Abstract
Markov Chain Monte Carlo (MCMC) methods for sampling probability density functions (combined with abundant computational resources) have transformed the sciences, especially in performing ...probabilistic inferences, or fitting models to data. In this primarily pedagogical contribution, we give a brief overview of the most basic MCMC method and some practical advice for the use of MCMC in real inference problems. We give advice on method choice, tuning for performance, methods for initialization, tests of convergence, troubleshooting, and use of the chain output to produce or report parameter estimates with associated uncertainties. We argue that autocorrelation time is the most important test for convergence, as it directly connects to the uncertainty on the sampling estimate of any quantity of interest. We emphasize that sampling is a method for doing integrals; this guides our thinking about how MCMC output is best used.
Cold stellar streams-produced by tidal disruptions of globular clusters-are long-lived, coherent dynamical features in the halo of the Milky Way. They hold the promise of delivering precise ...information about the gravitational potential, including constraints on the shape of the dark matter halo. Because of their different ages and different positions in phase space, different streams tell us different things about the Galaxy. Here we employ a Cramér-Rao lower bound (CRLB) or Fisher-matrix approach to understand the quantitative information content in (toy versions of) 11 known streams: ATLAS, GD-1, Hermus, Kwando, Orinoco, PS1A, PS1C, PS1D, PS1E, Sangarius, and Triangulum. This approach depends on a generative model, which we have developed previously, and which permits calculation of derivatives of predicted stream properties with respect to Galaxy and stream parameters. We find that in simple analytic models of the Milky Way, streams on eccentric orbits contain the most information about the halo shape. For each stream, there are near degeneracies between dark matter halo properties and parameters of the bulge, the disk, and the stream progenitor itself, but simultaneous fitting of multiple streams will constrain all parameters at the percent level. At this precision, simulated dark matter halos deviate from simple analytic parameterizations, so we add an expansion of basis functions as a first step in giving the gravitational potential more freedom. As freedom increases, the information about the halo reduces overall, and it becomes more localized to the current position of the stream. In the limit of high model freedom, a stellar stream appears to measure the local acceleration at its current position; this motivates thinking about future nonparametric approaches. The CRLB formalism also permits us to assess the value of future measurements of stellar velocities, distances, and proper motions. We show that velocities of stream stars are essential for producing competitive constraints on the distribution of dark matter.
ABSTRACT We present photometry of images from the Wide-Field Infrared Survey Explorer (WISE) of over 400 million sources detected by the Sloan Digital Sky Survey (SDSS). We use a "forced photometry" ...technique, using measured SDSS source positions, star-galaxy classification, and galaxy profiles to define the sources whose fluxes are to be measured in the WISE images. We perform photometry with The Tractor image modeling code, working on our "unWISE" coaddds and taking account of the WISE point-spread function and a noise model. The result is a measurement of the flux of each SDSS source in each WISE band. Many sources have little flux in the WISE bands, so often the measurements we report are consistent with zero given our uncertainties. However, for many sources we get 3 or 4 measurements; these sources would not be reported by the "official" WISE pipeline and will not appear in the WISE catalog, yet they can be highly informative for some scientific questions. In addition, these small-signal measurements can be used in stacking analyses at the catalog level. The forced photometry approach has the advantage that we measure a consistent set of sources between SDSS and WISE, taking advantage of the resolution and depth of the SDSS images to interpret the WISE images; objects that are resolved in SDSS but blended together in WISE still have accurate measurements in our photometry. Our results, and the code used to produce them, are publicly available at http://unwise.me.
