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Selection effects in correlated observations with application to distance-ladder observations

This paper derives a Bayesian likelihood for selection biases in correlated observables and applies it to Cepheid variable data, finding only weak evidence that such unmodelled effects significantly contribute to the Hubble tension, though they may partially lower the inferred Hubble constant depending on distance priors.

Original authors: Will J. Percival (Waterloo Centre for Astrophysics, University of Waterloo)

Published 2026-07-27

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CC BY 4.0

Original authors: Will J. Percival (Waterloo Centre for Astrophysics, University of Waterloo)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to measure the size of the entire universe. To do this, astronomers use a cosmic "ladder." The bottom rung of this ladder involves measuring the distance to nearby stars using a technique called parallax (watching them shift position as Earth orbits the Sun). The next rung up involves a special type of pulsating star called a Cepheid variable. These stars are like cosmic lighthouses: the slower they pulse, the brighter they truly are. By measuring how fast they pulse and how bright they look from Earth, astronomers can calculate how far away they are. Once they know the distance to these stars in galaxies that have exploded as Type Ia supernovae, they can use those supernovae as even brighter lighthouses to measure distances across the vast cosmos. By combining these distances with how fast those galaxies are moving away from us, scientists can calculate the Hubble constant ( $H_0$ ), which tells us the current expansion rate of the universe.

Here is the problem: everyone wants to know the exact number for this expansion rate. But there is a stubborn disagreement. Measurements made by looking at the early universe (the Cosmic Microwave Background) give one answer, while measurements using this cosmic ladder give a slightly different, faster answer. This disagreement is called the "Hubble tension." It's like two very smart friends measuring the same room with different tape measures and getting different results. For years, scientists have wondered if one of the tape measures is broken, or if there is a hidden trick in how they are reading the numbers. One major concern is "selection bias." This happens when the way you choose which stars to study accidentally skews your results. For example, if you only look at the brightest stars in a crowd, you might think the average person is taller than they really are.

This paper asks a very specific question: Could a subtle, hidden selection bias be messing up the measurements of those Cepheid stars? Specifically, the author investigates a scenario where the stars are selected based on how easy they are to see in visible light, but their brightness is measured in infrared light. Because the background "noise" in both types of light is related, picking stars that are easy to see in visible light might accidentally pick stars that look slightly different in infrared, creating a tiny, systematic error. The paper builds a statistical model to see if this specific type of bias exists in the data and, if it does, how much it changes the calculated expansion rate of the universe.

The author, Will J. Percival, starts by creating a mathematical model to describe exactly how this "correlated selection" works. Imagine you are trying to weigh apples in a basket, but you can only pick the apples that look shiny enough to be seen under a dim light. If the shininess of the apple skin is related to its actual weight, your selection process might accidentally favor heavier (or lighter) apples, skewing your average weight calculation even if you weigh them perfectly later. The paper derives a formula to correct for this, showing that even if the selection seems random, it can shift the average result if the variables are linked.

When the author applies this model to the actual data used in recent measurements of the Hubble constant, the results are a mix of "maybe" and "it depends." The study finds only weak evidence that this specific type of selection bias is actually happening. The statistical significance is low, hovering between 1.2σ and 1.9σ. In the language of science, this is like hearing a faint whisper in a noisy room; it suggests something might be there, but it's not a shout that confirms it. The data does not prove that this bias is the culprit behind the Hubble tension.

However, the paper does something very interesting: it asks, "What if we assume this bias exists and correct for it anyway?" When the author adds a correction factor to the model to account for this potential bias, the calculated value of the Hubble constant drops. Depending on the assumptions made about how stars are distributed in space (the "prior"), the expansion rate drops by 0.7 km s⁻¹ Mpc⁻¹ to 1.1 km s⁻¹ Mpc⁻¹. If the author allows for a different correction for every single galaxy, the drop can be even larger, but this makes the result very sensitive to the initial assumptions.

The paper explicitly rules out the idea that this selection effect is the sole solution to the Hubble tension. The shift it causes is real but too small to completely bridge the gap between the different measurements of the universe's expansion. The author concludes that while this specific bias might not be the main villain, it could be one small piece of a larger puzzle. The study highlights that the choice of mathematical assumptions (priors) about where stars are located has a huge impact on the final answer, sometimes more than the selection bias itself.

In the end, this paper doesn't solve the mystery of the Hubble tension, but it provides a new tool for the detective work. It shows that even if we can't prove a bias exists, we should still check how much it would change our answer if it did. The author suggests that future observations with the James Webb Space Telescope (JWST), which can see these stars with much sharper precision, will help test these ideas. If the bias is real, better measurements should reduce its effect. For now, the Hubble tension remains a stubborn puzzle, but this paper adds a new, slightly more cautious perspective to the ongoing cosmic debate.

