cover page

Systems Dependability Assessment Set

coordinated by
Jean-Francois Aubry

Volume 2

Benefits of Bayesian Network Models

Philippe Weber

Christophe Simon


Foreword by J.-F. Aubry

Systems Dependability Assessment is the title of a series of books, of which this is the third. The preface to the first series described the reasons why the authors embarked upon writing these books: in recent decades, they have made significant contributions to recent approaches to the predictive dependability of systems by considering concepts developed in other scientific fields but not yet applied to account of dependability. All these authors belong to the Automatic Control Research Center (CRAN, Centre de Recherches en Automatique de Nancy) of the University of Lorraine, France, a research laboratory whose activities are widely oriented towards the diagnosis, reliability, maintainability and safety of systems, which can be described in one word: dependability.

Assessment must be understood as the set of means, methods and tools to provide quantitative measures of dependability, and in these books we are interested in providing predictive measures by using probabilistic approaches. The first two books were dedicated to methods based on the frequentist knowledge of basic elements of a system and on models describing how the failure of this system depends on those of its components. These models are essentially of state-transition type, such as finite state automata and petri nets, and results were obtained by analytic or simulation approaches according to the level of complexity introduced, and data inputs for these models were probabilistic distributions of elementary events that were derived from strong feedback. This is within the approach called the frequentist or objective of probability theory.

The present book is different as it is based on Bayesian networks. Frequentist and Bayesian approaches to probabilities were both developed at about the same time in the 18th Century, but the first approach culminated in industrial development in the 19th Century, eclipsing the second. The original mathematical formulation of subjective probabilities was made almost simultaneously by Laplace and Bayes, the latter having given his name to the theorem of probability of causes. The Bayesian approach supposes an a priori knowledge, even if only approximate, of an event probability. From the knowledge of the a priori probabilities of the event and of its cause, the Bayes formula gives an a posteriori probability of the event, its likelihood function somehow describing the causal dependency. It is, in fact, a means of improving the knowledge of the event probability.

Bayes theorem can be materialized by a causal decision tree and extended to represent chained causality in a system. This is the base of Bayesian networks. With the development of computing technologies, the second half of the 20th Century saw the return of Bayesian methods, being an efficient tool to aid decision making in an uncertain environment. They provided solutions to problems such as climate change prediction or, more recently, the detection of spam in data communications.

Dependability assessment problems do not escape from Bayesian approaches. CRAN was among the first to promote these techniques, proposing methods and tools in association with industrial users and developers. Original works have been conducted, especially in the dependability of multi-state systems, integration of environmental and operating constraints in reliability computation (dynamic context) and interaction between the dependability and control of systems. The reader will find in this book a clear presentation of all these advances and I do not doubt that he/she can find a substantial benefit.

Jean-François AUBRY
Professor Emeritus
Lorraine University
Nancy, France

Foreword by L. Portinale

Probabilistic graphical models and Bayesian belief networks, in particular, have definitely become a reference formalism in dependability modeling and assessment. The graphical structure, together with the compact representation of the joint distribution of the system variables of interest, provides the reliability engineer with a powerful tool at both the modeling and analytical levels.

The dependency structure, induced by the graph component of the formalism, allows the modeler to make explicit a set of reasonable independence assumptions that may lead to huge simplification at the computational level, as well as with respect to the problem of probability elicitation, without compromising the suitability of the model produced to the actual real-world application.

Standard dependability models usually fit into two categories: 1) combinatorial models (as fault trees or reliability block diagrams) – they determine the occurrence of an undesired event through a combinatorial composition of sub-events; this class of model is very easy to analyze, but it cannot model situations involving complex dependencies among system components and sub-systems; 2) state–space models (such as Markov chains or petri nets) – they allow complex interactions among system parts to be modeled, but they may incur the “state-explosion” problem; this usually means that the analysis has to be performed by considering the cross-product of the system variables, producing a potentially huge number of states.

Bayesian networks and related models allow for efficient factorization of the set of system states, without the need for an explicit representation of the whole joint distribution; moreover it has the additional advantage of inference algorithms available for the analysis of any a posteriori situation of interest (i.e. evidence can be gathered by a monitoring system and fed into a dependability framework for fault detection and identification). Finally, when time is explicitly taken into account, models such as dynamic Bayesian networks result in a factored representation of a Markov chain, providing a framework with the modeling advantages of state–space models, without the drawbacks at the analytical level.

The present book, written by some of the most respected researchers and practitioners in the field, provides a comprehensive presentation and analysis of the probabilistic graphical model approach to dependability, providing a view of the different facets involved in real-world dependability applications: system reliability, maintenance and risk evaluation. The main objective is to devise a principled approach to the modeling of complex dependable systems, with the aim of supporting decisions in an uncertain and evolving setting. This supports and promotes Bayesian networks and probabilistic graphical models as some of the most relevant and important formalisms in modern dependability analysis.

Computer Science Institute (DiSIT)
University of Eastern Piedmont
Alessandria, Italy


It is not easy to thank all those who have participated in and contributed to the research mentioned in this book.

I want to thank my scientific mentors Professor Benoît Iung and Professor Didier Theilliol, who have guided and supported my activity during all these years we have worked together. Special thanks to Christophe Simon, my co-author, my scientific partner and friend for all our joint contributions.

I thank my industry partners, Carole Duval (EDF), Paul Munteanu and Lionel Jouffe (Bayesia) for their trust and the various joint projects.

I cannot conclude the acknowledgments without extending my warmest thanks to my family, especially my wife Carole, who preserved the equilibrium in my life, and to my children Loïc and Manon, who have filled my family life with happiness.

Philippe WEBER
June 2016