Bayesian networks are a formalism for probabilistic reasoning that have grown increasingly popular for tasks such as classification in data-mining In some situations, the structure of the Bayesian network can be given by an expert If not, retrieving it automatically from a database of cases is a NP-hard problem; notably because of the complexity of the search space In the last decade, numerous methods have been introduced to learn the network’s structure automatically, by simplifying the search space or by using an heuristic in the search space Most methods deal with completely observed data, but some can deal with incomplete data The Bayes Net Toolbox for Matlab, introduced by Murphy (2004), offers functions for both using and learning Bayesian Networks But this toolbox is not ’state of the art’ as regards structural learning methods This is why we propose the SLP package

/pdf/bnt-structure-learning-package-documentation-and-experiments-442b1pt81s.pdf

BNT STRUCTURE LEARNING PACKAGE : Documentation and Experiments

In this paper we investigate the geometry of a discrete Bayesian
network whose graph is a tree all of whose variables are binary and the
only observed variables are those labeling its leaves. We provide the full
geometric description of these models which is given by a set of polynomial
equations together with a set of complementary implied inequalities induced
by the positivity of probabilities on hidden variables. The phylogenetic
invariants given by the equations can be useful in the construction of simple
diagnostic tests. However, in this paper we point out the importance of also
incorporating the associated inequalities into any statistical analysis. The
full characterization of these inequality constraints derived in this paper
helps us determine now why routine statistical methods can break down
for this model class.

/pdf/implicit-inequality-constraints-in-a-binary-tree-model-54fzk3ynoh.pdf

Implicit inequality constraints in a binary tree model

Resume : Les reseaux bayesiens sont un formalisme de raisonnement probabiliste de plus en plus utilise pour des tâches aussi diverses que le diagnostic medical, la fouille de texte ou encore la robotique. Dans certains cas, la structure du reseau bayesien est fournie a priori par un expert. Par contre, la determination de cette structure a partir de donnees est une problematique NP-difficile pour laquelle de nombreuses methodes d’apprentissage automatique ont ete proposees ces dernieres annees (recherche d’un arbre optimal, recherche gloutonne, prise en compte de donnees incompletes, etc). Apres un bref rappel des principales methodes existantes, nous proposons deux series de tests destines tout d’abord a evaluer la precision de ces methodes en essayant de retrouver un graphe connu, et ensuite a tester leur efficacite face a des problemes de classification. Les resultats obtenus nous permettent d’etudier par exemple la robustesse des methodes pour la detection de relations "faibles" entre les variables, ou encore leur comportement en fonction du nombre d’exemples. Mots-cles : Reseaux Bayesiens, Apprentissage Statistique, Raisonnement Probabiliste, Classification, Aide a la Decision

/pdf/etude-comparative-d-algorithmes-d-apprentissage-de-structure-ixfz16pjra.pdf

Étude Comparative d’Algorithmes d’Apprentissage de Structure dans les Réseaux Bayésiens

In this paper we investigate undirected discrete graphical tree models when all the variables in the system are binary, where leaves represent the observable variables and where all the inner nodes are unobserved. A novel approach based on the theory of partially ordered sets allows us to obtain a convenient parametrization of this model class. The construction of the proposed coordinate system mirrors the combinatorial definition of cumulants. A simple product-like form of the resulting parametrization gives insight into identifiability issues associated with this model class. In particular, we provide necessary and sufficient conditions for such a model to be identified up to the switching of labels of the inner nodes. When these conditions hold, we give explicit formulas for the parameters of the model. Whenever the model fails to be identified, we use the new parametrization to describe the geometry of the unidentified parameter space. We illustrate these results using a simple example.

