Jeffreys priors for mixture estimation

While Jeffreys priors usually are well-defined for the parameters of mixtures of distributions, they are not available in closed form. Furthermore, they often are improper priors. Hence, they have never been used to draw inference on the mixture parameters. We study in this paper the implementation and the properties of Jeffreys priors in several mixture settings, show that the associated posterior distributions most often are improper, and then propose a noninformative alternative for the analysis of mixtures

Approximate Integrated Likelihood via ABC methods

We propose a novel use of a recent new computational tool for Bayesian inference, namely the Approximate Bayesian Computation (ABC) methodology. ABC is a way to handle models for which the likelihood function may be intractable or even unavailable and/or too costly to evaluate; in particular, we consider the problem of eliminating the nuisance parameters from a complex statistical model in order to produce a likelihood function depending on the quantity of interest only. Given a proper prior for the entire vector parameter, we propose to approximate the integrated likelihood by the ratio of kernel estimators of the marginal posterior and prior for the quantity of interest. We present several examples.Comment: 28 pages, 8 figure

Constraining the Warm Dark Matter Particle Mass through Ultra-Deep UV Luminosity Functions at z=2

We compute the mass function of galactic dark matter halos for different values of the Warm Dark Matter (WDM) particle mass m_X and compare it with the abundance of ultra-faint galaxies derived from the deepest UV luminosity function available so far at redshift z~2. The magnitude limit M_UV=-13 reached by such observations allows us to probe the WDM mass functions down to scales close to or smaller than the half-mass mode mass scale ~10^9 M_sun. This allowed for an efficient discrimination among predictions for different m_X which turn out to be independent of the star formation efficiency adopted to associate the observed UV luminosities of galaxies to the corresponding dark matter masses. Adopting a conservative approach to take into account the existing theoretical uncertainties in the galaxy halo mass function, we derive a robust limit m_X>1.8 keV for the mass of thermal relic WDM particles when comparing with the measured abundance of the faintest galaxies, while m_X>1.5 keV is obtained when we compare with the Schechter fit to the observed luminosity function. The corresponding lower limit for sterile neutrinos depends on the modeling of the production mechanism; for instance m_sterile > 4 keV holds for the Shi-Fuller mechanism. We discuss the impact of observational uncertainties on the above bound on m_X. As a baseline for comparison with forthcoming observations from the HST Frontier Field, we provide predictions for the abundance of faint galaxies with M_UV=-13 for different values of m_X and of the star formation efficiency, valid up to z~4.Comment: 14 pages, 3 figures. Accepted for publication in The Astrophysical Journa

Jeffreys priors for mixture estimation: properties and alternatives

While Jeffreys priors usually are well-defined for the parameters of mixtures of distributions, they are not available in closed form. Furthermore, they often are improper priors. Hence, they have never been used to draw inference on the mixture parameters. The implementation and the properties of Jeffreys priors in several mixture settings are studied. It is shown that the associated posterior distributions most often are improper. Nevertheless, the Jeffreys prior for the mixture weights conditionally on the parameters of the mixture components will be shown to have the property of conservativeness with respect to the number of components, in case of overfitted mixture and it can be therefore used as a default priors in this context.Comment: arXiv admin note: substantial text overlap with arXiv:1511.0314

A high space density of L* Active Galactic Nuclei at z~4 in the COSMOS field

Identifying the source population of ionizing radiation, responsible for the reionization of the universe, is currently a hotly debated subject with conflicting results. Studies of faint, high-redshift star-forming galaxies, in most cases, fail to detect enough escaping ionizing radiation to sustain the process. Recently, the capacity of bright quasi-stellar objects to ionize their surrounding medium has been confirmed also for faint active galactic nuclei (AGNs), which were found to display an escaping fraction of ~74% at z~4. Such levels of escaping radiation could sustain the required UV background, given the number density of faint AGNs is adequate. Thus, it is mandatory to accurately measure the luminosity function of faint AGNs (L~L*) in the same redshift range. For this reason we have conducted a spectroscopic survey, using the wide field spectrograph IMACS at the 6.5m Baade Telescope, to determine the nature of our sample of faint AGN candidates in the COSMOS field. This sample was assembled using photometric redshifts, color, and X-ray information. We ended up with 16 spectroscopically confirmed AGNs at 3.6<z<4.2 down to a magnitude of i$_{AB}$=23.0 for an area of 1.73 deg$^{2}$. This leads to an AGN space density of ~1.6$\times10^{-6} Mpc^{-3}$ (corrected) at z~4 for an absolute magnitude of M$_{1450}$=-23.5. This is higher than previous measurements and seems to indicate that AGNs could make a substantial contribution to the ionizing background at z~4. Assuming that AGN physical parameters remain unchanged at higher redshifts and fainter luminosities, these sources could be regarded as the main drivers of cosmic reionization.Comment: 10 pages, 3 figures, accepted for publication by Ap