We consider structural equation models in which variables can be written as a function of their parents and noise terms, which are assumed to be jointly independent. Corresponding to each structural equation model, there is a directed acyclic graph describing the relationships between the variables. In Gaussian structural equation models with linear functions, the graph can be identified from the joint distribution only up to Markov equivalence classes, assuming faithfulness. In this work, we prove full identifiability if all noise variables have the same variances: the directed acyclic graph can be recovered from the joint Gaussian distribution. Our result has direct implications for causal inference: if the data follow a Gaussian structural equation model with equal error variances and assuming that all variables are observed, the causal structure can be inferred from observational data only. We propose a statistical method and an algorithm that exploit our theoretical findings.

Classifiers based on probabilistic graphical models are very effective. In continuous domains, maximum likelihood is usually used to assess the predictions of those classifiers. When data is scarce, this can easily lead to overfitting. In any probabilistic setting, Bayesian averaging (BA) provides theoretically optimal predictions and is known to be robust to overfitting. In this work we introduce Bayesian Conditional Gaussian Network Classifiers, which efficiently perform exact Bayesian averaging over the parameters. We evaluate the proposed classifiers against the maximum likelihood alternatives proposed so far over standard UCI datasets, concluding that performing BA improves the quality of the assessed probabilities (conditional log likelihood) whilst maintaining the error rate. Overfitting is more likely to occur in domains where the number of data items is small and the number of variables is large. These two conditions are met in the realm of bioinformatics, where the early diagnosis of cancer from mass spectra is a relevant task. We provide an application of our classification framework to that problem, comparing it with the standard maximum likelihood alternative, where the improvement of quality in the assessed probabilities is confirmed.

A Bayesian factor graph reduced to normal form consists in the interconnection of diverter units (or equal constraint units) and Single-Input/Single-Output (SISO) blocks. In this framework localized adaptation rules are explicitly derived from a constrained maximum likelihood (ML) formulation and from a minimum KL-divergence criterion using KKT conditions. The learning algorithms are compared with two other updating equations based on a Viterbi-like and on a variational approximation respectively. The performance of the various algorithm is verified on synthetic data sets for various architectures. The objective of this paper is to provide the programmer with explicit algorithms for rapid deployment of Bayesian graphs in the applications.

A new family of penalty functions, adaptive to likelihood, is introduced for model selection in general regression models. It arises naturally through assuming certain types of prior distribution on the regression parameters. To study stability properties of the penalized maximum likelihood estimator, two types of asymptotic stability are defined. Theoretical properties, including the parameter estimation consistency, model selection consistency, and asymptotic stability, are established under suitable regularity conditions. An efficient coordinate-descent algorithm is proposed. Simulation results and real data analysis show that the proposed method has competitive performance in comparison with existing ones.

Decision tree learning is a popular approach for classification and regression in machine learning and statistics, and Bayesian formulations---which introduce a prior distribution over decision trees, and formulate learning as posterior inference given data---have been shown to produce competitive performance. Unlike classic decision tree learning algorithms like ID3, C4.5 and CART, which work in a top-down manner, existing Bayesian algorithms produce an approximation to the posterior distribution by evolving a complete tree (or collection thereof) iteratively via local Monte Carlo modifications to the structure of the tree, e.g., using Markov chain Monte Carlo (MCMC). We present a sequential Monte Carlo (SMC) algorithm that instead works in a top-down manner, mimicking the behavior and speed of classic algorithms. We demonstrate empirically that our approach delivers accuracy comparable to the most popular MCMC method, but operates more than an order of magnitude faster, and thus represents a better computation-accuracy tradeoff.

This supplementary material includes three parts: some preliminary results, four examples, an experiment, three new algorithms, and all proofs of the results in the paper "Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs".

We review the challenges of Bayesian network learning, especially parameter learning, and specify the problem of learning with sparse data. We explain how it is possible to incorporate both qualitative knowledge and data with a multinomial parameter learning method to achieve more accurate predictions with sparse data.

This paper proposes a simple linear Bayesian approach to reinforcement learning. We show that with an appropriate basis, a Bayesian linear Gaussian model is sufficient for accurately estimating the system dynamics, and in particular when we allow for correlated noise. Policies are estimated by first sampling a transition model from the current posterior, and then performing approximate dynamic programming on the sampled model. This form of approximate Thompson sampling results in good exploration in unknown environments. The approach can also be seen as a Bayesian generalisation of least-squares policy iteration, where the empirical transition matrix is replaced with a sample from the posterior.

A novel proposal for object recognition based on relational grammars and Bayesian Networks is presented. Based on this grammar an object is represented as a hierarchy of features and spatial relations. This representation is transformed to a Bayesian network structure which parameters are learned from examples. Thus, recognition is based on probabilistic inference in the Bayesian network representation. Preliminary results in modeling natural objects are presented.

This paper concerns building probabilistic models with an underlying ontology that defines the classes and properties used in the model. In particular, it considers the problem of reasoning with properties that may not always be defined. Furthermore, we may even be uncertain about whether a property is defined for a given individual. One approach is to explicitly add a value "undefined" to the range of random variables, forming extended belief networks; however, adding an extra value to a random variable's range has a large computational overhead. In this paper, we propose an alternative, ontologically-based belief networks, where all properties are only used when they are defined, and we show how probabilistic reasoning can be carried out without explicitly using the value "undefined" during inference. We prove this is equivalent to reasoning with the corresponding extended belief network and empirically demonstrate that inference becomes more efficient.