An Unbiased Method of Measuring the Ratio of Two Data Sets

Abstract

In certain cases of astronomical data analysis, the meaningful physical quantity to extract is the ratio R between two data sets. Examples include the lensing ratio, the interloper rate in spectroscopic redshift samples, and the decay rate of gravitational potential and E _G to test gravity. However, simply taking the ratio of the two data sets is biased, since it renders (even statistical) errors in the denominator into systematic errors in R . Furthermore, it is not optimal in minimizing statistical errors of R . Based on Bayesian analysis and the usual assumption of Gaussian error in the data, we derive an analytical expression of the posterior probability density function P ( R ). This result enables fast and unbiased R measurement, with minimal statistical errors. Furthermore, it relies on no underlying model other than the proportionality relation between the two data sets. Even more generally, it applies to cases where the proportionality relation holds for the underlying physics/statistics instead of the two data sets directly. It also applies to the case of multiple ratios ( R → R = ( R _1 , R _2 , ⋯ )). We take the lensing ratio as an example to demonstrate our method. We take lenses as DESI imaging survey galaxies, and sources as DECaLS cosmic shear and Planck cosmic microwave background (CMB) lensing. We restrict the analysis to the ratio between CMB lensing and cosmic shear. The resulting P ( R ) values, for multiple lens–shear pairs, are all nearly Gaussian. The signal-to-noise ratio of measured R ranges from 4.9 to 8.4. We perform several tests to verify the robustness of the above result.

Keywords

• Weak gravitational lensing
• Large-scale structure of the universe
• Astrostatistics
• Observational cosmology
• Cosmic microwave background radiation