Mitigating Eddington and Malmquist Biases in Latent-Inclination Inference of the Tully-Fisher Relation

Abstract

The Tully-Fisher relation is a vital distance indicator, but its precise inference is challenged by selection bias, statistical bias, and uncertain inclination corrections. This study presents a Bayesian framework that simultaneously addresses these issues. To eliminate the need for individual inclination corrections, inclination is treated as a latent variable with a known probability distribution. To correct for the distance-dependent Malmquist bias arising from sample selection, the model incorporates Gaussian scatter in the dependent variable, the distribution of the independent variable, and the observational selection function into the data likelihood. To mitigate the statistical bias -- termed the ``general Eddington bias'' -- caused by Gaussian scatter and the non-uniform distribution of the independent variable, two methods are introduced: (1) analytical bias corrections applied to the dependent variable before likelihood computation, and (2) a dual-scatter model that accounts for Gaussian scatter in the independent variable within the likelihood function. The effectiveness of these methods is demonstrated using simulated datasets. By rigorously addressing selection and statistical biases in a latent-variable regression analysis, this work provides a robust approach for unbiased distance estimates from standardizable candles, which is critical for improving the accuracy of Hubble constant determinations.