Andrés Cano

Importance sampling in Bayesian networks using probability trees

In this paper a new Monte-Carlo algorithm for the propagation of probabilities in Bayesian networks is proposed. This algorithm has two stages: in the first one an approximate propagation is carried out by means of a deletion sequence of the variables. In the second stage a sample is obtained using as sampling distribution the calculations of the first step. The different configurations of the sample are weighted according to the importance sampling technique. We show how the use of probability trees to store and to approximate probability potentials, and a careful selection of the deletion sequence, make this algorithm able to propagate over large networks with extreme probabilities.

Using probability trees to compute marginals with imprecise probabilities

This paper presents an approximate algorithm to obtain a posteriori intervals of probability, when available information is also given with intervals. The algorithm uses probability trees as a means of representing and computing with the convex sets of probabilities associated to the intervals.

Heuristic Algorithms for the Triangulation of Graphs

Different uncertainty propagation algorithms in graphical structures can be viewed as a particular case of propagation in a joint tree, which can be obtained from different triangulations of the original graph. The complexity of the resulting propagation algorithms depends on the size of the resulting triangulated graph. The problem of obtaining an optimum graph triangulation is known to be NP-complete. Thus approximate algorithms which find a good triangulation in reasonable time are of particular interest. This work describes and compares several heuristic algorithms developed for this purpose.

Lazy evaluation in penniless propagation over join trees

In this paper, we investigate the application of the ideas behind Lazy propagation to the Penniless propagation scheme. Probabilistic potentials attached to the messages and the nodes of the join tree are represented in a factorized way as a product of (approximate) probability trees, and the combination operations are postponed until they are compulsory for the deletion of a variable. We tested two variations of the basic Lazy scheme: One is based on keeping a hash table for the operations with probabilistic potentials that are carried out more than once during the propagation, to avoid repeating computations; the other uses a heuristic method to determine the order of the operations when combining a set of potentials.

A Method for Integrating Expert Knowledge When Learning Bayesian Networks From Data

Automatic learning of Bayesian networks from data is a challenging task, particularly when the data are scarce and the problem domain contains a high number of random variables. The introduction of expert knowledge is recognized as an excellent solution for reducing the inherent uncertainty of the models retrieved by automatic learning methods. Previous approaches to this problem based on Bayesian statistics introduce the expert knowledge by the elicitation of informative prior probability distributions of the graph structures. In this paper, we present a new methodology for integrating expert knowledge, based on Monte Carlo simulations and which avoids the costly elicitation of these prior distributions and only requests from the expert information about those direct probabilistic relationships between variables which cannot be reliably discerned with the help of the data.

Strong Conditional Independence for Credal Sets

This paper investigates the concept of strong conditional independence for sets of probability measures. Couso, Moral and Walley [7] have studied different possible definitions for unconditional independence in imprecise probabilities. Two of them were considered as more relevant: epistemic independence and strong independence. In this paper, we show that strong independence can have several extensions to the case in which a conditioning to the value of additional variables is considered. We will introduce simple examples in order to make clear their differences. We also give a characterization of strong independence and study the verification of semigraphoid axioms.

Editorial: Reasoning with imprecise probabilities

This special issue of the International Journal of Approximate Reasoning (IJAR) grew out of the 4th International Symposium on Imprecise Probabilities and Their Applications (ISIPTA’05), held in Pittsburgh, USA, in July 2005 (http://www.sipta.org/isipta05). The symposium was organized by Teddy Seidenfeld, Robert Nau, and Fabio G. Cozman, and brought together researchers from various branches interested in imprecision in probabilities. Research in artificial intelligence, economics, engineering, psychology, philosophy, statistics, and other fields was presented at the meeting, in a lively atmosphere that fostered communication and debate. Invited talks by Isaac Levi and Arthur Dempster enlightened the attendants, while tutorials by Gert de Cooman, Paolo Vicig, and Kurt Weichselberger introduced basic (and advanced) concepts; finally, the symposium ended with a workshop on financial risk assessment, organized by Teddy Seidenfeld. The ISIPTA series started in 1999; the first one was held in Ghent, Belgium – followed by symposia held in Cornell, USA (in 2001), in Lugano, Switzerland (in 2003), and in Pittsburgh, USA (in 2005). The next edition of this biennial event will take place in Prague, Czech Republic, in July 2007 (http://www.sipta.org/isipta07). Selected papers from the first three symposia appeared in special issues of IJAR in 2000 and 2005, in a special issue of Risk, Decision and Policy in 2000, and in a special issue of Annals of Mathematics and Artificial Intelligence in 2005. This special issue of IJAR contains ten articles; the first eight of them are revised versions of selected papers from ISIPTA’05. The first four papers deal with independence and graphical models; they are followed by two papers on probabilistic logic, and by two papers on decision–theoretic and combinatorial results. We close this special issue with a very special treat – the publication of Peter Williams’ essay Notes on Coherent Previsions, a fundamental paper that appeared in 1975 as a technical report, and that has widely circulated since then. ISIPTA’05 marked the 30th anniversary of this paper, and we were fortunate to obtain a revised version from its author for this special issue. Williams’ essay deals with foundations of probability, and addresses many profound questions that are basic to reasoning under uncertainty. The paper requires substantial background; for this reason, it is preceded by a short paper by Vicig, Zaffalon, and Cozman. This short paper offers commentary and guidance on Williams’ influential work.

Hill-climbing and branch-and-bound algorithms for exact and approximate inference in credal networks

This paper proposes two new algorithms for inference in credal networks. These algorithms enable probability intervals to be obtained for the states of a given query variable. The first algorithm is approximate and uses the hill-climbing technique in the Shenoy-Shafer architecture to propagate in join trees; the second is exact and is a modification of Rocha and Cozmans branch-and-bound algorithm, but applied to general directed acyclic graphs.

A forward-backward Monte Carlo method for solving influence diagrams

Although influence diagrams are powerful tools for representing and solving complex decision-making problems, their evaluation may require an enormous computational effort and this is a primary issue when processing real-world models. We shall propose an approximate inference algorithm to deal with very large models. For such models, it may be unfeasible to achieve an exact solution. This anytime algorithm returns approximate solutions which are increasingly refined as computation progresses, producing knowledge that offers insight into the decision problem.

Approximate inference in Bayesian networks using binary probability trees

The present paper introduces a new kind of representation for the potentials in a Bayesian network: Binary Probability Trees. They enable the representation of context-specific independences in more detail than probability trees. This enhanced capability leads to more efficient inference algorithms for some types of Bayesian networks. This paper explains the procedure for building a binary probability tree from a given potential, which is similar to the one employed for building standard probability trees. It also offers a way of pruning a binary tree in order to reduce its size. This allows us to obtain exact or approximate results in inference depending on an input threshold. This paper also provides detailed algorithms for performing the basic operations on potentials (restriction, combination and marginalization) directly to binary trees. Finally, some experiments are described where binary trees are used with the variable elimination algorithm to compare the performance with that obtained for standard probability trees.