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