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REVIEW 2 major objections 4 minor 44 references

A correlated-observable selection bias can shift the measured Hubble constant by about 1 km/s/Mpc, and the data show weak evidence it is there.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 04:45 UTC pith:EYWP6ENC

load-bearing objection A careful, transparent application of a standard selection-correction model to the Cepheid distance ladder; the H0 shift is real in the model but rests on an acknowledged approximation. the 2 major comments →

arxiv 2607.22425 v1 pith:EYWP6ENC submitted 2026-07-24 astro-ph.CO

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

Will J. Percival (Waterloo Centre for Astrophysics , University of Waterloo) This is my paper
classification astro-ph.CO
keywords selection biascorrelated observablesHubble constantdistance ladderCepheid variablesBayesian inferenceMalmquist biasHeckman correction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives the correct likelihood for measurements made on one observable when the sample was selected on another, correlated observable. Applied to the Cepheid distance ladder, it shows that residual selection of this form would shift the recovered Hubble constant downward by 0.7 to 1.1 km/s/Mpc, depending on the distance prior. The evidence for a non-zero effect is weak, at 1.2 to 1.9 sigma, but the shift is comparable to the current systematic budget of H0 measurements. The paper argues that such selection corrections should be included in distance-ladder systematics even when not strongly detected.

Core claim

The central result is the selection-corrected likelihood f(y|x<xlim) = phi(z)/(sigma_y Phi(alpha)) * Phi((alpha - rho z)/sqrt(1-rho^2)), where the sample is truncated in x and the measured variable y is correlated with x. In the weak-correlation regime this reduces to a mean shift proportional to rho sigma_y lambda(alpha), with only higher-order skewness. Because each Cepheid has a different photometric uncertainty, the predicted correction scales with the per-star error, which lets the data constrain the selection amplitude at the same time as the distance scale. Fitting this to the R22 Cepheid data with a single selection parameter lowers H0 by -0.7 to -1.1 km/s/Mpc depending on the distan

What carries the argument

Equation (4), the probit-truncated correlated-Normal likelihood. It is the Bayes-theorem product of the marginal Normal for y and the conditional probability of passing the x cut, expressed through the standard Normal density phi and cumulative distribution Phi. The mean-shift approximation (Eq. 6), E[y|x<xlim] = mu_y - rho sigma_y lambda(alpha), is what is implemented in the distance-ladder fits by adding a term s_i sigma_j to each Cepheid magnitude.

Load-bearing premise

Residual Cepheid selection is adequately described by the mean-shift approximation, adding a term s_i sigma_j to the magnitude model — that is, the selection correction scales linearly with each star's photometric error and other terms in the full likelihood can be neglected.

What would settle it

Take the exactly known bivariate-Normal model of Section 2.1, impose a known cut in x with a known rho, then fit the data with the mean-shift correction alone; if the recovered s_i and H0 shift deviate from the analytic predictions by more than the statistical error, the mean-shift approximation or the assumption of constant rho per host is false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If residual selection exists in this form, each Cepheid's correction scales with its measured photometric error, so heteroscedastic errors partially break the degeneracy between selection and distance.
  • Fitting a single shared selection parameter reduces H0 by -0.7 km/s/Mpc (uniform distance-modulus prior) to -1.1 km/s/Mpc (uniform-r^3 prior).
  • The combination of the uniform-r^3 distance prior and the shared selection correction yields H0 = 69.8 +/- 1.2 km/s/Mpc, reducing the Hubble tension.
  • The 38-parameter per-host selection model overfits: chi^2 drops by 23.2 for 38 extra parameters, so the paper adopts the single-parameter model as the conservative baseline.
  • Measures with smaller photometric errors, such as JWST photometry of extragalactic Cepheids, would shrink the predicted size of this class of selection effect.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same likelihood applies to any survey that cuts on one observable and analyzes another correlated one — e.g., SN Ia host selection, weak-lensing shape measurements, or cluster cosmology — so the machinery is a general tool, not only a H0 correction.
  • Because the detection significance is 1.2-1.9σ, the true size of the shift, if any, is largely unknown; the paper's H0 shifts should be read as the sensitivity of the current data to this class of effect, not as a measured systematic value.
  • A decisive test would be to forward-model the Cepheid selection in the R22 pipeline and check whether the resulting correction scales with sigma_j as the model assumes; if it does not, the fitted s_i would be an artifact.
  • The 2-step correction procedure is not equivalent to joint Bayesian inference; the appendix shows it removes the prior-selection coupling, so future analyses that treat selection corrections as fixed should propagate their uncertainty.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper derives the likelihood for a measurement y when selection is applied to a correlated observable x, f(y|x<x_lim) = φ(z)/(σ_y Φ(α)) Φ((α−ρz)/√(1−ρ²)) (Eq. 4), and notes that in the weak-correlation, mild-truncation regime this reduces to a nearly Gaussian likelihood with a mean shift proportional to the measurement uncertainty (Eq. 6). It then applies this model to the SH0ES/R22 Cepheid distance ladder by adding per-host or shared selection parameters s_i σ_j to the magnitude model. Fitting to the R22 data, the paper reports weak evidence for nonzero residual selection (1.2–1.9σ depending on the distance prior) and a corresponding reduction in H0 of −0.7 to −1.1 km/s/Mpc for a single shared selection parameter. Larger shifts are found for per-host selection parameters, particularly with a uniform-r³ distance prior, but the paper argues these overfit. The paper also examines the impact of alternative distance priors on the Milky Way Cepheid calibration and the extragalactic distance ladder.

