A difficult task in modeling with Bayesian networks is the elicitation of numerical parameters of Bayesian networks. A large number of parameters is needed to specify a conditional probability table (CPT) that has a larger parent set. In this paper we show that, most CPTs from real applications of Bayesian networks can actually be very well approximated by tables that require substantially less parameters. This observation has practical consequence not only for model elicitation but also for efficient probabilistic reasoning with these networks.

Variational inference is a powerful concept that underlies many iterative approximation algorithms; expectation propagation, mean-field methods and belief propagations were all central themes at the school that can be perceived from this unifying framework. The lectures of Manfred Opper introduce the archetypal example of Expectation Propagation, before establishing the connection with the other approximation methods. Corrections by expansion about the expectation propagation are then explained. Finally some advanced inference topics and applications are explored in the final sections.

A new nonparametric approach for system identification has been recently proposed where the impulse response is seen as the realization of a zero--mean Gaussian process whose covariance, the so--called stable spline kernel, guarantees that the impulse response is almost surely stable. Maximum entropy properties of the stable spline kernel have been pointed out in the literature. In this paper we provide an independent proof that relies on the theory of matrix extension problems in the graphical model literature and leads to a closed form expression for the inverse of the first order stable spline kernel as well as to a new factorization in the form $UWU^\top$ with $U$ upper triangular and $W$ diagonal. Interestingly, all first--order stable spline kernels share the same factor $U$ and $W$ admits a closed form representation in terms of the kernel hyperparameter, making the factorization computationally inexpensive. Maximum likelihood properties of the stable spline kernel are also highlighted. These results can be applied both to improve the stability and to reduce the computational complexity associated with the computation of stable spline estimators.

Gibbs sampling is a workhorse for Bayesian inference but has several limitations when used for parameter estimation, and is often much slower than non-sampling inference methods. SAME (State Augmentation for Marginal Estimation) \cite{Doucet99,Doucet02} is an approach to MAP parameter estimation which gives improved parameter estimates over direct Gibbs sampling. SAME can be viewed as cooling the posterior parameter distribution and allows annealed search for the MAP parameters, often yielding very high quality (lower loss) estimates. But it does so at the expense of additional samples per iteration and generally slower performance. On the other hand, SAME dramatically increases the parallelism in the sampling schedule, and is an excellent match for modern (SIMD) hardware. In this paper we explore the application of SAME to graphical model inference on modern hardware. We show that combining SAME with factored sample representation (or approximation) gives throughput competitive with the fastest symbolic methods, but with potentially better quality. We describe experiments on Latent Dirichlet Allocation, achieving speeds similar to the fastest reported methods (online Variational Bayes) and lower cross-validated loss than other LDA implementations. The method is simple to implement and should be applicable to many other models.

In this study, we enrich the framework of nonparametric kernel Bayesian inference via the flexible incorporation of certain probabilistic models, such as additive Gaussian noise models. Nonparametric inference expressed in terms of kernel means, which is called kernel Bayesian inference, has been studied using basic rules such as the kernel sum rule (KSR), kernel chain rule, kernel product rule, and kernel Bayes' rule (KBR). However, the current framework used for kernel Bayesian inference deals only with nonparametric inference and it cannot allow inference when combined with probabilistic models. In this study, we introduce a novel KSR, called model-based KSR (Mb-KSR), which exploits the knowledge obtained from some probabilistic models of conditional distributions. The incorporation of Mb-KSR into nonparametric kernel Bayesian inference facilitates more flexible kernel Bayesian inference than nonparametric inference. We focus on combinations of Mb-KSR, Non-KSR, and KBR, and we propose a filtering algorithm for state space models, which combines nonparametric learning of the observation process using kernel means and additive Gaussian noise models of the transition dynamics. The idea of the Mb-KSR for additive Gaussian noise models can be extended to more general noise model cases, including a conjugate pair with a positive-definite kernel and a probabilistic model.

Automatic anomaly detection is a major issue in various areas. Beyond mere detection, the identification of the origin of the problem that produced the anomaly is also essential. This paper introduces a general methodology that can assist human operators who aim at classifying monitoring signals. The main idea is to leverage expert knowledge by generating a very large number of indicators. A feature selection method is used to keep only the most discriminant indicators which are used as inputs of a Naive Bayes classifier. The parameters of the classifier have been optimized indirectly by the selection process. Simulated data designed to reproduce some of the anomaly types observed in real world engines.

Abstract-- Automatic anomaly detection is a major issue in various areas. Beyond mere detection, the identification of the source of the problem that produced the anomaly is also essential. This is particularly the case in aircraft engine health monitoring where detecting early signs of failure (anomalies) and helping the engine owner to implement efficiently the adapted maintenance operations (fixing the source of the anomaly) are of crucial importance to reduce the costs attached to unscheduled maintenance. This paper introduces a general methodology that aims at classifying monitoring signals into normal ones and several classes of abnormal ones. The main idea is to leverage expert knowledge by generating a very large number of binary indicators. Each indicator corresponds to a fully parametrized anomaly detector built from parametric anomaly scores designed by experts. A feature selection method is used to keep only the most discriminant indicators which are used at inputs of a Naive Bayes classifier. This give an interpretable classifier based on interpretable anomaly detectors whose parameters have been optimized indirectly by the selection process. The proposed methodology is evaluated on simulated data designed to reproduce some of the anomaly types observed in real world engines.

Agent programming is mostly a symbolic discipline and, as such, draws little benefits from probabilistic areas as machine learning and graphical models. However, the greatest objective of agent research is the achievement of autonomy in dynamical and complex environments --- a goal that implies embracing uncertainty and therefore the entailed representations, algorithms and techniques. This paper proposes an innovative and conflict free two layer approach to agent programming that uses already established methods and tools from both symbolic and probabilistic artificial intelligence. Moreover, this framework is illustrated by means of a widely used agent programming example, GoldMiners.

We propose a parsimonious topic model for text corpora. In related models such as Latent Dirichlet Allocation (LDA), all words are modeled topic-specifically, even though many words occur with similar frequencies across different topics. Our modeling determines salient words for each topic, which have topic-specific probabilities, with the rest explained by a universal shared model. Further, in LDA all topics are in principle present in every document. By contrast our model gives sparse topic representation, determining the (small) subset of relevant topics for each document. We derive a Bayesian Information Criterion (BIC), balancing model complexity and goodness of fit. Here, interestingly, we identify an effective sample size and corresponding penalty specific to each parameter type in our model. We minimize BIC to jointly determine our entire model -- the topic-specific words, document-specific topics, all model parameter values, {\it and} the total number of topics -- in a wholly unsupervised fashion. Results on three text corpora and an image dataset show that our model achieves higher test set likelihood and better agreement with ground-truth class labels, compared to LDA and to a model designed to incorporate sparsity.

We present a novel, scalable and Bayesian approach to modelling the occurrence of pairs of symbols (i, j) drawn from a large vocabulary. Observed pairs are assumed to be generated by a simple popularity based selection process followed by censoring using a preference function. By basing inference on the well-founded principle of variational bounding, and using new site-independent bounds, we show how a scalable inference procedure can be obtained for large data sets. State of the art results are presented on real-world movie viewing data.