The Kullback–Leibler Divergence and the Convergence Rate of Fast Covariance Matrix Estimators in Galaxy Clustering Analysis

Abstract

We present a method to quantify the convergence rate of the fast estimators of the covariance matrices in the large-scale structure analysis. Our method is based on the Kullback–Leibler (KL) divergence, which describes the relative entropy of two probability distributions. As a case study, we analyze the delete-d jackknife estimator for the covariance matrix of the galaxy correlation function. We introduce the information factor or the normalized KL divergence with the help of a set of baseline covariance matrices to diagnose the information contained in the jackknife covariance matrix. Using a set of quick particle mesh mock catalogs designed for the Baryon Oscillation Spectroscopic Survey DR11 CMASS galaxy survey, we find that the jackknife resampling method succeeds in recovering the covariance matrix with 10 times fewer simulation mocks than that of the baseline method at small scales (s ≤ 40 h −1 Mpc). However, the ability to reduce the number of mock catalogs is degraded at larger scales due to the increasing bias on the jackknife covariance matrix. Note that the analysis in this paper can be applied to any fast estimator of the covariance matrix for galaxy clustering measurements.