Significance. If the empirical result holds, the paper makes a useful contribution to the Hubble-tension literature by identifying a plausible ~1 km/s/Mpc systematic from correlated-observable selection that is not currently included in distance-ladder error budgets. The theoretical derivation in Section 2 is clean, standard (Heckman-type), and clearly presented, and the analytic MAP solution for the augmented linear model (Eq. 16) is elegant, fast, and validated against MCMC. The paper also contains good empirical practice: posterior predictive checks, explicit discussion of prior sensitivity, and a null test showing that the selection-augmented model does not bias H0 when selection is absent. The reported detection significance is honestly low, and the authors are transparent about the model being an effective description rather than a physical pipeline model. The main weaknesses are that the data analysis uses only the mean-shift approximation to Eq. 4, and the checks performed cannot validate that approximation against the full selection likelihood.

major comments (2)
  1. [§5.1, Eq. (15)] The data fit replaces the full selection likelihood of Eq. (4) with a simple mean shift s_i σ_j added to a Gaussian likelihood, i.e. Eq. (6) with the variance and higher-order terms omitted. This is valid only in the regime ρ≪1, α≫1, where the corrections in Eq. (7) are negligible. However, the paper never demonstrates that the actual Cepheid selection is in this regime; the fitted s_i = ρλ(α) is a degenerate product, so ρ and α are not separately constrained. The posterior predictive tests in §5.4–§5.8 are generated from the same approximate model, so they cannot establish the adequacy of the mean-shift approximation. Since the reported ΔH0 values are conditional on this approximation, this is a load-bearing assumption. I would like to see either a direct fit using the full likelihood of Eq. (4), or a simulation-based sensitivity study showing that the mean-shift model recovers the same
  2. [§5.7–§5.8] The null tests are described as posterior predictive tests, but the synthetic data are drawn from the same model that is being fit — namely, the approximate mean-shift model. These tests therefore verify internal consistency of the approximate model, not the external validity of the approximation relative to the true selection process. The paper acknowledges this in §6 ('we cannot be certain that the simple model is appropriate'), but the limitation should be stated directly next to the PPT results in §5.7 and §5.8. As written, a reader could easily overinterpret the PPT agreement as evidence that the selection model is correct.
minor comments (4)
  1. [General] There are several typos: 'paramters' in the Table 2 caption, 'corelation' in §6, 'difficutly' in §6, and an unclosed parenthesis in the introduction ('Section 4.').
  2. [§4] The text refers to 'H¨og˚as and M¨ortsell (2025)' and 'H¨og¨as and M¨ortsell (2026)' with inconsistent accent rendering; the reference list should be checked for consistency.
  3. [Table 2] The notation 'individual (42)' versus 'individual' is not defined until later in the text; consider adding a footnote in the table caption clarifying that (42) includes M31, SMC, and LMC.
  4. [§5.3] The sentence 'The signal-to-noise is then Z=1ᵀCₛ⁻¹ŝ/√(1ᵀCₛ⁻¹1)' uses notation that is not fully defined (e.g., ŝ, Cₛ). It is clear from context, but a brief definition would improve readability.

