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

Monthly Notices of the Royal Astronomical Society, vol. 510, pp. 3207–3221

Abstract

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 are 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.

Authors 4

1. Will J. Percival corresponding

   University of Waterloo · Perimeter Institute

   Affiliation as printed

   Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada

   Perimeter Institute for Theoretical Physics, 31 Caroline St. North, Waterloo, ON N2L 2Y5, Canada

   Waterloo Centre for Astrophysics, University of Waterloo, Waterloo, ON N2L 3G1, Canada

2. Oliver Friedrich

   University of Cambridge

   Affiliation as printed

   Churchill College, University of Cambridge, Cambridge CB3 0DS, UK

   Kavli Institute for Cosmology, University of Cambridge, Cambridge CB3 0HA, UK

3. Elena Sellentin Aachen Leiden Observatory Mathematical Institute

   Leiden University · Leiden Observatory

   Affiliation as printed

   Leiden Observatory, Leiden University, Oort Gebouw, Niels Bohrweg 2, NL-2333 CA Leiden, the Netherlands

   Mathematical Institute, Leiden University, Snellius Gebouw, Niels Bohrweg 1, NL-2333 CA Leiden, the Netherlands

4. Alan Heavens

   Imperial College London

   Affiliation as printed

   Imperial Centre for Inference and Cosmology (ICIC), Department of Physics, Imperial College London, Blackett Laboratory, Prince Consort Road, London SW7 2AZ, UK