum) are usually measured and inferences made by fitting models to these rather than the raw data, assuming that the likelihood for these statistics has multivariate Gaussian form. The covariance matrix used to calculate the likelihood is often estimated from simulations, such that it is itself a random variable. This is a standard problem in Bayesian statistics, which requires a prior to be placed on the true model parameters and covariance matrix, influencing the joint posterior distribution. As an alternative to the commonly-used Independence-Jeffreys prior, we introduce a prior that leads to a posterior that has approximately frequentist matching coverage. This is achieved by matching the covariance of the posterior to that of the distribution of true values of the parameters around the maximum likelihood values in repeated trials, under certain assumptions. Using this prior, credible intervals derived from a Bayesian analysis can be interpreted approximately as confidence intervals, containing the truth a certain proportion of the time for repeated trials. Linking frequentist and Bayesian approaches that have previously appeared in the astronomical literature, this offers a consistent and conservative approach for credible intervals quoted on model parameters for problems where the covariance matrix is itself an estimate.