CMB likelihood approximation for banded probability distributions

TL;DR

This paper introduces a new likelihood approximation method for CMB data assuming banded correlations, enabling more accurate hybrid likelihoods and faster convergence of estimators, with minimal impact on cosmological parameters.

Contribution

It presents a novel likelihood approximation exploiting banded correlation structures, improving hybrid likelihood construction and estimator convergence in CMB analysis.

Findings

Hybrid likelihood estimator effectively merges low- and high-l likelihoods.

Correlation effects are negligible for WMAP cosmological parameters.

Method reduces Monte Carlo samples needed for Blackwell-Rao estimator by orders of magnitude.

Abstract

We investigate sets of random variables that can be arranged sequentially such that a given variable only depends conditionally on its immediate predecessor. For such sets, we show that the full joint probability distribution may be expressed exclusively in terms of uni- and bivariate marginals. Under the assumption that the CMB power spectrum likelihood only exhibits correlations within a banded multipole range, \Delta l, we apply this expression to two outstanding problems in CMB likelihood analysis. First, we derive a statistically well-defined hybrid likelihood estimator, merging two independent (e.g., low- and high-l) likelihoods into a single expression that properly accounts for correlations between the two. Applying this expression to the WMAP likelihood, we verify that the effect of correlations on cosmological parameters in the transition region is negligible in terms of cosmological parameters for WMAP; the largest relative shift seen for any parameter is 0.06\sigma. However, because this may not hold for other experimental setups (e.g., for different instrumental noise properties or analysis masks), but must rather be verified on a case-by-case basis, we recommend our new hybridization scheme for future experiments for statistical self-consistency reasons. Second, we use the same expression to improve the convergence rate of the Blackwell-Rao likelihood estimator, reducing the required number of Monte Carlo samples by several orders of magnitude, and thereby extend it to high-l applications.