The paper introduces mixed networks, a new framework for expressing and reasoning with probabilistic and deterministic information. The framework combines belief networks with constraint networks, defining the semantics and graphical representation. We also introduce the AND/OR search space for graphical models, and develop a new linear space search algorithm. This provides the basis for understanding the benefits of processing the constraint information separately, resulting in the pruning of the search space. When the constraint part is tractable or has a small number of solutions, using the mixed representation can be exponentially more effective than using pure belief networks which model constraints as conditional probability tables.

Sparse linear (or generalized linear) models combine a standard likelihood function with a sparse prior on the unknown coefficients. These priors can conveniently be expressed as a maximization over zero-mean Gaussians with different variance hyperparameters. Standard MAP estimation (Type I) involves maximizing over both the hyperparameters and coefficients, while an empirical Bayesian alternative (Type II) first marginalizes the coefficients and then maximizes over the hyperparameters, leading to a tractable posterior approximation. The underlying cost functions can be related via a dual-space framework from Wipf et al. (2011), which allows both the Type I or Type II objectives to be expressed in either coefficient or hyperparmeter space. This perspective is useful because some analyses or extensions are more conducive to development in one space or the other. Herein we consider the estimation of a trade-off parameter balancing sparsity and data fit. As this parameter is effectively a variance, natural estimators exist by assessing the problem in hyperparameter (variance) space, transitioning natural ideas from Type II to solve what is much less intuitive for Type I. In contrast, for analyses of update rules and sparsity properties of local and global solutions, as well as extensions to more general likelihood models, we can leverage coefficient-space techniques developed for Type I and apply them to Type II. For example, this allows us to prove that Type II-inspired techniques can be successful recovering sparse coefficients when unfavorable restricted isometry properties (RIP) lead to failure of popular L1 reconstructions. It also facilitates the analysis of Type II when non-Gaussian likelihood models lead to intractable integrations.

In many applications that require matrix solutions of minimal rank, the underlying cost function is non-convex leading to an intractable, NP-hard optimization problem. Consequently, the convex nuclear norm is frequently used as a surrogate penalty term for matrix rank. The problem is that in many practical scenarios there is no longer any guarantee that we can correctly estimate generative low-rank matrices of interest, theoretical special cases notwithstanding. Consequently, this paper proposes an alternative empirical Bayesian procedure build upon a variational approximation that, unlike the nuclear norm, retains the same globally minimizing point estimate as the rank function under many useful constraints. However, locally minimizing solutions are largely smoothed away via marginalization, allowing the algorithm to succeed when standard convex relaxations completely fail. While the proposed methodology is generally applicable to a wide range of low-rank applications, we focus our attention on the robust principal component analysis problem (RPCA), which involves estimating an unknown low-rank matrix with unknown sparse corruptions. Theoretical and empirical evidence are presented to show that our method is potentially superior to related MAP-based approaches, for which the convex principle component pursuit (PCP) algorithm (Candes et al., 2011) can be viewed as a special case.

This article describes different models based on Bayesian networks RB modeling expertise in the diagnosis of brain tumors. Indeed, they are well adapted to the representation of the uncertainty in the process of diagnosis of these tumors. In our work, we first tested several structures derived from the Bayesian network reasoning performed by doctors on the one hand and structures generated automatically on the other. This step aims to find the best structure that increases diagnostic accuracy. The machine learning algorithms relate MWST-EM algorithms, SEM and SEM + T. To estimate the parameters of the Bayesian network from a database incomplete, we have proposed an extension of the EM algorithm by adding a priori knowledge in the form of the thresholds calculated by the first phase of the algorithm RBE . The very encouraging results obtained are discussed at the end of the paper

Prediction markets are used in real life to predict outcomes of interest such as presidential elections. This paper presents a mathematical theory of artificial prediction markets for supervised learning of conditional probability estimators. The artificial prediction market is a novel method for fusing the prediction information of features or trained classifiers, where the fusion result is the contract price on the possible outcomes. The market can be trained online by updating the participants' budgets using training examples. Inspired by the real prediction markets, the equations that govern the market are derived from simple and reasonable assumptions. Efficient numerical algorithms are presented for solving these equations. The obtained artificial prediction market is shown to be a maximum likelihood estimator. It generalizes linear aggregation, existent in boosting and random forest, as well as logistic regression and some kernel methods. Furthermore, the market mechanism allows the aggregation of specialized classifiers that participate only on specific instances. Experimental comparisons show that the artificial prediction markets often outperform random forest and implicit online learning on synthetic data and real UCI datasets. Moreover, an extensive evaluation for pelvic and abdominal lymph node detection in CT data shows that the prediction market improves adaboost's detection rate from 79.6% to 81.2% at 3 false positives/volume.

