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Clarifying the Hubble constant tension with a Bayesian hierarchical model of the local distance ladder

Stephen M. Feeney, Daniel J. Mortlock, Niccolò Dalmasso

TL;DR

This paper reframes the local distance ladder as a Bayesian hierarchical model to robustly infer the Hubble constant $H_0$ by propagating all uncertainties from anchors to Cepheids to SNe. It demonstrates that non-Gaussian, heavy-tailed likelihoods for anchors, Cepheids, and SNe are essential to accurately capture tails of the $H_0$ posterior, which in turn affects model comparison with ΛCDM. Through Hamiltonian Monte Carlo sampling of a ~3000-parameter joint posterior, the authors find $H_0$ near $72.7$ km s$^{-1}$ Mpc$^{-1}$ for outlier-clean data and about $73.1$ km s$^{-1}$ Mpc$^{-1}$ when SN outliers are included, with Bayesian evidence indicating only modest support for deviations from ΛCDM depending on Planck datasets. The work provides a principled, extensible framework that reduces ad hoc data cuts, enables tail-aware hypothesis testing, and can be extended to incorporate additional datasets (e.g., Gaia) and more realistic outlier models.

Abstract

Estimates of the Hubble constant, $H_0$, from the distance ladder and the cosmic microwave background (CMB) differ at the $\sim$3-$σ$ level, indicating a potential issue with the standard $Λ$CDM cosmology. Interpreting this tension correctly requires a model comparison calculation depending on not only the traditional `$n$-$σ$' mismatch but also the tails of the likelihoods. Determining the form of the tails of the local $H_0$ likelihood is impossible with the standard Gaussian least-squares approximation, as it requires using non-Gaussian distributions to faithfully represent anchor likelihoods and model outliers in the Cepheid and supernova (SN) populations, and simultaneous fitting of the full distance-ladder dataset to correctly propagate uncertainties. We have developed a Bayesian hierarchical model that describes the full distance ladder, from nearby geometric anchors through Cepheids to Hubble-Flow SNe. This model does not rely on any distributions being Gaussian, allowing outliers to be modeled and obviating the need for arbitrary data cuts. Sampling from the $\sim$3000-parameter joint posterior using Hamiltonian Monte Carlo, we find $H_0$ = (72.72 $\pm$ 1.67) ${\rm km\,s^{-1}\,Mpc^{-1}}$ when applied to the outlier-cleaned Riess et al. (2016) data, and ($73.15 \pm 1.78$) ${\rm km\,s^{-1}\,Mpc^{-1}}$ with SN outliers reintroduced. Our high-fidelity sampling of the low-$H_0$ tail of the distance-ladder likelihood allows us to apply Bayesian model comparison to assess the evidence for deviation from $Λ$CDM. We set up this comparison to yield a lower limit on the odds of the underlying model being $Λ$CDM given the distance-ladder and Planck XIII (2016) CMB data. The odds against $Λ$CDM are at worst 10:1 or 7:1, depending on whether the SNe outliers are cut or modeled, or 60:1 if an approximation to the Planck Int. XLVI (2016) likelihood is used.

Clarifying the Hubble constant tension with a Bayesian hierarchical model of the local distance ladder

TL;DR

This paper reframes the local distance ladder as a Bayesian hierarchical model to robustly infer the Hubble constant by propagating all uncertainties from anchors to Cepheids to SNe. It demonstrates that non-Gaussian, heavy-tailed likelihoods for anchors, Cepheids, and SNe are essential to accurately capture tails of the posterior, which in turn affects model comparison with ΛCDM. Through Hamiltonian Monte Carlo sampling of a ~3000-parameter joint posterior, the authors find near km s Mpc for outlier-clean data and about km s Mpc when SN outliers are included, with Bayesian evidence indicating only modest support for deviations from ΛCDM depending on Planck datasets. The work provides a principled, extensible framework that reduces ad hoc data cuts, enables tail-aware hypothesis testing, and can be extended to incorporate additional datasets (e.g., Gaia) and more realistic outlier models.

Abstract

Estimates of the Hubble constant, , from the distance ladder and the cosmic microwave background (CMB) differ at the 3- level, indicating a potential issue with the standard CDM cosmology. Interpreting this tension correctly requires a model comparison calculation depending on not only the traditional `-' mismatch but also the tails of the likelihoods. Determining the form of the tails of the local likelihood is impossible with the standard Gaussian least-squares approximation, as it requires using non-Gaussian distributions to faithfully represent anchor likelihoods and model outliers in the Cepheid and supernova (SN) populations, and simultaneous fitting of the full distance-ladder dataset to correctly propagate uncertainties. We have developed a Bayesian hierarchical model that describes the full distance ladder, from nearby geometric anchors through Cepheids to Hubble-Flow SNe. This model does not rely on any distributions being Gaussian, allowing outliers to be modeled and obviating the need for arbitrary data cuts. Sampling from the 3000-parameter joint posterior using Hamiltonian Monte Carlo, we find = (72.72 1.67) when applied to the outlier-cleaned Riess et al. (2016) data, and () with SN outliers reintroduced. Our high-fidelity sampling of the low- tail of the distance-ladder likelihood allows us to apply Bayesian model comparison to assess the evidence for deviation from CDM. We set up this comparison to yield a lower limit on the odds of the underlying model being CDM given the distance-ladder and Planck XIII (2016) CMB data. The odds against CDM are at worst 10:1 or 7:1, depending on whether the SNe outliers are cut or modeled, or 60:1 if an approximation to the Planck Int. XLVI (2016) likelihood is used.

Paper Structure

This paper contains 23 sections, 25 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Hubble constant discrepancies
  3. Gaussian likelihoods
  4. Non-Gaussian likelihoods
  5. Gaussian MASER distance
  6. Heavy-tailed distribution
  7. A generative model of local distance ladder data
  8. Existing algorithms for estimating $H_0$
  9. Riess et al. (2016) approach
  10. Cardona et al. (2016) approach
  11. Bayesian parameter inference
  12. The posterior distribution of $H_0$
  13. Bayesian hierarchical model
  14. Hamiltonian Monte Carlo sampling scheme
  15. Data selection
  16. ...and 8 more sections

Figures (14)

  • Figure 1: Example likelihoods used in pedagogical model comparison calculations: Gaussian local $h$ measurement (purple solid), Gaussian MASER distance (yellow long-dash), student-t local $h$ measurement with $\nu=10$ (pink short-dash) and Planck CMB estimate (gray dot-dash).
  • Figure 2: Top: posterior probability of our standard model, $\Lambda$, versus 'tension' for increasingly heavy-tailed likelihoods (coloured solid lines) and Gaussian MASER distance likelihood (gray dot-dashed line). Bottom: as above, but plotted as a function of the width of the prior on the difference between the local and cosmological $h$ values.
  • Figure 3: The complete Bayesian network for the simultaneous analysis of the anchor, Cepheid and SN data within the model described by Equations \ref{['equation:anchor']}-\ref{['equation:z2mu']}, shown as a directed acyclic graph. Data are plotted as double black circles, model parameters as single black circles. Probability distributions are indicated by orange rectangles, Cepheid populations with red boxes and host populations with blue boxes.
  • Figure 4: The Bayesian network equivalent of the Riess_etal:2016 generalized least squares estimator for the analysis of the anchor, Cepheid and compressed SN data. Green double circles represent data measured without uncertainty, filled black circles indicate non-stochastic variables.
  • Figure 5: The Bayesian network sampled from in this study. This is somewhat simplified from the full treatment to allow the Riess_etal:2016 data to be processed.
  • ...and 9 more figures