In astronomical and cosmological studies one often wishes to infer some properties of an infinite-dimensional field indexed within a finite-dimensional metric space given only a finite collection of noisy observational data. Bayesian inference offers
an increasingly-popular strategy to overcome the inherent ill-posedness of this signal reconstruction challenge. However, there remains a great deal of confusion within the astronomical community regarding the appropriate mathematical devices for framing such analyses and the diversity of available computational procedures for recovering posterior functionals. In this brief research note I will attempt to clarify both these issues from an applied statistics perpective, with insights garnered from my post-astronomy experiences as a computational Bayesian / epidemiological geostatistician.
In this brief research note I present a generalized version of the Savage-Dickey Density Ratio for representation of the Bayes factor (or marginal likelihood ratio) of nested statistical models; the new version takes the form of a Radon-Nikodym deriv
ative and is thus applicable to a wider family of probability spaces than the original (restricted to those admitting an ordinary Lebesgue density). A derivation is given following the measure-theoretic construction of Marin & Robert (2010), and the equivalent estimator is demonstrated in application to a distributional modeling problem.
In the second paper of this series we extend our Bayesian reanalysis of the evidence for a cosmic variation of the fine structure constant to the semi-parametric modelling regime. By adopting a mixture of Dirichlet processes prior for the unexplained
errors in each instrumental subgroup of the benchmark quasar dataset we go some way towards freeing our model selection procedure from the apparent subjectivity of a fixed distributional form. Despite the infinite-dimensional domain of the error hierarchy so constructed we are able to demonstrate a recursive scheme for marginal likelihood estimation with prior-sensitivity analysis directly analogous to that presented in Paper I, thereby allowing the robustness of our posterior Bayes factors to hyper-parameter choice and model specification to be readily verified. In the course of this work we elucidate various similarities between unexplained error problems in the seemingly disparate fields of astronomy and clinical meta-analysis, and we highlight a number of sophisticated techniques for handling such problems made available by past research in the latter. It is our hope that the novel approach to semi-parametric model selection demonstrated herein may serve as a useful reference for others exploring this potentially difficult class of error model.
We review the evidence behind recent claims of spatial variation in the fine structure constant deriving from observations of ionic absorption lines in the light from distant quasars. To this end we expand upon previous non-Bayesian analyses limited
by the assumptions of an unbiased and strictly Normal distribution for the unexplained errors of the benchmark quasar dataset. Through the technique of reverse logistic regression we estimate and compare marginal likelihoods for three competing hypotheses---(i) the null hypothesis (no cosmic variation), (ii) the monopole hypothesis (a constant Earth-to-quasar offset), and (iii) the monopole+dipole hypothesis (a cosmic variation manifest to the Earth-bound observer as a North-South divergence)---under a variety of candidate parametric forms for the unexplained error term. Our analysis reveals weak support for a skeptical interpretation in which the apparent dipole effect is driven solely by systematic errors of opposing sign inherent in measurements from the two telescopes employed to obtain these observations. Throughout we seek to exemplify a best practice approach to Bayesian model selection with prior-sensitivity analysis; in a companion paper we extend this methodology to a semi-parametric framework using the infinite-dimensional Dirichlet process.
Approximate Bayesian Computation (ABC) represents a powerful methodology for the analysis of complex stochastic systems for which the likelihood of the observed data under an arbitrary set of input parameters may be entirely intractable-the latter co
ndition rendering useless the standard machinery of tractable likelihood-based, Bayesian statistical inference (e.g. conventional Markov Chain Monte Carlo simulation; MCMC). In this article we demonstrate the potential of ABC for astronomical model analysis by application to a case study in the morphological transformation of high redshift galaxies. To this end we develop, first, a stochastic model for the competing processes of merging and secular evolution in the early Universe; and second, through an ABC-based comparison against the observed demographics of massive (M_gal > 10^11 M_sun) galaxies (at 1.5 < z < 3) in the CANDELS/EGS dataset we derive posterior probability densities for the key parameters of this model. The Sequential Monte Carlo (SMC) implementation of ABC exhibited herein, featuring both a self-generating target sequence and self-refining MCMC kernel, is amongst the most efficient of contemporary approaches to this important statistical algorithm. We highlight as well through our chosen case study the value of careful summary statistic selection, and demonstrate two modern strategies for assessment and optimisation in this regard. Ultimately, our ABC analysis of the high redshift morphological mix returns tight constraints on the evolving merger rate in the early Universe and favours major merging (with disc survival or rapid reformation) over secular evolution as the mechanism most responsible for building up the first generation of bulges in early-type disks.
