The Parameter-Level Performance of Covariance Matrix Conditioning in Cosmic Microwave Background Data Analyses

Abstract

Empirical estimates of the band power covariance matrix are commonly used in cosmic microwave background (CMB) power spectrum analyses. While this approach easily captures correlations in the data, noise in the resulting covariance estimate can systematically bias the parameter fitting. Conditioning the estimated covariance matrix, by applying prior information on the shape of the eigenvectors, can reduce these biases and ensure the recovery of robust parameter constraints. In this work, we use simulations to benchmark the performance of four different conditioning schemes, motivated by contemporary CMB analyses. The simulated surveys measure the T T , TE, and EE power spectra over the angular multipole range 300 ≤ ≤ 3500 in ∆ = 50 wide bins, for temperature map-noise levels of 10, 6.4 and 2 µK-arcmin. We divide the survey data into N real = 30, 50, or 100 uniform subsets. We show the results of different conditioning schemes on the errors in the covariance estimate, and how these uncertainties on the covariance matrix propagate to the best-fit parameters and parameter uncertainties. The most significant effect we find is an additional scatter in the best-fit point, beyond what is expected from the data likelihood. For a minimal conditioning strategy, N real = 30, and a temperature map-noise level of 10 µK-arcmin, we find the uncertainty on the recovered best fit parameter to be ×1.3 larger than the apparent posterior width from the likelihood (×1.2 larger than the uncertainty when the true covariance is used). Stronger priors on the covariance matrix reduce the misestimation of parameter uncertainties to < 1%. As expected, empirical estimates perform better with higher N real , ameliorating the adverse effects on parameter constraints.

Bias on Tensor-to-Scalar Ratio Inference With Estimated Covariance Matrices

“…The MAP estimates are unbiased, but the scatter is larger when fewer simulations are used in the covariance-matrix estimate. A similar effect has been described in Balkenhol & Reichardt (2021) as an additional scatter in parameter constraints that is unaccounted for. Note that this effect depends on the particular realization of the (simulated) data used to compute bandpowers and, therefore, is not corrected by the SH likelihood approximation, which peaks at the same value of 𝑟 as the HL likelihood.…”

“…Successful strategies in preventing this have employed some form of covariance-matrix conditioning (Balkenhol & Reichardt 2021;Dutcher et al 2021;BICEP/Keck Collaboration 2021;Sayre et al 2020;Polarbear Collaboration 2020). As an example of how MC noise, as in the off-diagonal elements of Fig.…”

“…In this paper, we address the problem of using a limited number of simulations to estimate a covariance matrix to account for statistical and systematic uncertainties in the data. From previous studies with CMB temperature and 𝐸-mode data, it is known that an ill-formed covariance matrix can cause parameter estimates to be biased and uncertainties to be underestimated (Hartlap et al 2006;Hamimeche & Lewis 2008;Taylor et al 2013;Dodelson & Schneider 2013;Balkenhol & Reichardt 2021;Percival et al 2021). A common remedy is the conditioning of the covariance matrix, which can involve masking most off-diagonal matrix elements or applying some semi-analytical corrections (see Balkenhol & Reichardt (2021) for a review).…”

“…From previous studies with CMB temperature and 𝐸-mode data, it is known that an ill-formed covariance matrix can cause parameter estimates to be biased and uncertainties to be underestimated (Hartlap et al 2006;Hamimeche & Lewis 2008;Taylor et al 2013;Dodelson & Schneider 2013;Balkenhol & Reichardt 2021;Percival et al 2021). A common remedy is the conditioning of the covariance matrix, which can involve masking most off-diagonal matrix elements or applying some semi-analytical corrections (see Balkenhol & Reichardt (2021) for a review). These conditioning strategies are followed to varying extents in 𝑟 inferences using data from ground-based experiments (Polarbear Collaboration 2020;Sayre et al 2020;SPIDER Collaboration 2021;BICEP/Keck Collaboration 2021).…”