The subtle statistics of the distance ladder: On the distance prior and selection effects

Abstract

Statistical methodology is rarely considered significant in distance-ladder studies or a potential contributor to the Hubble tension. We suggest it should be, highlighting two appreciable issues. First, astronomical distances are inferred latent parameters, requiring a prior. We show that the (often implicit) uniform priors on distance moduli common to Bayesian distance-ladder analyses bias distances low due to objects being uniformly distributed in volume, which biases the Hubble constant high. Frequentist $\chi^2$ methods are unbiased for volume- or redshift-limited samples only if the redshift uncertainty (including peculiar velocities) vanishes, though simulation-based calibration can correct the bias. Second, in a Bayesian framework, selection effects introduce additional posterior factors describing the probability of objects entering the sample under the model. These partly counteract the volume prior, depending on the nature of the selection. After detailed analytic and mock-based studies, we quantify the volume-prior effect in the CosmicFlows-4 and SH0ES samples. Both use frequentist methods, so the effect appears as a potential estimator bias rather than a missing prior. The implied Hubble constant shifts are significant but must not be applied na\"{\i}vely -- principled selection modelling is also required, as we investigate explicitly for CosmicFlows-4. Both effects should already be captured by the SH0ES pipeline's simulation-based bias corrections. Our work highlights the crucial need to model both distances and selection accurately, either directly in a Bayesian forward model, or via post-hoc simulation-based corrections with realistic source and selection distributions. Such modelling requires samples with known, homogeneous selection criteria, which future surveys should prioritise.