I present a critical review of techniques for estimating confidence intervals on binomial population proportions inferred from success counts in small-to-intermediate samples. Population proportions arise frequently as quantities of interest in astro
nomical research; for instance, in studies aiming to constrain the bar fraction, AGN fraction, SMBH fraction, merger fraction, or red sequence fraction from counts of galaxies exhibiting distinct morphological features or stellar populations. However, two of the most widely-used techniques for estimating binomial confidence intervals--the normal approximation and the Clopper & Pearson approach--are liable to misrepresent the degree of statistical uncertainty present under sampling conditions routinely encountered in astronomical surveys, leading to an ineffective use of the experimental data (and, worse, an inefficient use of the resources expended in obtaining that data). Hence, I provide here an overview of the fundamentals of binomial statistics with two principal aims: (i) to reveal the ease with which (Bayesian) binomial confidence intervals with more satisfactory behaviour may be estimated from the quantiles of the beta distribution using modern mathematical software packages (e.g. R, matlab, mathematica, IDL, python); and (ii) to demonstrate convincingly the major flaws of both the normal approximation and the Clopper & Pearson approach for error estimation.
We use the high angular resolution in the near-infrared of the WFC3 on HST to determine YHVz color-color selection criteria to identify and characterize 1.5<z<3.5 galaxies in the HUDF09 and ERS (GOODS-South) fields. The WFC3 NIR images reveal galaxie
s at these redshifts that were undetected in the rest-frame UV HUDF/GOODS images, as well as true centers and regular disks in galaxies classified as highly irregular in rest-frame UV light. Across the 1.5<z<2.15 redshift range, regular disks are unveiled in the WFC3 images of ~25% of both intermediate and high mass galaxies, i.e., above 10^10 Msun. Meanwhile, galaxies maintaining diffuse and/or irregular morphologies in the rest-frame optical light---i.e., not yet dynamically settled---at these epochs are almost entirely restricted to masses below 10^11 Msun. In contrast at 2.25 < z < 3.5 these diffuse and/or irregular structures overwhelmingly dominate the morphological mix in both the intermediate and high mass regimes, while no regular disks, and only a small fraction (25%) of smooth spheroids, are evident above 10^11 Msun. Strikingly, by 1.5 < z < 2.25 roughly 2 out of every 3 galaxies at the highest masses are spheroids. In our small sample, the fraction of star-forming galaxies at these mass scales decreases concurrently from ~60% to ~5%. If confirmed, this indicates that z~2 is the epoch of both the morphological transformation and quenching of star-formation which assemble the first substantial population of massive ellipticals.
We investigate the (large-scale) bar fraction in a mass-complete sample of M > 10^10.5 Msun disk galaxies at 0.2 < z < 0.6 in the COSMOS field. The fraction of barred disks strongly depends on mass, disk morphology, and specific star formation rate (
SSFR). At intermediate stellar mass (10^10.5 < M < 10^11 Msun) the bar fraction in early-type disks is much higher, at all redshifts, by a factor ~2, than that in late-type disks. This trend is reversed at higher stellar mass (M > 10^11 Msun), where the fraction of bars in early-type disks becomes significantly lower, at all redshifts, than that in late-type disks. The bar fractions for galaxies with low and high SSFRs closely follow those of the morphologically-selected early-type and late-type populations, respectively. This indicates a close correspondence between morphology and SSFR in disk galaxies at these earlier epochs. Interestingly, the total bar fraction in 10^10.5 < M < 10^11 Msun disks is built up by a factor of ~2 over the redshift interval explored, while for M > 10^11 Msun disks it remains roughly constant. This indicates that, already by z ~ 0.6, spectral and morphological transformations in the most massive disk galaxies have largely converged to the familiar Hubble sequence that we observe in the local Universe, while for intermediate mass disks this convergence is ongoing until at least z ~ 0.2. Moreover, these results highlight the importance of employing mass-limited samples for quantifying the evolution of barred galaxies. Finally, the evolution of the barred galaxy populations investigated does not depend on the large-scale environmental density (at least, on the scales which can be probed with the available photometric redshifts).
