1204.4724 (Julien Carron)

On the assumption of Gaussian likelihoods for estimators of cosmological power spectra and their information content    [PDF]

Julien Carron

In this note we revisit the Fisher information content of cosmological power spectra of Gaussian fields, when based on the assumption of a multivariate Gaussian likelihood for estimators. We discuss that while the assumption of a Gaussian likelihood is motivated by the central limit theorem, it leads if used consistently to a Fisher information content that violates the Cramer-Rao inequality, due to the presence of independent information from the parameter dependent covariance matrix. At any fixed multipole, this term is shown to become dominant in the limit 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 the covariance matrix never carries independent information content. The reason why the information content of the spectra is correctly described by the usual formula (i.e. without the covariance term) in this estimator perspective is precisely the fact the the estimators have a chi-squared like distribution, and not a Gaussian distribution. The assumption of a Gaussian estimators likelihood is thus from the point of view of the information neither necessary nor really adequate, and we warn against the use of Gaussian likelihoods with parameter dependent covariance matrices for parameter inference from such spectra.

View original: http://arxiv.org/abs/1204.4724