On Default Priors for Robust Bayesian Estimation with Divergences

This paper presents objective priors for robust Bayesian estimation against outliers based on divergences. The minimum γ-divergence estimator is well-known to work well in estimation against heavy contamination. The robust Bayesian methods by using quasi-posterior distributions based on divergences have been also proposed in recent years. In the objective Bayesian framework, the selection of default prior distributions under such quasi-posterior distributions is an important problem. In this study, we provide some properties of reference and moment matching priors under the quasi-posterior distribution based on the γ-divergence. In particular, we show that the proposed priors are approximately robust under the condition on the contamination distribution without assuming any conditions on the contamination ratio. Some simulation studies are also presented.

Galaxy disc scaling relations: A tight linear galaxy–halo connection challenges abundance matching

In ΛCDM cosmology, to first order, galaxies form out of the cooling of baryons within the virial radius of their dark matter halo. The fractions of mass and angular momentum retained in the baryonic and stellar components of disc galaxies put strong constraints on our understanding of galaxy formation. In this work, we derive the fraction of angular momentum retained in the stellar component of spirals, fj, the global star formation efficiency fM, and the ratio of the asymptotic circular velocity (Vflat) to the virial velocity fV, and their scatter, by fitting simultaneously the observed stellar mass-velocity (Tully–Fisher), size–mass, and mass–angular momentum (Fall) relations. We compare the goodness of fit of three models: (i) where the logarithm of fj, fM, and fV vary linearly with the logarithm of the observable Vflat; (ii) where these values vary as a double power law; and (iii) where these values also vary as a double power law but with a prior imposed on fM such that it follows the expectations from widely used abundance matching models. We conclude that the scatter in these fractions is particularly small (∼0.07 dex) and that the linear model is by far statistically preferred to that with abundance matching priors. This indicates that the fundamental galaxy formation parameters are small-scatter single-slope monotonic functions of mass, instead of being complicated non-monotonic functions. This incidentally confirms that the most massive spiral galaxies should have turned nearly all the baryons associated with their haloes into stars. We call this the failed feedback problem.