This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and f-divergence functionals such as Hellinger and Kullback-Leibler distances on the space of mixture distributions is investigated in detail using various identifiability conditions. Convergence in Wasserstein metrics for discrete measures implies convergence of individual atoms that provide support for the measures, thereby providing a natural interpretation of convergence of clusters in clustering applications where mixture models are typically employed. Convergence rates of posterior distributions for latent mixing measures are established, for both finite mixtures of multivariate distributions and infinite mixtures based on the Dirichlet process.

In experimental tests of human behavior in unstructured bargaining games, typically many joint utility outcomes are found to occur, not just one. This suggests we predict the outcome of such a game as a probability distribution. This is in contrast to what is conventionally done (e.g, in the Nash bargaining solution), which is predict a single outcome. We show how to translate Nash's bargaining axioms to provide a distribution over outcomes rather than a single outcome. We then prove that a subset of those axioms forces the distribution over utility outcomes to be a power-law distribution. Unlike Nash's original result, our result holds even if the feasible set is finite. When the feasible set is convex and comprehensive, the mode of the power law distribution is the Harsanyi bargaining solution, and if we require symmetry it is the Nash bargaining solution. However, in general these modes of the joint utility distribution are not the experimentalist's Bayes-optimal predictions for the joint utility. Nor are the bargains corresponding to the modes of those joint utility distributions the modes of the distribution over bargains in general, since more than one bargain may result in the same joint utility. After introducing distributional bargaining solution concepts, we show how an external regulator can use them to optimally design an unstructured bargaining scenario. Throughout we demonstrate our analysis in computational experiments involving flight rerouting negotiations in the National Airspace System. We emphasize that while our results are formulated for unstructured bargaining, they can also be used to make predictions for noncooperative games where the modeler knows the utility functions of the players over possible outcomes of the game, but does not know the move spaces the players use to determine those outcomes.

There has been much recent interest in application of the pool-adjacent-violators (PAV) algorithm for the purpose of calibrating the probabilistic outputs of automatic pattern recognition and machine learning algorithms. Special cost functions, known as proper scoring rules form natural objective functions to judge the goodness of such calibration. We show that for binary pattern classifiers, the non-parametric optimization of calibration, subject to a monotonicity constraint, can be solved by PAV and that this solution is optimal for all regular binary proper scoring rules. This extends previous results which were limited to convex binary proper scoring rules. We further show that this result holds not only for calibration of probabilities, but also for calibration of log-likelihood-ratios, in which case optimality holds independently of the prior probabilities of the pattern classes.

The Dirichlet process (DP) is a fundamental mathematical tool for Bayesian nonparametric modeling, and is widely used in tasks such as density estimation, natural language processing, and time series modeling. Although MCMC inference methods for the DP often provide a gold standard in terms asymptotic accuracy, they can be computationally expensive and are not obviously parallelizable. We propose a reparameterization of the Dirichlet process that induces conditional independencies between the atoms that form the random measure. This conditional independence enables many of the Markov chain transition operators for DP inference to be simulated in parallel across multiple cores. Applied to mixture modeling, our approach enables the Dirichlet process to simultaneously learn clusters that describe the data and superclusters that define the granularity of parallelization. Unlike previous approaches, our technique does not require alteration of the model and leaves the true posterior distribution invariant. It also naturally lends itself to a distributed software implementation in terms of Map-Reduce, which we test in cluster configurations of over 50 machines and 100 cores. We present experiments exploring the parallel efficiency and convergence properties of our approach on both synthetic and real-world data, including runs on 1MM data vectors in 256 dimensions.

The Swedish Institute of Computer Science (SICS) has for several years developed methods for statistical anomaly detection based on a framework called Bayesian principal anomaly (Holst and Ekman 2011). In this article we describe a novel application Addtrack is a tool developed originally by Bombardier domain for the anomaly-detection method: condition Transportation for general analysis, monitoring, monitoring of trains (Holst, Ekman, and and visualization of train conditions and Larsen 2006). It is "intelligent" in statistical models. There are currently many the sense that analysis modules, such as the one popular anomaly-detection methods based on described in this article, can be used to preprocess nonparametric models (see, for example, Ahmed, and visualize data sets. Addtrack, including the anomalydetection model is very general since the parametric module described in this article, is forms of the distributions need not be currently deployed in Sweden, India, China, and known.