Circularity Check

0 steps flagged

No significant circularity: the selection likelihood is an external (Heckman) result, and the selection parameters are fitted nuisance parameters; residual self-citations are auxiliary and not load-bearing.

full rationale

The paper's derivation chain is self-contained on the statistical side: Eq. (4) is re-derived within the paper (and explicitly credited to Heckman 1979), and Eqs. (6)-(7) follow by Taylor expansion, not by assuming the conclusion. The distance-ladder application (§5.1) adds a correction s_i σ_j that corresponds to the combination ρλ(α) from Eq. (6); s_i is a free parameter estimated from the same data and marginalized over, which is standard joint inference rather than a predicted output. The reported ΔH0 values are posterior shifts from fits, not out-of-sample predictions. The posterior predictive tests (§5.4-5.8) are explicitly described as necessary-consistency checks, and the paper states "Posterior predictive agreement is a necessary condition for a good model but not evidence that the model is uniquely correct." The paper also acknowledges the model is "an effective statistical description, not a physical model of the SH0ES pipeline" and that "we cannot be certain that the simple model is appropriate" — these are model-validity limitations, not circular reasoning. Self-citations (Paradiso et al. 2024, Nguyen et al. 2025, Crespi et al. 2025, Krolewski et al. 2025) appear only as auxiliary references for methods or context and do not carry the argument. No step reduces by construction to its input.

Axiom & Free-Parameter Ledger

4 free parameters · 7 axioms · 0 invented entities

The central derivation rests on standard Heckman selection math; the application rests on several domain assumptions about the SH0ES selection process that are acknowledged but not independently verified. The main fitted free parameters are the selection coefficients s/s_i, with a hyperparameter variant, plus an adopted intrinsic scatter. No new physical entities are introduced.

free parameters (4)
  • shared selection coefficient s (or per-host s_i) = not tabulated directly; implied mean magnitude correction +0.03 to +0.05 mag
    Added to Wesenheit model as s σ_j; fitted to Cepheid residuals and drives the H0 shifts in Table 2.
  • per-host selection coefficients s_i (38 or 42) = individual values plotted in Fig. 2; mean +0.03 to +0.05 mag
    Allows different selection per host; in the individual model with uniform-r3 prior this couples strongly to the distance prior and lowers H0 to 66–69 km/s/Mpc.
  • hierarchical hyperparameters (μ_s, σ_s) = not reported
    Used in hierarchical model; constrain per-host s_i and reduce the prior coupling.
  • Cepheid intrinsic scatter σ_intr = 0.06 mag (adopted, not fitted)
    Chosen for consistency with R22 (§4); enters σ_j and thus the amplitude s σ_j of the selection correction; preferred values from Högas & Mörtsell (2025) are 0.044 mag.
axioms (7)
  • standard math Heckman selection model: bivariate Normal (x,y) with probit threshold on x (Eqs. 2–5)
    Derivation follows Heckman (1979); cited in §2; mathematical content is standard and externally verified by econometrics literature.
  • ad hoc to paper Mean-shift approximation Eq. 6 replaces full selection likelihood Eq. 4 in the data fit
    §5.1 adds s_i σ_j corrections without propagating the variance/skewness terms of Eq. 4; validity requires small ρ and/or large α, which are not checked.
  • domain assumption Constant selection parameters per host (same α, ρ; only s=ρλ(α) identified)
    §2.2: 'we can model this correlation with a fixed correlation coefficient ρ for each host'; the actual SH0ES selection involves multiple criteria and period cuts, so constancy is an assumption.
  • domain assumption Selection in visible correlates with NIR photometric uncertainty via crowding
    §2.2: 'we have selection and observation correlated through the background crowding'; no direct measurement of ρ provided; the correction is assumed to scale with σ_j.
  • domain assumption R22 data vector and covariance matrix are correct and complete
    §5.1: 'If we use the R22 matrices as provided, we match their measurement of H0'; the paper re-uses R22 data without re-deriving photometry.
  • domain assumption Uniform-r3 distance prior for extragalactic hosts is an admissible modeling choice
    §3: 'there being no preferred location in the universe'; this prior choice is central to the larger H0 shifts and is contested in literature.
  • domain assumption Adopted intrinsic scatter σ_intr = 0.06 mag for Cepheids
    §4: 'for consistency with R22 we adopt σ_intr = 0.06 mag'; affects error bars and hence the σ_j weighting of the selection correction.