/pdf/tree-cumulants-and-the-geometry-of-binary-tree-models-4e1f1ib2bt.pdf

Tree cumulants and the geometry of binary tree models

Durant ces travaux de these, une comparaison empirique de differentes techniques d'apprentissage de structure de reseaux bayesiens a ete effectuee, car meme s'il peut en exister tres ponctuellement, il n'existe pas de comparaisons plus globales de ces algorithmes. De multiples phases de tests nous ont permis d'identifier quelles methodes souffraient de difficultes d'initialisation et nous avons propose une technique pour les resoudre. Nous avons ensuite adapte differentes methodes d'apprentissage de structure aux bases de donnees incompletes et avons notamment introduit une technique pour apprendre efficacement une structure arborescente. Cette methode est ensuite adaptee a la problematique plus specifique de la classification et permet d'apprendre efficacement et en toute generalite un classifieur de Bayes Naif augmente. Un formalisme original permettant de generer des bases de donnees incompletes ayant des donnees manquantes verifiant les hypotheses MCAR ou MAR est egalement introduit. De nombreuses bases synthetiques ou reelles ont alors ete utilisees pour tester ces methodes d'apprentissage de structure a partir de bases incompletes.

De l'identification de structure de réseaux bayésiens à la reconnaissance de formes à partir d'informations complètes ou incomplètes

The problem of learning Markov equivalence classes of Bayesian network structures may be solved by searching for the maximum of a scoring metric in a space of these classes. This paper deals with the definition and analysis of one such search space. We use a theoretically motivated neighbourhood, the inclusion boundary, and represent equivalence classes by essential graphs. We show that this search space is connected and that the score of the neighbours can be evaluated incrementally. We devise a practical way of building this neighbourhood for an essential graph that is purely graphical and does not explicitely refer to the underlying independences. We find that its size can be intractable, depending on the complexity of the essential graph of the equivalence class. The emphasis is put on the potential use of this space with greedy hillclimbing search.

/pdf/on-the-construction-of-the-inclusion-boundary-neighbourhood-49z0zdxufr.pdf

On the construction of the inclusion boundary neighbourhood for markov equivalence classes of bayesian network structures

The problem of learning Markov equivalence classes of Bayesian network structures may be solved by searching for the maximum of a scoring metric in a space of these classes. This paper deals with the definition and analysis of one such search space. We use a theoretically motivated neighbourhood, the inclusion boundary, and represent equivalence classes by essential graphs. We show that this search space is connected and that the score of the neighbours can be evaluated incrementally. We devise a practical way of building this neighbourhood for an essential graph that is purely graphical and does not explicitely refer to the underlying independences. We find that its size can be intractable, depending on the complexity of the essential graph of the equivalence class. The emphasis is put on the potential use of this space with greedy hill -climbing search

On the Construction of the Inclusion Boundary Neighbourhood for Markov Equivalence Classes of Bayesian Network Structures

Chordal graphs can be used to encode dependency models that are representable by both directed acyclic and undirected graphs. This paper discusses a very simple and efficient algorithm to learn the chordal structure of a probabilistic model from data. The algorithm is a greedy hill-climbing search algorithm that uses the inclusion boundary neighborhood over chordal graphs. In the limit of a large sample size and under appropriate hypotheses on the scoring criterion, we prove that the algorithm will find a structure that is inclusion-optimal when the dependency model of the data-generating distribution can be represented exactly by an undirected graph. The algorithm is evaluated on simulated datasets.

/pdf/learning-inclusion-optimal-chordal-graphs-1tq5lg7eal.pdf

Learning inclusion-optimal chordal graphs

Learning Inclusion-Optimal Chordal Graphs

Discrete Bayesian network models with hidden variables de ne an important class of statistical models. These models are usually de ned parametrically, but can also be described semi-algebraically as the solutions in the probability simplex of a nite set of polynomial equations and inequations. In this paper we present a semi-algebraic description of discrete Naive Bayes models with two hidden classes and a nite number of observable variables. The identi ability of the parameters is also studied. Our derivations are based on an alternative parametrization of the Naive Bayes models with an arbitrary number of hidden classes.

/pdf/a-semi-algebraic-description-of-discrete-naive-bayes-models-28rfdt3sag.pdf

A Semi-Algebraic Description of Discrete Naive Bayes Models with Two Hidden Classes

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