adsabs/AstroCLIMB huggingface dataset:
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: (Left) Relationship between the depth of the minimum between the primary and secondary maxima of the i-band light curves and the color stretch parameter, s  BV , for a combined sample of SNe Ia from the CSP-I and CSP-II with Branch classifications. Solid symbols correspond to objects whose light curves are plotted in Figures 13(b) and 14(a) and (b). The inset plot illustrates an example of how the depth (the red line) is measured for each SN. (Right) B-band Hubble residuals in magnitudes (upper panel) and the decline rate parameter, Δm 15(B), (lower panel) plotted vs. the depth of the minimum between the primary and secondary maxima of the i-band light curves for 91T-like, 99aa-like, and slow-declining CN SNe observed by the CSP-II. The larger symbols in the lower diagram correspond to SNe whose values of Δm 15(B) were measured directly; the smaller symbols correspond to those where SNooPy template fits were used to measure the decline rate. Linear regression fits are shown in both panels as dashed lines. Values of the Pearson correlation coefficient, ρ, are also indicated.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: (a) BVri light curves of 12 99aa-like SNe observed by the CSP-I. K-corrections have been applied using the Hsiao et al. (2007) spectral template. The light curves in each filter were then shifted vertically to the same magnitude at maximum light after correction for time dilation. Decline rates, Δm 15(B), for each SN are given in parentheses. (b) Same as (a), but for BVri light curves of 11 slow-declining Branch CN SNe observed by the CSP-II.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Evolution of the apparent (B − V) and (r − i) colors of 91T-like (left), 99aa-like (middle), and slow-declining CN SNe (right) observed by the CSP-II. These are the same objects whose BVri light curves are plotted in Figures 13(b) and 14(a) and (b). The color measurements have been K-corrected and also corrected for Milky Way reddening. The numbers in parentheses after the names of the SNe in the legends are the values of E(B − V)host estimated using SNooPy. The dashed lines in the (B − V) plots are the Lira relations for unreddened SNe as given in Equation (6) of Burns et al. (2014).
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Color evolution in (B − V) and (r − i) of the 91T-like (magenta symbols), 99aa-like (cyan symbols), and slow-declining CN SNe (black symbols) with low host reddenings E(B − V) ≤ 0.10 mag observed by the CSP-II. The dashed line in the (B − V) plot is the Lira relation for unreddened SNe as given in Equation (6) of Burns et al. (2014).
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Comparison of the NIR spectra of the 91T-like event LSQ12gdj and the Branch CN SN 2011fe at three different phases. Identifications of the strongest features are given. The spectra of SN 2011fe have been combined with the optical spectrophotometry published by Pereira et al. (2013) in order to include the Ca II λλ8498, 8542, 8662 triplet feature. These extensions are denoted by the dashed black lines. Wavelength regions of the strongest telluric features are shaded in gray and indicated by the Earth symbol.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Spectra covering the wavelength region of the [Fe II] 1.644 μm lines of SN 1991T and the slow-declining Branch CN SN 2013aa, both obtained approximately 1 yr past B maximum. References for the spectra are Spyromilio et al. (1992) for SN 1991T and Kumar et al. (2023) for SN 2013aa.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Velocity measurements of the edge of the H-band break plotted vs. the color stretch parameter, s  BV . The values for the Branch CN, BL (“broad-line”), and CL (“cool”) SNe are reproduced from Figure 4 of Ashall et al. (2019). New measurements for three 99aa-like events (SN 2012G, SN 2013hh, and ASASSN-14me) and two 91T-like SNe (LSQ12gdj and SN 2014eg) from the CSP-II have been added.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Comparison of the 0.8–1.3 μm spectra of the 91T-like SN LSQ12gdj, the 99aa-like events LSQ12hzj and SN 2013hh, and the Branch CN SN 2011fe a few days before and after maximum light. Identifications of the strongest features are given. See text for further details.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Color evolution in (uvw1 − v) and (uvm2 − uvw1) of the 91T-like (upper left), 99aa-like (upper right), and slow-declining CN SNe (lower). As detailed in the text, the colors have been corrected for both Milky Way and host galaxy dust extinction. The near-UV red and blue groups defined by Milne et al. (2013) are indicated by the red and blue regions, respectively.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Cumulative histograms of the host galaxy masses of the CSP-I (black) and CSP-II (red) SNe Ia samples. Note the heavy weighting to larger masses of the CSP-I sample, which was mostly drawn from targeted SN searches, compared to the CSP-II sample, which came nearly entirely from untargeted searches. Also plotted is the host mass distribution of the updated PISCO sample (see Section 7.2), which has been shifted to lower mass values by 0.12 dex to account for differences in the Hubble constants and stellar initial mass functions (Rana & Basu 1992; vs. Salpeter 1955) assumed in computing the masses for the CSP-I and CSP-II samples (Uddin et al. 2020, 2023) and the PISCO sample (Galbany et al. 2018).
