Abstract not found

Consider the problem of joint parameter estimation and prediction in a Markov random field: that is, the model parameters are estimated on the basis of an initial set of data, and then the fitted model is used to perform prediction (e.g., smoothing, denoising, interpolation) on a new noisy observation.

I propose a common framework that combines three different paradigms in machine learning: generative, discriminative and imitative learning. A generative probabilistic distribution is a principled way to model many machine learning and machine perception problems. Therein, one provides domain specific knowledge in terms of structure and parameter priors over the joint space of variables. Bayesian networks and Bayesian statistics provide a rich and flexible language for specifying this knowledge and subsequently refining it with data and observations. The final result is a distribution that is a good generator of novel exemplars.

The Group-Lasso method for finding important explanatory factors suffers from the potential non-uniqueness of solutions and also from high computational costs. We formulate conditions for the uniqueness of Group-Lasso solutions which lead to an easily implementable test procedure that allows us to identify all potentially active groups. These results are used to derive an efficient algorithm that can deal with input dimensions in the millions and can approximate the solution path efficiently. The derived methods are applied to large-scale learning problems where they exhibit excellent performance and where the testing procedure helps to avoid misinterpretations of the solutions. 1.

Curved exponential family models are a useful generalization of exponential random graph models (ERGMs). In particular, models involving the alternating k-star, alternating k-triangle, and alternating ktwopath statistics of Snijders et al. [Snijders, T.A.B., Pattison, P.E., Robins, G.L., Handcock, M.S., in press. New specifications for exponential random graph models. Sociological Methodology] may be viewed as curved exponential family models. This article unifies recent material in the literature regarding curved exponential family models for networks in general and models involving these alternating statistics in particular. It also discusses the intuition behind rewriting the three alternating statistics in terms of the degree distribution and the recently introduced shared partner distributions. This intuition suggests a redefinition of the alternating k-star statistic. Finally, this article demonstrates the use of the statnet package in R for fitting models of this sort, comparing new results on an oft-studied network dataset with results found in the literature.

Abstract. We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions. Key words and phrases: Gibbs sampler, running time analyses, exponential families, conjugate priors, location families, orthogonal polynomials, singular value decomposition. 1.

divergence

Abstract not found

Abstract: Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the parameter space of a reference model with nice properties, such as for example a regular exponential family. This observation leads to the definition of an ‘algebraic exponential family’. This new definition provides a unified framework for the study of statistical models with algebraic structure. In this paper we review the ingredients to this definition and illustrate in examples how computational algebraic geometry can be used to solve problems arising in statistical inference in algebraic models. Key words and phrases: Algebraic statistics, computational algebraic geometry, exponential family, maximum likelihood estimation, model invariants, singularities. 1.

Underlying a variety of techniques for approximate inference---among them mean field, sum-product, and cluster variational methods---is a classical variational principle from statistical physics, which involves a "free energy" optimization problem over the set of all distributions. Working within the framework of exponential families, we describe an alternative view, in which the optimization takes place over the (typically) much lowerdimensional space of mean parameters. The associated constraint set consists of all mean parameters that are globally realizable; for discrete random variables, we refer to this set as a marginal polytope. As opposed to the classical formulation, the representation given here clarifies that there are two distinct components to variational inference algorithms: (a) an approximation to the entropy function; and (b) an approximation to the marginal polytope.