Determining $H_0$ with Bayesian hyper-parameters

TL;DR

The paper tackles the determination of the local Hubble constant $H_0$ by employing Bayesian hyper-parameters (HPs) to Cepheid and SN Ia distance data, using anchors from NGC 4258, the LMC, MW, and M31. By marginalizing over per-point HPs, the method down-weights data points with potential unrecognized systematics without subjective outlier rejection, allowing a dataset-consistency check via effective weights. Applying this framework to the Riess et al. 2011 data yields $H_0 = 75.0 \pm 3.9$ km s$^{-1}$ Mpc$^{-1}$, while the larger Riess et al. 2016 data gives $H_0 = 73.75 \pm 2.11$ km s$^{-1}$ Mpc$^{-1}$, with anchor choices (especially including MW) significantly influencing the result. The HP approach proves robust across several assumptions and provides a transparent means to probe unknown systematics, highlighting the impact of anchors and suggesting a path toward reconciling local calibrations with CMB-based inferences in light of future Gaia data.

Abstract

We re-analyse recent Cepheid data to estimate the Hubble parameter $H_0$ by using Bayesian hyper-parameters (HPs). We consider the two data sets from Riess et al 2011 and 2016 (labelled R11 and R16, with R11 containing less than half the data of R16) and include the available anchor distances (megamaser system NGC4258, detached eclipsing binary distances to LMC and M31, and MW Cepheids with parallaxes), use a weak metallicity prior and no period cut for Cepheids. We find that part of the R11 data is down-weighted by the HPs but that R16 is mostly consistent with expectations for a Gaussian distribution, meaning that there is no need to down-weight the R16 data set. For R16, we find a value of $H_0 = 73.75 \pm 2.11 \, \mathrm{km} \, \mathrm{s}^{-1} \, \mathrm{Mpc}^{-1}$ if we use HPs for all data points (including Cepheid stars, supernovae type Ia, and the available anchor distances), which is about 2.6 $σ$ larger than the Planck 2015 value of $H_0 = 67.81 \pm 0.92 \,\mathrm{km}\, \mathrm{s}^{-1} \, \mathrm{Mpc}^{-1}$ and about 3.1 $σ$ larger than the updated Planck 2016 value $66.93 \pm 0.62 \,\mathrm{km}\, \mathrm{s}^{-1} \, \mathrm{Mpc}^{-1}$. We test the effect of different assumptions, and find that the choice of anchor distances affects the final value significantly. If we exclude the Milky Way from the anchors, then the value of $H_0$ decreases. We find however no evident reason to exclude the MW data. The HP method used here avoids subjective rejection criteria for outliers and offers a way to test datasets for unknown systematics.

Figures (18)

