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 common assumption of (perhaps implicit) uniform priors on distance moduli biases distances low due to objects being intrinsically uniformly distributed in volume. At fixed measured redshifts, this biases the Hubble constant high. Second, selection effects introduce additional factors in the posterior. These typically counteract the effect of the volume prior to some extent, but depend significantly on the nature of the selection. Typical assumptions place $H_0$ at the top of the plausible range, corresponding to a redshift-selected sample. After a detailed analytic and mock-based study of these effects, we apply them to the CosmicFlows-4 sample, where introducing the distance prior causes an approximately 12~per cent increase in distances and $>8$ km/s/Mpc (55 statistical $\sigma$) decrease in the Hubble constant for the case of volume or magnitude selection. Redshift selection would fully undo this shift and is the more likely scenario, as a phenomenological model shows. We also investigate the SH0ES sample, where the volume-prior effect is modest ($1.6\sigma$) and is likely already accounted for within the SH0ES pipeline. Our work highlights the crucial need to model both the distance prior and selection accurately for robust distance ladders and derived parameters. The latter requires samples with known, homogeneous selection criteria, which should be prioritised in future surveys.