We measure the circular velocity curve vc(R) of the Milky Way with the highest precision to date across Galactocentric distances of 5 ≤ R ≤ 25 kpc. Our analysis draws on the six-dimensional ...phase-space coordinates of 23,000 luminous red giant stars, for which we previously determined precise parallaxes using a data-driven model that combines spectral data from APOGEE with photometric information from WISE, 2MASS, and Gaia. We derive the circular velocity curve with the Jeans equation assuming an axisymmetric gravitational potential. At the location of the Sun we determine the circular velocity with its formal uncertainty to be = with systematic uncertainties at the ∼2%-5% level. We find that the velocity curve is gently but significantly declining at (−1.7 0.1) km s−1 kpc−1, with a systematic uncertainty of 0.46 km s−1 kpc−1, beyond the inner 5 kpc. We exclude the inner 5 kpc from our analysis due to the presence of the Galactic bar, which strongly influences the kinematic structure and requires modeling in a nonaxisymmetric potential. Combining our results with external measurements of the mass distribution for the baryonic components of the Milky Way from other studies, we estimate the Galaxy's dark halo mass within the virial radius to be = and a local dark matter density of = .
We present a model for the interaction of the GD-1 stellar stream with a massive perturber that naturally explains many of the observed stream features, including a gap and an off-stream spur of ...stars. The model involves an impulse by a fast encounter, after which the stream grows a loop of stars at different orbital energies. At specific viewing angles, this loop appears offset from the stream track. A quantitative comparison of the spur-and-gap features prefers models where the perturber is in the mass range of 106-108 M . Orbit integrations back in time show that the stream encounter could not have been caused by any known globular cluster or dwarf galaxy with a determined orbit, and mass, size, and impact parameter arguments show that it could not have been caused by a molecular cloud in the Milky Way disk. The most plausible explanation for the gap-and-spur structure is an encounter with a dark matter substructure, like those predicted to populate galactic halos in ΛCDM cosmology. However, the expected densities of ΛCDM subhalos in this mass range and in this part of the Milky Way are 2 -3 lower than the inferred density of the GD-1 perturber. This observation opens up the possibility that detailed observations of streams could measure the mass spectrum of dark matter substructures and even identify individual substructures and their orbits in the Galactic halo.
No true extrasolar Earth analog is known. Hundreds of planets have been found around Sun-like stars that are either Earth-sized but on shorter periods, or else on year-long orbits but somewhat ...larger. Under strong assumptions, exoplanet catalogs have been used to make an extrapolated estimate of the rate at which Sun-like stars host Earth analogs. These studies are complicated by the fact that every catalog is censored by non-trivial selection effects and detection efficiencies, and every property (period, radius, etc.) is measured noisily. Here we present a general hierarchical probabilistic framework for making justified inferences about the population of exoplanets, taking into account survey completeness and, for the first time, observational uncertainties. We are able to make fewer assumptions about the distribution than previous studies; we only require that the occurrence rate density be a smooth function of period and radius (employing a Gaussian process). By applying our method to synthetic catalogs, we demonstrate that it produces more accurate estimates of the whole population than standard procedures based on weighting by inverse detection efficiency. We apply the method to an existing catalog of small planet candidates around G dwarf stars. We confirm a previous result that the radius distribution changes slope near Earth's radius. We find that the rate density of Earth analogs is about 0.02 (per star per natural logarithmic bin in period and radius) with large uncertainty. This number is much smaller than previous estimates made with the same data but stronger assumptions.
emcee: The MCMC Hammer Foreman-Mackey, Daniel; Hogg, David W.; Lang, Dustin ...
Publications of the Astronomical Society of the Pacific,
03/2013, Volume:
125, Issue:
925
Journal Article
Peer reviewed
Open access
ABSTRACT We introduce a stable, well tested Python implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) proposed by Goodman & Weare (2010). The code is open ...source and has already been used in several published projects in the astrophysics literature. The algorithm behind emcee has several advantages over traditional MCMC sampling methods and it has excellent performance as measured by the autocorrelation time (or function calls per independent sample). One major advantage of the algorithm is that it requires hand-tuning of only 1 or 2 parameters compared to ∼N2 for a traditional algorithm in an N-dimensional parameter space. In this document, we describe the algorithm and the details of our implementation. Exploiting the parallelism of the ensemble method, emcee permits any user to take advantage of multiple CPU cores without extra effort. The code is available online at http://dan.iel.fm/emcee under the GNU General Public License v2.
