Frequentist versus Bayesian analyses: Cross-correlation as an (approximate) sufficient statistic for LIGO-Virgo stochastic background searches

TL;DR

This paper demonstrates that the frequency integrand of the cross-correlation statistic and its variance serve as approximate sufficient statistics for LIGO-Virgo stochastic background searches, enabling more efficient Bayesian analysis.

Contribution

It proves that a hybrid frequentist-Bayesian analysis using these statistics is equivalent to a fully Bayesian approach, improving analysis efficiency.

Findings

Frequency integrand and variance are approximate sufficient statistics.

Hybrid analysis is equivalent to fully Bayesian analysis.

Work closes a gap in LIGO-Virgo stochastic background research.

Abstract

Sufficient statistics are combinations of data in terms of which the likelihood function can be rewritten without loss of information. Depending on the data volume reduction, the use of sufficient statistics as a preliminary step in a Bayesian analysis can lead to significant increases in efficiency when sampling from posterior distributions of model parameters. Here we show that the frequency integrand of the cross-correlation statistic and its variance are approximate sufficient statistics for ground-based searches for stochastic gravitational-wave backgrounds. The sufficient statistics are approximate because one works in the weak-signal approximation and uses measured estimates of the auto-correlated power in each detector. Using analytic and numerical calculations, we prove that LIGO-Virgo's hybrid frequentist-Bayesian parameter estimation analysis is equivalent to a fully Bayesian analysis. This work closes a gap in the LIGO-Virgo literature, and suggests directions for additional searches.