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: (Upper) Equivalent width of the host galaxy component of Na I D absorption, EW(Na I D), plotted vs. the host reddening, E(B − V)Host, for the 91T-like, 99aa-like, and slow-declining Branch CN SNe in the CSP-II sample. Plotted as a dashed line is the relation given by Poznanski et al. (2012), while the gray area corresponds to the 1σ dispersion observed for lines of sight through the Milky Way (Phillips et al. 2013). (Lower) Cumulative distributions of host galaxy equivalent widths for the 91T-like, 99aa-like, and slow-declining Branch CN SNe in the CSP-II sample.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Swift U, UVW1, UVM2, and UVW2 light curves of 91T-like (magenta), 99aa-like (cyan), and slow-declining Branch CN (black) SNe. The observations are corrected for time dilation and normalized to the same magnitude at maximum light in each filter. The time axis is the time (in days) with respect to the phase of B max.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Cumulative histograms of the host galaxy masses of 91T-like (magenta), 99aa-like (cyan), and Branch slow-declining CN SNe (black) observed by the CSP-II. Also plotted is the host mass distribution of the Branch SS subsample (brown), which is the combination of the 91T-like and 99aa-like SNe. The number of SNe in each of these subsamples is indicated in the figure legend. The cumulative host mass distribution for the full sample of 193 CSP-II SNe Ia is plotted with a dashed blue line. Shown in the lower right corner are the p-values derived from two-sample K-S tests with the full CSP-II SNe Ia sample serving as the reference. These are color-coded to indicate the different CSP-II subsamples (CN = slow-declining Branch CN; 99aa = 99aa-like; 91T = 91T-like; SS = 99aa-like + 91T-like).
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Cumulative histograms of local environmental properties derived from IFU observations. Shown in the upper left corner of each plot are the p​​​​​​-​values derived from two-sample K-S tests with the updated PISCO sample serving as the reference sample. These are color-coded to indicate the different CSP-II subsamples (CN = slow-declining Branch CN; 99aa = 99aa-like; 91T = 91T-like; SS = 99aa-like + 91T-like).
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Pseudobolometric light curves of the 91T-like SN LSQ12gdj, the 99aa-like iPTF14bdn, and the slow-declining Branch CN SN ASASSN-14lp integrated between 2000 and 9000 Å. The data are plotted as a function of time in days since first light. The gray shading indicates the luminosity uncertainty for ASASSN-14lp due to the large error in its distance (see Table 3).
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Spectra of three SNe Ia–CSM (SN 2002ic, PTF11kx, and SN 2018evt). In each panel, comparison is made with a spectrum of SN 1991T at a comparable epoch with respect to B maximum.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Evolution of the pEW(Si II λ6355) measurements for Branch SS-class SNe Ia in the Blondin et al. (2012) spectroscopic sample classified as Wang-class “91T” (a) and “N” (b). The trajectories of SN 1991T and SN 1999aa in these diagrams are plotted for reference.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: Same as Figure A1, but for 91T-like events taken from other sources in the literature.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: (a) Same as Figure A1, but for Branch SS-class SNe Ia in the Folatelli et al. (2013) spectroscopic sample. (b) Same as Figure A1, but for SNe Ia in the Silverman et al. (2012) spectroscopic sample, which these authors classified as 91T-like and 99aa-like.
Paper: 1991T-like Supernovae 
Authors: M. M. Phillips, C. Ashall, Peter J. Brown, L. Galbany, M. A. Tucker, Christopher R. Burns, Carlos Contreras, P. Hoeflich, E. Y. Hsiao, S. Kumar, Nidia Morrell, Syed A. Uddin, E. Baron, Wendy L. Freedman, Kevin Krisciunas, S. E. Persson, Anthony L. Piro, B. J. Shappee, Maximilian Stritzinger, Nicholas B. Suntzeff, Sudeshna Chakraborty, R. P. Kirshner, J. Lu, G. H. Marion, Abigail Polin, M. Shahbandeh
Figure caption: (a) The spectrum of SN 2005eu at −10 days is compared with spectra at similar epochs of SN 1991T, SN 1999aa, and the 03fg-like SN 2006gz in the upper panel. In the lower panel, a similar comparison at approximately +7 days is shown. (b) The BVRI photometry of SN 2005eu published by Ganeshalingam et al. (2010) is plotted along with best SNooPy fits.
Paper: Pulsar Timing Arrays Require Hierarchical Models
Authors: Rutger van Haasteren
Figure caption: Posteriors and optimal statistic signal-to-noise ratio (S/N) for a realization for each of our four scenarios. Many realizations were made and analyzed (see Figure 3); the graphs shown here are for the first realization of each scenario. Left columns are the ﹩P({\mathrm{log}}_{10}{h}^{2}| d)﹩ posterior distributions for the model with hierarchical, log-uniform, and uniform priors on the detector noise amplitudes. Right columns are the normalized distributions that one gets for the values of the optimal detection statistic S/N, evaluated using noise parameters from random samples drawn from the posterior distribution. The rows are for a specific realization of the “Normal,” “Box,” and “Identical” scenarios. The signal prior is always the log-uniform prior given by Equation (5).