Technical Summary: Selection Effects in Correlated Observations with Application to Distance-Ladder Observations

Problem Statement
Astronomical inference is frequently complicated by selection biases, where the detected sample of objects is not a representative subset of the underlying population. While standard selection effects (e.g., Malmquist bias) occur when selection acts directly on the measured quantity, this paper addresses a subtler class of bias: selection effects arising when objects are selected using one observable ( $x$ ) but analyzed using a different, correlated observable ( $y$ ). This scenario is prevalent in distance-ladder measurements of the Hubble constant ( $H_0$ ), specifically in the SH0ES project. In these measurements, Cepheid variable stars are selected based on visible-wavelength observations (to identify periodic variability) but their magnitudes are measured in the Near-Infrared (NIR) to minimize extinction and metallicity effects. Because background crowding and noise affect both bands, the selection and measurement observables are correlated. If unmodeled, this correlation can induce a systematic shift in the mean of the measured magnitudes, potentially biasing the inferred distance scale and $H_0$ .

Methodology
The author develops a statistical framework within a Bayesian context to model and correct for these effects.

Theoretical Derivation: The paper derives the likelihood function for a variable $y$ given that a correlated variable $x$ has been truncated (e.g., $x < x_{\text{lim}}$ ). Assuming a bivariate Normal distribution for the true values and measurement errors, the conditional distribution $f(y | x < x_{\text{lim}})$ is derived. The analysis shows that while the shape of the likelihood remains approximately Normal (with suppressed skewness in the weak-correlation regime), the mean is shifted by an amount linear in the correlation coefficient $\rho$ and dependent on the selection threshold.
Application to Cepheids: The model is applied to the Cepheid data used in the Riess et al. (2022, R22) distance ladder. The selection correction is parameterized to scale with the independently measured photometric uncertainty of each Cepheid ( $\sigma_j$ ), reflecting the expectation that selection bias is more pronounced for fainter or noisier detections.
Bayesian Implementation: The study performs a joint Bayesian analysis of the distance ladder, simultaneously inferring $H_0$ $H_{0}$ and nuisance parameters describing the selection effects. Three parameterization schemes for the selection correction are tested:
- Individual: A unique correction parameter for each host galaxy.
- Hierarchical: Correction parameters drawn from a shared Normal distribution.
- Shared: A single global correction parameter for all hosts.
Prior Sensitivity: The analysis rigorously tests the impact of prior choices on distance moduli. It contrasts the standard uniform-in-distance-modulus prior ( $\text{uniform-}\mu$ ) with a uniform-in-volume prior ( $\text{uniform-}r^3$ ), which accounts for the spatial density of galaxies in a volume-limited sample.
Validation: The results are validated using Posterior Predictive Tests (PPTs) to ensure the model does not overfit the data and that the inclusion of selection parameters does not introduce artificial biases in the absence of true selection effects.

Key Results

Evidence for Selection Effects: The data show weak evidence for non-zero residual selection corrections. Depending on the adopted distance prior, the significance of a non-zero correction ranges from $1.2\sigma$ to $1.9\sigma$ .
Impact on $H_0$ :
- Prior Sensitivity: Changing the prior from uniform-in-distance-modulus to uniform-in-volume ( $\text{uniform-}r^3$ ) reduces the inferred $H_0$ by approximately $1.7 \text{ km s}^{-1} \text{ Mpc}^{-1}$ even without selection corrections.
- Selection Correction Impact: Including a single shared selection correction parameter lowers the recovered $H_0$ by an additional $\Delta H_0 = -0.7 \text{ km s}^{-1} \text{ Mpc}^{-1}$ (for $\text{uniform-}\mu$ ) to $-1.1 \text{ km s}^{-1} \text{ Mpc}^{-1}$ (for $\text{uniform-}r^3$ ).
- Combined Effect: When combining the $\text{uniform-}r^3$ prior with the shared selection correction, the inferred $H_0$ drops to $69.8 \pm 1.2 \text{ km s}^{-1} \text{ Mpc}^{-1}$ .
- Overfitting Risks: Allowing independent selection corrections for each host galaxy increases the shift in $H_0$ further (e.g., to $66.04 \text{ km s}^{-1} \text{ Mpc}^{-1}$ with $\text{uniform-}r^3$ ), but the paper notes this model overfits the data and is highly sensitive to the prior, making the single-parameter model the conservative baseline.
Posterior Predictive Checks: PPTs confirm that the observed magnitude distributions and goodness-of-fit statistics are consistent with the model assumptions. The data lie within the range of synthetic realizations generated under the null hypothesis of no selection effects, consistent with the weak statistical significance of the detected corrections.

Significance and Claims
The paper claims that while introducing a new selection correction cannot by itself explain the "Hubble tension" (the discrepancy between local and CMB measurements of $H_0$ ), it may be a significant component of a multi-faceted solution. The study demonstrates that:

Selection effects in correlated observables can induce mean offsets that are not easily detected by standard goodness-of-fit tests but can shift cosmological inferences.
The inferred value of $H_0$ is highly sensitive to the choice of priors on distance parameters and the treatment of selection effects.
The coupling between the distance prior and the selection model is non-trivial; marginalizing over selection uncertainties allows the distance prior to exert greater influence on the global fit.
Future improvements in photometric precision (e.g., via JWST observations of extragalactic Cepheids) will alter the magnitude of these selection effects, providing a critical test of the inferred corrections.

The work provides a formal statistical framework for incorporating correlated-observable selection effects into Bayesian inference, applicable beyond Cepheids to any scenario where sample selection and measurement rely on correlated variables.

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