Multi-epoch radial velocity measurements of stars can be used to identify stellar, substellar, and planetary-mass companions. Even a small number of observation epochs can be informative about ...companions, though there can be multiple qualitatively different orbital solutions that fit the data. We have custom-built a Monte Carlo sampler (The Joker) that delivers reliable (and often highly multimodal) posterior samplings for companion orbital parameters given sparse radial velocity data. Here we use The Joker to perform a search for companions to 96,231 red giant stars observed in the APOGEE survey (DR14) with ≥3 spectroscopic epochs. We select stars with probable companions by making a cut on our posterior belief about the amplitude of the variation in stellar radial velocity induced by the orbit. We provide (1) a catalog of 320 companions for which the stellar companion's properties can be confidently determined, (2) a catalog of 4898 stars that likely have companions, but would require more observations to uniquely determine the orbital properties, and (3) posterior samplings for the full orbital parameters for all stars in the parent sample. We show the characteristics of systems with confidently determined companion properties and highlight interesting systems with candidate compact object companions.
M dwarfs have enormous potential for our understanding of structure and formation on both Galactic and exoplanetary scales through their properties and compositions. However, current atmosphere ...models have limited ability to reproduce spectral features in stars at the coolest temperatures (Teff < 4200 K) and to fully exploit the information content of current and upcoming large-scale spectroscopic surveys. Here we present a catalog of spectroscopic temperatures, metallicities, and spectral types for 5875 M dwarfs in the Apache Point Observatory Galactic Evolution Experiment (APOGEE) and Gaia-DR2 surveys using The Cannon: a flexible, data-driven spectral-modeling and parameter-inference framework demonstrated to estimate stellar-parameter labels ( , , , and detailed abundances) to high precision. Using a training sample of 87 M dwarfs with optically derived labels spanning calibrated with bolometric temperatures, and dex calibrated with FGK binary metallicities, we train a two-parameter model with predictive accuracy (in cross-validation) to 77 K and 0.09 dex respectively. We also train a one-dimensional spectral classification model using 51 M dwarfs with Sloan Digital Sky Survey optical spectral types ranging from M0 to M6, to predictive accuracy of 0.7 types. We find Cannon temperatures to be in agreement to within 60 K compared to a subsample of 1702 sources with color-derived temperatures, and Cannon metallicities to be in agreement to within 0.08 dex metallicity compared to a subsample of 15 FGK+M or M+M binaries. Finally, our comparison between Cannon and APOGEE pipeline (ASPCAP DR14) labels finds that ASPCAP is systematically biased toward reporting higher temperatures and lower metallicities for M dwarfs.
ABSTRACT The mass of a star is arguably its most fundamental parameter. For red giant stars, tracers luminous enough to be observed across the Galaxy, mass implies a stellar evolution age. It has ...proven to be extremely difficult to infer ages and masses directly from red giant spectra using existing methods. From the Kepler and apogee surveys, samples of several thousand stars exist with high-quality spectra and asteroseismic masses. Here we show that from these data we can build a data-driven spectral model using The Cannon, which can determine stellar masses to ∼0.07 dex from apogee dr12 spectra of red giants; these imply age estimates accurate to ∼0.2 dex (40%). We show that The Cannon constrains these ages foremost from spectral regions with CN absorption lines, elements whose surface abundances reflect mass-dependent dredge-up. We deliver an unprecedented catalog of 70,000 giants (including 20,000 red clump stars) with mass and age estimates, spanning the entire disk (from the Galactic center to kpc). We show that the age information in the spectra is not simply a corollary of the birth-material abundances and , and that, even within a monoabundance population of stars, there are age variations that vary sensibly with Galactic position. Such stellar age constraints across the Milky Way open up new avenues in Galactic archeology.
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