• Figure 1: The hyper-parameter marginalized probability distribution function (pdf) of Eq. (\ref{['Eq:hyper-likelihood']}) for $\sigma = 1$, in orange. Close to the origin, $x=0$, it is similar to a Gaussian pdf with $\sigma = \sqrt{5/3}$ (green), except that its amplitude at the peak is 16% lower than a normalised Gaussian with that variance. Asymptotically for $|x|\rightarrow\infty$ it decreases as $1/x^3$ and looks like a student's t distribution with 2 degrees of freedom (blue), but the latter is narrower at small $x$.
• Figure 2: Different determinations of the Hubble constant. The top panel includes direct measurements that used R16 data; indirect measurements are also shown. From top to bottom (less precise to more precise): the H0LiCOW distance measurement Bonvin:2016crt, the indirect determination by the WMAP team Hinshaw:2012aka, our 'baseline analysis' for the R16 data set in Eq. \ref{['Eq:primary-best-fit-R16']}, the Riess et al. measurement Riess:2016jrr (R16), the indirect measurement by the Planck collaboration Ade:2015xua and the revised indirect measurement by the Planck collaboration Aghanim:2016yuo. The bottom panel includes direct measurements that used R11 and the same indirect measurements as in the top panel. From less precise to more precise: our 'baseline analysis' for the R11 data set in Eq. \ref{['Eq:H0-value-standard-analysis']}, H0LiCOW, Efstathiou's measurement using three anchor distances and a $60$ days period cut-off Efstathiou:2013via, measurement by Riess et al.Riess:2011yx (R11), WMAP 2009, Planck 2015, Planck 2016.
• Figure 3: Posterior constraints for fit $(29)$ (red contours) and fit $(44)$ (blue contours), marginalizing over HPs. The analysis with R11 data uses a strong prior on the metallicity $Z_w$ but is otherwise weaker because it uses less data than R16. Green, red and black vertical dashed lines in $\mu_{0,4258}$ column indicate NGC 4258 distance modulus from Polshaw:2015ika, Riess:2016jrr and Humphreys:2013eja, respectively. The black dashed vertical line in $\mu_{0,\mathrm{LMC}}$ column shows the LMC distance modulus from Pietrzynski:2013gia. The black dotted vertical line in the $H_0$ column indicates the updated Planck 2016 value for the base six-parameter $\Lambda$CDM model Aghanim:2016yuo. Black, green, and red dashed vertical lines in $H_0$ column respectively indicate the values derived by the Planck collaboration for the base six-parameter $\Lambda$CDM model Ade:2015xua, Efstathiou's value Efstathiou:2013via used by the Planck collaboration as a prior, and the $3\%$ measurement reported by Riess:2011yx; The red dotted vertical line indicates the best estimate from the analysis in Riess:2016jrr. The numbers of HPs is 722 for fit $(29)$ (red contours here) and 2298 for fit $(44)$ (blue contours here).
• Figure 4: Effective hyper-parameters for the R11 Cepheid sample used in fit $(29)$. Out of the 646 Cepheid variables in the 8 $\mathrm{SNe\,Ia}$ host galaxies and in the $\mathrm{NGC4258}$ megamaser system, 348 have $\alpha^{\mathrm{eff}}=1$, 263 $10^{-1}\leq \alpha^{\mathrm{eff}} < 1$, 34 $10^{-2}\leq \alpha^{\mathrm{eff}} < 10^{-1}$, and 1 $10^{-3} \leq \alpha^{\mathrm{eff}} < 10^{-2}$. Out of the 53 LMC Cepheid variables, 25 have $\alpha^{\mathrm{eff}}=1$, 24 $10^{-1}\leq \alpha^{\mathrm{eff}} < 1$, 4 $10^{-2}\leq \alpha^{\mathrm{eff}} < 10^{-1}$. Out of the 13 MW Cepheid stars, 10 stars have $\alpha^{\mathrm{eff}}=1$ and 3 stars with $10^{-1}\leq \alpha^{\mathrm{eff}} < 1$ (compare with Fig. \ref{['Fig:MW-Cepheid-variables']}). Overall, $23\%$ of the MW Cepheids are down-weighted; the fraction reaches $46\%$ for extragalactic Cepheids in Riess:2011yx; as for the LMC Cepheid variables, the analysis down-weights $53\%$ of the stars.
• Figure 5: Relative distances from Cepheids and SNe Ia. We plot the peak apparent visual magnitudes of each $\mathrm{SNe\,Ia}$ (from Table 3 in Riess:2011yx) with error bars rescaled by HPs (colour code is the same as in Fig. \ref{['Fig:LMC-Cepheid-variables-fit-c']}) against the relative distances between hosts determined from the 'baseline analysis' fit $(29)$ in Tables \ref{['Table:details-fits']}--\ref{['Table:Constraints-main-analysis']}. The solid line shows the corresponding best fit. The first point on the left corresponds to the expected reddening-free, peak magnitude of an $\mathrm{SNe\,Ia}$ appearing in the megamaser system NGC 4258 which is derived from the fit $(29)$.