Bayes and empirical Bayes: do they merge?

Bayesian inference is attractive for its coherence and good frequentist properties. However, it is a common experience that eliciting a honest prior may be difficult and, in practice, people often take an {\em empirical Bayes} approach, plugging empirical estimates of the prior hyperparameters into the posterior distribution. Even if not rigorously justified, the underlying idea is that, when the sample size is large, empirical Bayes leads to "similar" inferential answers. Yet, precise mathematical results seem to be missing. In this work, we give a more rigorous justification in terms of merging of Bayes and empirical Bayes posterior distributions. We consider two notions of merging: Bayesian weak merging and frequentist merging in total variation. Since weak merging is related to consistency, we provide sufficient conditions for consistency of empirical Bayes posteriors. Also, we show that, under regularity conditions, the empirical Bayes procedure asymptotically selects the value of the hyperparameter for which the prior mostly favors the "truth". Examples include empirical Bayes density estimation with Dirichlet process mixtures.Comment: 27 page

A new model for deflagration fronts in reactive fluids

We present a new way of modeling deflagration fronts in reactive fluids, the main emphasis being on turbulent thermonuclear deflagration fronts in white dwarfs undergoing a Type Ia supernova explosion. Our approach is based on a level set method which treats the front as a mathematical discontinuity and allows full coupling between the front geometry and the flow field. With only minor modifications, this method can also be applied to describe contact discontinuities. Two different implementations are described and their physically correct behaviour for simple testcases is shown. First results of the method applied to the concrete problems of Type Ia supernovae and chemical hydrogen combustion are briefly discussed; a more extensive analysis of our astrophysical simulations is given in (Reinecke et al. 1998, MPA Green Report 1122b).Comment: 11 pages, 13 figures, accepted by A&A, corrected and extended according to referee's comment

Merging of globular clusters within inner galactic regions. I. Do they survive the tidal interaction?

The main topic of this paper is the investigation of the modes of interaction of globular clusters (GCs) moving in the inner part of a galaxy. This is tackled by means of high-resolution N-body simulations, whose first results are presented in this article. Our simulations dealt with primordial very massive (order of 10^7 solar masses) GCs that were able to decay, because of dynamical friction, into the inner regions of triaxial galaxies on a time much shorter than their internal relaxation time. To check the disruptive role of both tidal forces and GC-GC collisions, we maximised the tidal interaction considering GCs on quasi-radial orbits. The available CPU resources allowed us to follow 8 oscillations of the GCs along their orbits and the main findings are: i) clusters with an initial high enough King concentration parameter (c>=1.2), preserve up to 50% of their initial mass; ii) the inner density distribution of the survived clusters keep a King model profile; iii) GC-GC collisions have a negligible effect with respect to that caused by the passage through the galactic center; iv) the orbital energy dissipation due to the tidal interaction is of the same order of that caused by dynamical friction; v) complex sub-structures like "ripples" and "clumps" formed, as observed around real clusters. These findings support the validity of the hypothesis of merging of GCs in the galactic central region, with modes that deserve further careful investigations.Comment: LaTeX 2e, AASTeX v5.x, 23 pages with 14 figures. Accepted for publication on the Astrophysical Journal. Final version with major change

An area law for entanglement from exponential decay of correlations

Area laws for entanglement in quantum many-body systems give useful information about their low-temperature behaviour and are tightly connected to the possibility of good numerical simulations. An intuition from quantum many-body physics suggests that an area law should hold whenever there is exponential decay of correlations in the system, a property found, for instance, in non-critical phases of matter. However, the existence of quantum data-hiding state--that is, states having very small correlations, yet a volume scaling of entanglement--was believed to be a serious obstruction to such an implication. Here we prove that notwithstanding the phenomenon of data hiding, one-dimensional quantum many-body states satisfying exponential decay of correlations always fulfil an area law. To obtain this result we combine several recent advances in quantum information theory, thus showing the usefulness of the field for addressing problems in other areas of physics.Comment: 8 pages, 3 figures. Short version of arXiv:1206.2947 Nature Physics (2013

Foundations for Behavioural Model Elaboration Using Modal Transition Systems

Modal Transition Systems (MTS) are an extension of Labelled Transition Systems (LTS) that have been shown to be useful to reason about system behaviour in the context of partial information. MTSs distinguish between required, proscribed and unknown behaviour and come equipped with a notion of refinement that supports incremental modelling where unknown behaviour is iteratively elaborated into required or proscribed behaviour. A particularly useful notion in the context of software and requirements engineering is that of “merge”. Merging two consistent models is a process that should result in a minimal common refinement of both models where consistency is defined as the existence of one common refinement. One of the current limitations of MTS merging is that a complete and correct algorithm for merging has not been developed. Hence, an engineer attempting to merge partial descriptions may be prevented to do so by overconstrained algorithms or algorithms that introduce behaviour that does not follow from the partial descriptions being merged. In this thesis we study the problems of consistency and merge for the existing MTSs semantics - strong and weak semantics - and provide a complete characterization of MTS consistency as well as a complete and correct algorithm for MTS merging using these semantics. Strong and weak semantics require MTS models to have the same communicating alphabet, the latter allowing the use of a distinguished unobservable action. In this work we show that the requirement of fixing the alphabet for MTS semantics and the treatment of observable actions are limiting if MTSs are to support incremental elaboration of partial behaviour models. We present a novel observational semantics for MTS, branching alphabet semantics, inspired by branching LTS equivalence, which supports the elaboration of model behaviour including the extension of the alphabet of the system to describe behaviour aspects that previously had not been taken into account. Furthermore, we show that some unintuitive refinements allowed by weak semantics are avoided, and prove a number of theorems that relate branching refinement with alphabet refinement and consistency. These theorems, which do not hold for other semantics, support the argument for considering branching alphabet as a sound semantics to support behaviour model elaboration

Dark matter self-interactions from a general spin-0 mediator

Dark matter particles interacting via the exchange of very light spin-0 mediators can have large self-interaction rates and obtain their relic abundance from thermal freeze-out. At the same time, these models face strong bounds from direct and indirect probes of dark matter as well as a number of constraints on the properties of the mediator. We investigate whether these constraints can be consistent with having observable effects from dark matter self-interactions in astrophysical systems. For the case of a mediator with purely scalar couplings we point out the highly relevant impact of low-threshold direct detection experiments like CRESST-II, which essentially rule out the simplest realization of this model. These constraints can be significantly relaxed if the mediator has CP-violating couplings, but then the model faces strong constraints from CMB measurements, which can only be avoided in special regions of parameter space.Comment: 20 pages, 5 figures + appendices. v2: Matches published version. v3: fixed typographical mistake in eq. (2.8). v4: fixed typographical mistake in eq. (C.4

Dark matter self-interactions from a general spin-0 mediator

Dark matter particles interacting via the exchange of very light spin-0 mediators can have large self-interaction rates and obtain their relic abundance from thermal freeze-out. At the same time, these models face strong bounds from direct and indirect probes of dark matter as well as a number of constraints on the properties of the mediator. We investigate whether these constraints can be consistent with having observable effects from dark matter self-interactions in astrophysical systems. For the case of a mediator with purely scalar couplings we point out the highly relevant impact of low-threshold direct detection experiments like CRESST-II, which essentially rule out the simplest realization of this model. These constraints can be significantly relaxed if the mediator has CP-violating couplings, but then the model faces strong constraints from CMB measurements, which can only be avoided in special regions of parameter space.Comment: 20 pages, 5 figures + appendices. v2: Matches published version. v3: fixed typographical mistake in eq. (2.8). v4: fixed typographical mistake in eq. (C.4