Last summer, I was at a conference having lunch with Hal Daume III when we got to talking about how "Bayesian" can be a funny and ambiguous term. It seems like the definition should be straightforward: "following the work of English mathematician Rev. Thomas Bayes," perhaps, or even "uses Bayes' theorem." But many methods bearing the reverend's name or using his theorem aren't even considered "Bayesian" by his most religious followers. Why is it that Bayesian networks, for example, aren't considered… y'know… Bayesian? As I've read more outside the fields of machine learning and natural language processing -- from psychometrics and environmental biology to hackers who dabble in data science -- I've noticed three distinct uses of the term "Bayesian."

Sum-Product Networks (SPNs) are recently introduced deep tractable probabilistic models by which several kinds of inference queries can be answered exactly and in a tractable time. Up to now, they have been largely used as black box density estimators, assessed only by comparing their likelihood scores only. In this paper we explore and exploit the inner representations learned by SPNs. We do this with a threefold aim: first we want to get a better understanding of the inner workings of SPNs; secondly, we seek additional ways to evaluate one SPN model and compare it against other probabilistic models, providing diagnostic tools to practitioners; lastly, we want to empirically evaluate how good and meaningful the extracted representations are, as in a classic Representation Learning framework. In order to do so we revise their interpretation as deep neural networks and we propose to exploit several visualization techniques on their node activations and network outputs under different types of inference queries. To investigate these models as feature extractors, we plug some SPNs, learned in a greedy unsupervised fashion on image datasets, in supervised classification learning tasks. We extract several embedding types from node activations by filtering nodes by their type, by their associated feature abstraction level and by their scope. In a thorough empirical comparison we prove them to be competitive against those generated from popular feature extractors as Restricted Boltzmann Machines. Finally, we investigate embeddings generated from random probabilistic marginal queries as means to compare other tractable probabilistic models on a common ground, extending our experiments to Mixtures of Trees.

This paper introduces a novel technique to track structures in time varying graphs. The method uses a maximum a posteriori approach for adjusting a three-dimensional co-clustering of the source vertices, the destination vertices and the time, to the data under study, in a way that does not require any hyper-parameter tuning. The three dimensions are simultaneously segmented in order to build clusters of source vertices, destination vertices and time segments where the edge distributions across clusters of vertices follow the same evolution over the time segments. The main novelty of this approach lies in that the time segments are directly inferred from the evolution of the edge distribution between the vertices, thus not requiring the user to make any a priori quantization. Experiments conducted on artificial data illustrate the good behavior of the technique, and a study of a real-life data set shows the potential of the proposed approach for exploratory data analysis.

Graphs are a common data modality, useful for modeling complex relationships between objects, with applications spanning fields as varied as biology (Jeong et al., 2001; Bullmore and Sporns, 2009), sociology (Wasserman and Faust, 1994), and computer vision (Foggia et al., 2014; Kandel et al., 2007), to name a few. For example, in neuroscience, vertices may be neurons and edges adjoin pairs of neurons that share a synapse (Bullmore and Sporns, 2009); in social networks, vertices may correspond to people and edges to friendships between them (Carrington et al., 2005; Yang and Leskovec, 2015); in computer vision, vertices may represent pixels in an image and edges may represent spatial proximity or multi-resolution mappings (Kandel et al., 2007). In many useful networks, vertices with similar attributes form densely-connected communities compared to vertices with highly disparate attributes, and uncovering these communities is an important step in understanding the structure of the network. There is an extensive literature devoted to uncovering this community structure in network data, including methods based on maximum modularity (Newman and Girvan, 2004; Newman, 2006b), spectral partitioning algorithms (Luxburg, 2007; Rohe et al., 2011; Sussman et al., 2012; Lyzinski et al., 2014b), and likelihood-based methods (Bickel and Chen, 2009), among others. In the setting of vertex nomination, one community in the network is of particular interest, and the inference task is to order the vertices into a nomination list with those vertices from the community of interest concentrating at the top of the list.

Expectation Maximization (EM) is among the most popular algorithms for estimating parameters of statistical models. However, EM, which is an iterative algorithm based on the maximum likelihood principle, is generally only guaranteed to find stationary points of the likelihood objective, and these points may be far from any maximizer. This article addresses this disconnect between the statistical principles behind EM and its algorithmic properties. Specifically, it provides a global analysis of EM for specific models in which the observations comprise an i.i.d. sample from a mixture of two Gaussians. This is achieved by (i) studying the sequence of parameters from idealized execution of EM in the infinite sample limit, and fully characterizing the limit points of the sequence in terms of the initial parameters; and then (ii) based on this convergence analysis, establishing statistical consistency (or lack thereof) for the actual sequence of parameters produced by EM.