One of the basic tasks for Bayesian networks (BNs) is that of learning a network structure from data. The BN-learning problem is NP-hard, so the standard solution is heuristic search. Many approaches have been proposed for this task, but only a very small number outperform the baseline of greedy hill-climbing with tabu lists; moreover, many of the proposed algorithms are quite complex and hard to implement. In this paper, we propose a very simple and easy-to-implement method for addressing this task. Our approach is based on the well-known fact that the best network (of bounded in-degree) consistent with a given node ordering can be found very efficiently. We therefore propose a search not over the space of structures, but over the space of orderings, selecting for each ordering the best network consistent with it. This search space is much smaller, makes more global search steps, has a lower branching factor, and avoids costly acyclicity checks. We present results for this algorithm on both synthetic and real data sets, evaluating both the score of the network found and in the running time. We show that ordering-based search outperforms the standard baseline, and is competitive with recent algorithms that are much harder to implement.

We propose a multi-wing harmonium model for mining multimedia data that extends and improves on earlier models based on two-layer random fields, which capture bidirectional dependencies between hidden topic aspects and observed inputs. This model can be viewed as an undirected counterpart of the two-layer directed models such as LDA for similar tasks, but bears significant difference in inference/learning cost tradeoffs, latent topic representations, and topic mixing mechanisms. In particular, our model facilitates efficient inference and robust topic mixing, and potentially provides high flexibilities in modeling the latent topic spaces. A contrastive divergence and a variational algorithm are derived for learning. We specialized our model to a dual-wing harmonium for captioned images, incorporating a multivariate Poisson for word-counts and a multivariate Gaussian for color histogram. We present empirical results on the applications of this model to classification, retrieval and image annotation on news video collections, and we report an extensive comparison with various extant models.

We introduce a novel latent grouping model for predicting the relevance of a new document to a user. The model assumes a latent group structure for both users and documents. We compared the model against a state-of-the-art method, the User Rating Profile model, where only users have a latent group structure. We estimate both models by Gibbs sampling. The new method predicts relevance more accurately for new documents that have few known ratings. The reason is that generalization over documents then becomes necessary and hence the twoway grouping is profitable.

In this paper, we present an algorithm for minimizing the difference between two submodular functions using a variational framework which is based on (an extension of) the concave-convex procedure [17]. Because several commonly used metrics in machine learning, like mutual information and conditional mutual information, are submodular, the problem of minimizing the difference of two submodular problems arises naturally in many machine learning applications. Two such applications are learning discriminatively structured graphical models and feature selection under computational complexity constraints. A commonly used metric for measuring discriminative capacity is the EAR measure which is the difference between two conditional mutual information terms. Feature selection taking complexity considerations into account also fall into this framework because both the information that a set of features provide and the cost of computing and using the features can be modeled as submodular functions. This problem is NP-hard, and we give a polynomial time heuristic for it. We also present results on synthetic data to show that classifiers based on discriminative graphical models using this algorithm can significantly outperform classifiers based on generative graphical models.

Boosted decision trees typically yield good accuracy, precision, and ROC area. However, because the outputs from boosting are not well calibrated posterior probabilities, boosting yields poor squared error and cross-entropy. We empirically demonstrate why AdaBoost predicts distorted probabilities and examine three calibration methods for correcting this distortion: Platt Scaling, Isotonic Regression, and Logistic Correction. We also experiment with boosting using log-loss instead of the usual exponential loss. Experiments show that Logistic Correction and boosting with log-loss work well when boosting weak models such as decision stumps, but yield poor performance when boosting more complex models such as full decision trees. Platt Scaling and Isotonic Regression, however, significantly improve the probabilities predicted by