Efficient Bayesian Inference for Generalized Bradley-Terry Models

The Bradley-Terry model is a popular approach to describe probabilities of the possible outcomes when elements of a set are repeatedly compared with one another in pairs. It has found many applications including animal behaviour, chess ranking and multiclass classification. Numerous extensions of the basic model have also been proposed in the literature including models with ties, multiple comparisons, group comparisons and random graphs. From a computational point of view, Hunter (2004) has proposed efficient iterative MM (minorization-maximization) algorithms to perform maximum likelihood estimation for these generalized Bradley-Terry models whereas Bayesian inference is typically performed using MCMC (Markov chain Monte Carlo) algorithms based on tailored Metropolis-Hastings (M-H) proposals. We show here that these MM\ algorithms can be reinterpreted as special instances of Expectation-Maximization (EM) algorithms associated to suitable sets of latent variables and propose some original extensions. These latent variables allow us to derive simple Gibbs samplers for Bayesian inference. We demonstrate experimentally the efficiency of these algorithms on a variety of applications

Bayesian nonparametric models for ranked data

We develop a Bayesian nonparametric extension of the popular Plackett-Luce choice model that can handle an infinite number of choice items. Our framework is based on the theory of random atomic measures, with the prior specified by a gamma process. We derive a posterior characterization and a simple and effective Gibbs sampler for posterior simulation. We develop a time-varying extension of our model, and apply it to the New York Times lists of weekly bestselling books.Comment: NIPS - Neural Information Processing Systems (2012

Scalable Bayesian nonparametric regression via a Plackett-Luce model for conditional ranks

We present a novel Bayesian nonparametric regression model for covariates X and continuous, real response variable Y. The model is parametrized in terms of marginal distributions for Y and X and a regression function which tunes the stochastic ordering of the conditional distributions F(y|x). By adopting an approximate composite likelihood approach, we show that the resulting posterior inference can be decoupled for the separate components of the model. This procedure can scale to very large datasets and allows for the use of standard, existing, software from Bayesian nonparametric density estimation and Plackett-Luce ranking estimation to be applied. As an illustration, we show an application of our approach to a US Census dataset, with over 1,300,000 data points and more than 100 covariates

A Hierarchical Bayesian Framework for Constructing Sparsity-inducing Priors

Variable selection techniques have become increasingly popular amongst statisticians due to an increased number of regression and classification applications involving high-dimensional data where we expect some predictors to be unimportant. In this context, Bayesian variable selection techniques involving Markov chain Monte Carlo exploration of the posterior distribution over models can be prohibitively computationally expensive and so there has been attention paid to quasi-Bayesian approaches such as maximum a posteriori (MAP) estimation using priors that induce sparsity in such estimates. We focus on this latter approach, expanding on the hierarchies proposed to date to provide a Bayesian interpretation and generalization of state-of-the-art penalized optimization approaches and providing simultaneously a natural way to include prior information about parameters within this framework. We give examples of how to use this hierarchy to compute MAP estimates for linear and logistic regression as well as sparse precision-matrix estimates in Gaussian graphical models. In addition, an adaptive group lasso method is derived using the framework.Comment: Submitted for publication; corrected typo

The Ties that Bind: Kymlicka and the Problem of Political Unity in Multination States

As asserted by Will Kymlicka, the recognition and accommodation of national minorities leads to a dilemma. Indeed, if denying them these rights can contribute to their willingness to secede, allowing them to self-govern can also ultimately lead to the weakening of their ties with the state in which they are integrated. This tension well described in Kymlicka’s Multicultural Citizenship and in his later works remains nonetheless without an explicit solution. This text addresses this question by suggesting that the dialogical dynamic behind the recognition and accommodation of national minorities hides a purely political patriotism stemming from the neo-republican tradition that is complementary to the nationalist sense of attachment that members of national minorities will inevitably come to feel toward their societal culture

Modelling individual migration patterns using a Bayesian nonparametric approach for capture-recapture data

We present a Bayesian nonparametric approach for modelling wildlife migration patterns using capture–recapture (CR) data. Arrival times of individuals are modelled in continuous time and assumed to be drawn from a Poisson process with unknown intensity function, which is modelled via a flexible nonparametric mixture model. The proposed CR framework allows us to estimate the following: (i) the total number of individuals that arrived at the site, (ii) their times of arrival and departure, and hence their stopover duration, and (iii) the density of arrival times, providing a smooth representation of the arrival pattern of the individuals at the site. We apply the model to data on breeding great crested newts (Triturus cristatus) and on migrating reed warblers (Acrocephalus scirpaceus). For the former, the results demonstrate the staggered arrival of individuals at the breeding ponds and suggest that males tend to arrive earlier than females. For the latter, they demonstrate the arrival of migrating flocks at the stopover site and highlight the considerable difference in stopover duration between caught and not-caught individuals

Bayesian nonparametrics for Sparse Dynamic Networks

We propose a Bayesian nonparametric prior for time-varying networks. To each node of the network is associated a positive parameter, modeling the sociability of that node. Sociabilities are assumed to evolve over time, and are modeled via a dynamic point process model. The model is able to (a) capture smooth evolution of the interaction between nodes, allowing edges to appear/disappear over time (b) capture long term evolution of the sociabilities of the nodes (c) and yield sparse graphs, where the number of edges grows subquadratically with the number of nodes. The evolution of the sociabilities is described by a tractable time-varying gamma process. We provide some theoretical insights into the model and apply it to three real world datasets.Comment: 10 pages, 8 figure

Classification

National audienceLa classification a pour objet de regrouper des données en classes possédant des caractéristiques similaires. La classification peut être supervisée lorsque l'on dispose d'un ensemble d'apprentissage labellisé, semi-supervisée ou non supervisée. Elle apparaît dans de nombreuses applications telles que la fouille de texte, la reconnaissance vocale ou l'analyse de données génomiques. L'objectif de cette session est d'offrir un panorama des approches statistiques pour la classification de données (modèles de mélange, SVM, processus de Dirichlet, etc.) et d'en présenter diverses applications