We present a unified approach for learning the parameters of Sum-Product networks (SPNs). We prove that any complete and decomposable SPN is equivalent to a mixture of trees where each tree corresponds to a product of univariate distributions. Based on the mixture model perspective, we characterize the objective function when learning SPNs based on the maximum likelihood estimation (MLE) principle and show that the optimization problem can be formulated as a signomial program. We construct two parameter learning algorithms for SPNs by using sequential monomial approximations (SMA) and the concave-convex procedure (CCCP), respectively. The two proposed methods naturally admit multiplicative updates, hence effectively avoiding the projection operation. With the help of the unified framework, we also show that, in the case of SPNs, CCCP leads to the same algorithm as Expectation Maximization (EM) despite the fact that they are different in general.

We derive fundamental sample complexity bounds for recovering sparse and structured signals for linear and nonlinear observation models including sparse regression, group testing, multivariate regression and problems with missing features. In general, sparse signal processing problems can be characterized in terms of the following Markovian property. We are given a set of $N$ variables $X_1,X_2,\ldots,X_N$, and there is an unknown subset of variables $S \subset \{1,\ldots,N\}$ that are relevant for predicting outcomes $Y$. More specifically, when $Y$ is conditioned on $\{X_n\}_{n\in S}$ it is conditionally independent of the other variables, $\{X_n\}_{n \not \in S}$. Our goal is to identify the set $S$ from samples of the variables $X$ and the associated outcomes $Y$. We characterize this problem as a version of the noisy channel coding problem. Using asymptotic information theoretic analyses, we establish mutual information formulas that provide sufficient and necessary conditions on the number of samples required to successfully recover the salient variables. These mutual information expressions unify conditions for both linear and nonlinear observations. We then compute sample complexity bounds for the aforementioned models, based on the mutual information expressions in order to demonstrate the applicability and flexibility of our results in general sparse signal processing models.

High costs of treatment, long waiting time, affordability of airfares to overseas destinations and favorable exchange rate change are crucial factors related to the fast growth of Medical Tourism (Connell, 2006). Rapid development of medical infrastructure with international standards and certification, easy availability of skilled manpower bring South Asian countries like Thailand, Malaysia, and India at the forefront in this area. With current annual growth of 13.0 percent, the Indian health care sector contributes about $ 23 billion (nearly 4 percent of GDP) to the Indian economy, with'foreign exchange earning around $1.8 billion' (Chakraborty, 2006). Although research studies are abundant focusing on social impacts of Medical Tourism, there is no proper methodology for customers, both foreign and domestic, to assess the medical tourist destination in any country. The problem can be solved by taking the interest of stakeholder's in assessing the weights of a multiple criteria set, namely medical infrastructure, logistics service providers, 1 government policy along with city demography. Therefore, assessment of desirable medical destination selection and evaluation problem can be considered decision making problem with multiple attributes varying from consumer demands to resource constraints of medical related industry. In this regard, MCDM has become a very crucial area of management research and decision theory with lots of methods developed, extended and modified in solving problems in the present and past few decades.

The aim of this paper is to provide some theoretical understanding of Bayesian non-negative matrix factorization methods. We derive an oracle inequality for a quasi-Bayesian estimator. This result holds for a very general class of prior distributions and shows how the prior affects the rate of convergence. We illustrate our theoretical results with a short numerical study along with a discussion on existing implementations.

We introduce a new criterion to determine the order of an autoregressive model fitted to time series data. It has the benefits of the two well-known model selection techniques, the Akaike information criterion and the Bayesian information criterion. When the data is generated from a finite order autoregression, the Bayesian information criterion is known to be consistent, and so is the new criterion. When the true order is infinity or suitably high with respect to the sample size, the Akaike information criterion is known to be efficient in the sense that its prediction performance is asymptotically equivalent to the best offered by the candidate models; in this case, the new criterion behaves in a similar manner. Different from the two classical criteria, the proposed criterion adaptively achieves either consistency or efficiency depending on the underlying true model. In practice where the observed time series is given without any prior information about the model specification, the proposed order selection criterion is more flexible and robust compared with classical approaches. Numerical results are presented demonstrating the adaptivity of the proposed technique when applied to various datasets.