On the assumption of Gaussianity for cosmological two-point statistics and parameter dependent covariance matrices

TL;DR

This paper critically examines the assumption of Gaussian likelihoods and parameter-dependent covariance matrices in cosmological two-point statistics, highlighting potential violations of statistical bounds and recommending fixed covariance matrices for unbiased inference.

Contribution

It demonstrates that using parameter-dependent covariance matrices with Gaussian likelihoods can artificially inflate Fisher information, and advocates for fixed covariance matrices to ensure conservative and unbiased parameter estimation.

Findings

Artificial information from parameter-dependent covariance matrices can violate Cramér-Rao bounds.

Fisher information does not increase with the number of modes due to non-Gaussian estimator distributions.

Using fixed covariance matrices guarantees conservative and unbiased parameter inference.

Abstract

In this brief paper we revisit the Fisher information content of cosmological power spectra or two-point functions of Gaussian fields in order to comment on the assumption of Gaussian estimators and the use of parameter-dependent covariance matrices for parameter inference in the context of precision cosmology. Even though the assumption of a Gaussian likelihood is motivated by the central limit theorem, we discuss that it leads to Fisher information content that violates the Cram\'er-Rao bound if used consistently, owing to independent but artificial information from the parameter-dependent covariance matrix. At any fixed multipole, this artificial term is shown to become dominant in the case of a large number of correlated fields. While the distribution of the estimators does indeed tend to a Gaussian with a large number of modes, it is shown, however, that its Fisher information content does not, in the sense that their covariance matrix never carries independent information content, precisely because of the non-Gaussian shape of the distribution. In this light, we discuss the use of parameter-dependent covariance matrices with Gaussian likelihoods for parameter inference from two-point statistics. As a rule of thumb, Gaussian likelihoods should always be used with a covariance matrix fixed in parameter space, since only this guarantees that conservative information content is assigned to the observables, and at the same time prevents biases appearing.