Network models have been popular for modeling and representing complex relationships and dependencies between observed variables. When data comes from a dynamic stochastic process, a single static network model cannot adequately capture transient dependencies, such as, gene regulatory dependencies throughout a developmental cycle of an organism. Kolar et al (2010b) proposed a method based on kernel-smoothing l1-penalized logistic regression for estimating time-varying networks from nodal observations collected from a time-series of observational data. In this paper, we establish conditions under which the proposed method consistently recovers the structure of a time-varying network. This work complements previous empirical findings by providing sound theoretical guarantees for the proposed estimation procedure. For completeness, we include numerical simulations in the paper.

In this paper we propose a model with a Dirichlet process mixture of gamma densities in the bulk part below threshold and a generalized Pareto density in the tail for extreme value estimation. The proposed model is simple and flexible allowing us posterior density estimation and posterior inference for high quantiles. The model works well even for small sample sizes and in the absence of prior information. We evaluate the performance of the proposed model through a simulation study. Finally, the proposed model is applied to a real environmental data.

The causal inference literature has provided a clear formal definition of confounding expressed in terms of counterfactual independence. The literature has not, however, come to any consensus on a formal definition of a confounder, as it has given priority to the concept of confounding over that of a confounder. We consider a number of candidate definitions arising from various more informal statements made in the literature. We consider the properties satisfied by each candidate definition, principally focusing on (i) whether under the candidate definition control for all "confounders" suffices to control for "confounding" and (ii) whether each confounder in some context helps eliminate or reduce confounding bias. Several of the candidate definitions do not have these two properties. Only one candidate definition of those considered satisfies both properties. We propose that a "confounder" be defined as a pre-exposure covariate C for which there exists a set of other covariates X such that effect of the exposure on the outcome is unconfounded conditional on (X,C) but such that for no proper subset of (X,C) is the effect of the exposure on the outcome unconfounded given the subset. We also provide a conditional analogue of the above definition; and we propose a variable that helps reduce bias but not eliminate bias be referred to as a "surrogate confounder." These definitions are closely related to those given by Robins and Morgenstern [Comput. Math. Appl. 14 (1987) 869-916]. The implications that hold among the various candidate definitions are discussed.

Many data sets contain rich information about objects, as well as pairwise relations between them. For instance, in networks of websites, scientific papers, and other documents, each node has content consisting of a collection of words, as well as hyperlinks or citations to other nodes. In order to perform inference on such data sets, and make predictions and recommendations, it is useful to have models that are able to capture the processes which generate the text at each node and the links between them. In this paper, we combine classic ideas in topic modeling with a variant of the mixed-membership block model recently developed in the statistical physics community. The resulting model has the advantage that its parameters, including the mixture of topics of each document and the resulting overlapping communities, can be inferred with a simple and scalable expectation-maximization algorithm. We test our model on three data sets, performing unsupervised topic classification and link prediction. For both tasks, our model outperforms several existing state-of-the-art methods, achieving higher accuracy with significantly less computation, analyzing a data set with 1.3 million words and 44 thousand links in a few minutes.

After experimenting with a number of non-probabilistic methods for dealing with uncertainty many researchers reaffirm a preference for probability methods [1] [2], although this remains controversial. The importance of being able to form decisions from incomplete data in diagnostic problems has highlighted probabilistic methods [5] which compute posterior probabilities from prior distributions in a way similar to Bayes Rule, and thus are called Bayesian methods. This paper documents the use of a Bayesian method in a real time problem which is similar to medical diagnosis in that there is a need to form decisions and take some action without complete knowledge of conditions in the problem domain. This particular method has a limitation which is discussed.