pith-pipeline@v1.3.0-alltime-deepseek · 17132 in / 16835 out tokens · 190211 ms · 2026-08-01T04:45:59.935961+00:00 · methodology

0 comments
read the original abstract

For over a century, following the work of Eddington, Kapteyn, Malmquist and others, astronomers have wrestled with selection biases when making inferences from samples of objects. Typically, selection is performed on the same observations used to make the measurements of interest. However, selecting objects using one observable while analyzing another can also lead to a selection bias when the observables are correlated. Within a Bayesian framework, unmodelled selection effects correspond to a misspecified generative model. We derive the likelihood for truncated selection in correlated observables and demonstrate its usefulness by searching for residual selection effects of this form in Cepheid variable star brightness measurements used in recent distance-ladder measurements of the Hubble constant H0. We specifically look for the form of bias where the selection correction depends on the photometric uncertainties, which the Cepheid data can constrain while simultaneously measuring H0. We find only weak evidence for non-zero residual selection corrections, at a significance of (1.2sigma) to (1.9sigma), depending on the distance prior adopted. Including a single extra parameter to model the unknown cut-off lowers the recovered H0 by -0.7km/s/Mpc to -1.1km/s/Mpc, again depending on the distance prior. Allowing for a different selection for each host galaxy can decrease H0 further, although this becomes very sensitive to the distance prior applied. While introducing a new selection correction cannot by itself explain the Hubble tension, it may be a component of a multi-faceted solution that includes the choice of priors on distances and other effects.

Figures

Figures reproduced from arXiv: 2607.22425 by University of Waterloo), Will J. Percival (Waterloo Centre for Astrophysics.

Figure 1
Figure 1. Figure 1: — The recovered distribution of measured magnitudes from a simple model of selection in a correlated, jointly Normal variable. In the absence of selection, the data follow the red “All” histogram, which is well described by a Normal distribution (black curve), as expected. After imposing selection, the observed distribution (blue histogram) is shifted toward brighter (lower magnitude) sources. In the weak-… view at source ↗
Figure 2
Figure 2. Figure 2: — The derived magnitude correction applied to Cepheid magnitudes in SN Ia host galaxies to account for selection effects. Data are plotted for galaxies hosting more than 40 observed Cepheids, colour-coded by each galaxy. The corrections are inferred from the data, as described in Section 5.1 allowing a different selection coefficient si for each host. Both positive and negative corrections are observed, an… view at source ↗
Figure 3
Figure 3. Figure 3: — Posterior predictive test (PPT) results comparing observed and simulated apparent magnitude distributions. Synthetic data sets are generated by drawing parameters from the posterior and sampling from the likelihood. We plot the distribution of offsets calculated from a set of 104 simulated samples and from the measured data. We consider four subsets of magnitude measurements: Cepheids in SN Ia host galax… view at source ↗
Figure 4
Figure 4. Figure 4: — The difference between the posterior mean values of H0 obtained from fits with and without galaxy-specific selection corrections, ∆H0 = Hˆ 0,sel − Hˆ0. The histograms show the results of 105 posterior predictive simulations generated under the null hypothesis of no residual selection effects, while the dashed vertical lines indicate the values inferred from the data. MAP values of H0 were calculated anal… view at source ↗
Figure 5
Figure 5. Figure 5: — Posterior predictive test (PPT) results comparing the amplitude of the selection corrections in the MAP model. We plot the distribution of offsets for both a set of 102 simulated samples (blue lines) and the actual data (thick black line). We only plot data for the Cepheids in SN Ia host galaxies and NGC 4258, as this is the most interesting sample for the corrections. Synthetic data sets are generated b… view at source ↗

discussion (0)

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