Parton distributions: determining probabilities in a space of functions

We discuss the statistical properties of parton distributions within the framework of the NNPDF methodology. We present various tests of statistical consistency, in particular that the distribution of results does not depend on the underlying parametrization and that it behaves according to Bayes' theorem upon the addition of new data. We then study the dependence of results on consistent or inconsistent datasets and present tools to assess the consistency of new data. Finally we estimate the relative size of the PDF uncertainty due to data uncertainties, and that due to the need to infer a functional form from a finite set of data.Comment: 11 pages, 8 figures, presented by Stefano Forte at PHYSTAT 2011 (to be published in the proceedings

Differential Calculus on Graphon Space

Recently, the theory of dense graph limits has received attention from multiple disciplines including graph theory, computer science, statistical physics, probability, statistics, and group theory. In this paper we initiate the study of the general structure of differentiable graphon parameters $F$. We derive consistency conditions among the higher G\^ateaux derivatives of $F$ when restricted to the subspace of edge weighted graphs $\mathcal{W}_{\bf p}$. Surprisingly, these constraints are rigid enough to imply that the multilinear functionals $\Lambda: \mathcal{W}_{\bf p}^n \to \mathbb{R}$ satisfying the constraints are determined by a finite set of constants indexed by isomorphism classes of multigraphs with $n$ edges and no isolated vertices. Using this structure theory, we explain the central role that homomorphism densities play in the analysis of graphons, by way of a new combinatorial interpretation of their derivatives. In particular, homomorphism densities serve as the monomials in a polynomial algebra that can be used to approximate differential graphon parameters as Taylor polynomials. These ideas are summarized by our main theorem, which asserts that homomorphism densities $t(H,-)$ where $H$ has at most $N$ edges form a basis for the space of smooth graphon parameters whose $(N+1)$st derivatives vanish. As a consequence of this theory, we also extend and derive new proofs of linear independence of multigraph homomorphism densities, and characterize homomorphism densities. In addition, we develop a theory of series expansions, including Taylor's theorem for graph parameters and a uniqueness principle for series. We use this theory to analyze questions raised by Lov\'asz, including studying infinite quantum algebras and the connection between right- and left-homomorphism densities.Comment: Final version (36 pages), accepted for publication in Journal of Combinatorial Theory, Series

Inferential models: A framework for prior-free posterior probabilistic inference

Posterior probabilistic statistical inference without priors is an important but so far elusive goal. Fisher's fiducial inference, Dempster-Shafer theory of belief functions, and Bayesian inference with default priors are attempts to achieve this goal but, to date, none has given a completely satisfactory picture. This paper presents a new framework for probabilistic inference, based on inferential models (IMs), which not only provides data-dependent probabilistic measures of uncertainty about the unknown parameter, but does so with an automatic long-run frequency calibration property. The key to this new approach is the identification of an unobservable auxiliary variable associated with observable data and unknown parameter, and the prediction of this auxiliary variable with a random set before conditioning on data. Here we present a three-step IM construction, and prove a frequency-calibration property of the IM's belief function under mild conditions. A corresponding optimality theory is developed, which helps to resolve the non-uniqueness issue. Several examples are presented to illustrate this new approach.Comment: 29 pages with 3 figures. Main text is the same as the published version. Appendix B is an addition, not in the published version, that contains some corrections and extensions of two of the main theorem

Towards a more realistic sink particle algorithm for the RAMSES code

We present a new sink particle algorithm developed for the Adaptive Mesh Refinement code RAMSES. Our main addition is the use of a clump finder to identify density peaks and their associated regions (the peak patches). This allows us to unambiguously define a discrete set of dense molecular cores as potential sites for sink particle formation. Furthermore, we develop a new scheme to decide if the gas in which a sink could potentially form, is indeed gravitationally bound and rapidly collapsing. This is achieved using a general integral form of the virial theorem, where we use the curvature in the gravitational potential to correctly account for the background potential. We detail all the necessary steps to follow the evolution of sink particles in turbulent molecular cloud simulations, such as sink production, their trajectory integration, sink merging and finally the gas accretion rate onto an existing sink. We compare our new recipe for sink formation to other popular implementations. Statistical properties such as the sink mass function, the average sink mass and the sink multiplicity function are used to evaluate the impact that our new scheme has on accurately predicting fundamental quantities such as the stellar initial mass function or the stellar multiplicity function.Comment: submitted to MNRAS, 24 pages, 19 figures, 5 table