Parameter inference with estimated covariance matrices

TL;DR

This paper develops a method for parameter inference using estimated covariance matrices, resulting in a multivariate t-distribution likelihood that accounts for covariance uncertainty, improving over previous Gaussian assumptions.

Contribution

It introduces a marginalization approach over the true covariance matrix, leading to a more accurate likelihood function for parameter inference with estimated covariances.

Findings

Likelihood becomes a multivariate t-distribution.

Method improves inference accuracy over Hartlap et al.'s approach.

Computational complexity remains similar to Gaussian likelihood.

Abstract

When inferring parameters from a Gaussian-distributed data set by computing a likelihood, a covariance matrix is needed that describes the data errors and their correlations. If the covariance matrix is not known a priori, it may be estimated and thereby becomes a random object with some intrinsic uncertainty itself. We show how to infer parameters in the presence of such an estimated covariance matrix, by marginalising over the true covariance matrix, conditioned on its estimated value. This leads to a likelihood function that is no longer Gaussian, but rather an adapted version of a multivariate t-distribution, which has the same numerical complexity as the multivariate Gaussian. As expected, marginalisation over the true covariance matrix improves inference when compared with Hartlap et al.'s method, which uses an unbiased estimate of the inverse covariance matrix but still assumes that the likelihood is Gaussian.