Paper: Pulsar Timing Arrays Require Hierarchical Models
Authors: Rutger van Haasteren
Figure caption: The four panels show the p–p plot as calculated when generating N  r  = 200 realizations of data for the four scenarios described in Section 2.1. The gray regions show the 1σ, 2σ, and 3σ regions one would get from the binomial distribution (p  r , N  r ). The priors used for inference are the hierarchical prior (solid), the log-uniform prior (dashed), and the uniform prior (dashed–dotted). The hierarchical prior performs best overall, but it is clear from the “Box” scenario (which cannot be exactly modeled by a log-normal) that even the hierarchical model will show slight posterior bias when the prior does not describe the data-generation process. Interestingly, even the “Bimodal” scenario—which is of course far from Gaussian—seems to be well described by the unimodal log-normal HBM.
Paper: Pulsar Timing Arrays Require Hierarchical Models
Authors: Rutger van Haasteren
Figure caption: Point estimators for the spin-noise parameters ﹩({\mathrm{log}}_{10}A,\gamma )﹩ for the “MLE” maximum-likelihood estimator log-uniform and the “HBM” model, plotted together with the “Injected” values. The posterior predictive for ﹩({\mathrm{log}}_{10}A,\gamma )﹩ is shown as contours. Although the point estimates were not very stable numerically due to light degeneracies in the model, the effects of shrinkage can be observed as the HBM point estimates are drawn toward the middle of the posterior predictive. For some pulsars we have also drawn lines between the MLE, HBM, and injection values, so that their relationship can be seen.
Paper: Pulsar Timing Arrays Require Hierarchical Models
Authors: Rutger van Haasteren
Figure caption: The posterior distribution of the GWB characteristic strain h  c  (yr−1) of a mock data set loosely modeled after the NANOGrav 12.5 yr data set. The injected spin noise came from a distribution of plausible parameters, and the injected GWB amplitude was conservatively set to h  c  = 10−15. The top and bottom plots show the same graph, but the scaling of the y-axis is linear and logarithmic for the top and bottom panels, respectively. The model with hierarchical priors (solid) for spin noise shows consistency with the injection, whereas the models with log-uniform (dashed) and uniform priors (dashed–dotted) for spin noise display significant posterior bias. The amount of posterior bias on display here is reason for concern regarding current PTA analyses in the literature.
Paper: Pulsar Timing Arrays Require Hierarchical Models
Authors: Rutger van Haasteren
Figure caption: The posterior distribution of the GWB characteristic strain h  c  (yr−1) of the First IPTA MDC, open data set 1 (Open1). Free model parameters were all timing model parameters, spin-noise amplitude for all pulsars, and the GWB amplitude h  c . Different lines are for different priors on the spin-noise amplitudes that have been marginalized over. The physical prior means the spin-noise amplitudes have been set to their injected value: 0. Visually, the hierarchical prior approximates the posterior of the “Physical” model the most. The posterior for uniform priors is visually inconsistent with the injection parameters.
Paper: Pulsar Timing Arrays Require Hierarchical Models
Authors: Rutger van Haasteren
Figure caption: Example of shrinkage for the “Normal” scenario of Figure 1. The top panel displays the hyperprior, the underlying distribution of the ensemble of detector noise ﹩{\mathrm{log}}_{10}{\sigma }_{a}^{2}﹩. Underneath, on the “True” line, the points represent the actual noise levels of 30 detectors in the ensemble. On the “MLE” line, the maximum-likelihood estimators of those detector noise levels are shown. On the “HBM” line, the noise estimates of the hierarchical Bayesian model are shown. Shrinkage is visible on the outer points, which are regressed slightly to the center of the hyperprior. The mean-squared error of the HBM is lower than for the MLE.
Paper: Pulsar Timing Arrays Require Hierarchical Models
Authors: Rutger van Haasteren
Figure caption: The detector noise values in the realization of data represented by the bimodal scenario of Figure 1. The solid gray histogram shows the true noise values of the population. The dashed line represents the true underlying population prior, from which the values in the histogram were drawn in this realization of the universe. The dotted line shows the (median) posterior predictive as inferred by the HBM: it is a Gaussian with both the mean and variance picked from the MCMC chain at their 50th percentile.
Paper: Pulsar Timing Arrays Require Hierarchical Models
Authors: Rutger van Haasteren
Figure caption: The detector noise σ  a  posterior distributions for the first four detectors when using log-uniform priors. True injected values are shown by the lines/cross. Detector noise σ 0 and σ 3 are detectable, whereas σ 1 and σ 2 are not. Note the difference in scale of the axes. The posterior distributions for σ 1 and σ 2 show support all the way to the lower boundary of the prior, since the data are consistent with no detector noise.
Paper: Pulsar Timing Arrays Require Hierarchical Models
Authors: Rutger van Haasteren
Figure caption: The detector noise σ  a  posterior distributions for the first four detectors when using hierarchical priors. True injected values are shown by the lines/cross. Compared to log-uniform priors, we now see that the estimates for σ 1 and σ 2 have borrowed strength from the ensemble, whereas the posterior distributions for σ 0 and σ 3 remain largely unchanged from the log-uniform case.