Paul-André Monney

Model-based diagnostics and probabilistic assumption-based reasoning

Abstract The mathematical foundations of model-based diagnostics or diagnosis from first principles have been laid by Reiter (1987). In this paper we extend Reiters ideas of model-based diagnostics by introducing probabilities into Reiters framework. This is done in a mathematically sound and precise way which allows one to compute the posterior probability that a certain component is not working correctly given some observations of the system. A straightforward computation of these probabilities is not efficient and in this paper we propose a new method to solve this problem. Our method is logic-based and borrows ideas from assumption-based reasoning and ATMS. We show how it is possible to determine arguments in favor of the hypothesis that a certain group of components is not working correctly. These arguments represent the symbolic or qualitative aspect of the diagnosis process. Then they are used to derive a quantitative or numerical aspect represented by the posterior probabilities. Using two new theorems about the relation between Reiters notion of conflict and our notion of argument, we prove that our so-called degree of support is nothing but the posterior probability that we are looking for. Furthermore, a model where each component may have more than two different operating modes is discussed and a new algorithm to compute posterior probabilities in this case is presented.

Theory of evidence — A survey of its mathematical foundations, applications and computational aspects

The mathematical theory of evidence has been introduced by Glenn Shafer in 1976 as a new approach to the representation of uncertainty. This theory can be represented under several distinct but more or less equivalent forms. Probabilistic interpretations of evidence theory have their roots in Arthur Dempsters multivalued mappings of probability spaces. This leads to random set and more generally to random filter models of evidence. In this probabilistic view evidence is seen as more or less probable arguments for certain hypotheses and they can be used to support those hypotheses to certain degrees. These degrees of support are in fact the reliabilities with which the hypotheses can be derived from the evidence. Alternatively, the mathematical theory of evidence can be founded axiomatically on the notion of belief functions or on the allocation of belief masses to subsets of a frame of discernment. These approaches aim to present evidence theory as an extension of probability theory. Evidence theory has been used to represent uncertainty in expert systems, especially in the domain of diagnostics. It can be applied to decision analysis and it gives a new perspective for statistical analysis. Among its further applications are image processing, project planning and scheduling and risk analysis. The computational problems of evidence theory are well understood and even though the problem is complex, efficient methods are available.

Probabilistic assumption-based reasoning

The classical propositional assumption-based model is extended to incorporate probabilities for the assumptions. Then it is placed into the framework of evidence theory. Several authors like Laskey, Lehner (1989) and Provan (1990) already proposed a similar point of view, but the first paper is not as much concerned with mathematical foundations, and Provans paper develops into a different direction. Here we thoroughly develop and present the mathematical foundations of this theory, together with computational methods adapted from Reiter, De Kleer (1987) and Inoue (1992). Finally, recently proposed techniques for computing degrees of support are presented.

Representation of evidence by hints

This paper introduces a mathematical model of a hint as a body of imprecise and uncertain information. Hints are used to judge hypotheses: the degree to which a hint supports a hypothesis and the degree to which a hypothesis appears as plausible in the light of a hint are defined. This leads in turn to support- and plausibility functions. Those functions are characterized as set functions which are normalized and monotone or alternating of order ∞. This relates the present work to G. Shafer’s mathematical theory of evidence. However, whereas Shafer starts out with an axiomatic definition of belief functions, the notion of a hint is considered here as the basic element of the theory. It is shown that a hint contains more information than is conveyed by its support function alone. Also hints allow for a straightforward and logical derivation of Dempster’s rule for combining independent and dependent bodies of information. This paper presents the mathematical theory of evidence for general, infinite frames of discernment from the point of view of a theory of hints.

An algebraic theory for statistical information based on the theory of hints

Statistical problems were at the origin of the mathematical theory of evidence, or Dempster-Shafer theory. It was also one of the major concerns of Philippe Smets, starting with his PhD dissertation. This subject is reconsidered here, starting with functional models, describing how data is generated in statistical experiments. Inference is based on these models, using probabilistic assumption-based reasoning. It results in posterior belief functions on the unknown parameters. Formally, the information used in the process of inference can be represented by hints. Basic operations on hints are combination, corresponding to Dempsters rule, and focussing. This leads to an algebra of hints. Applied to functional models, this introduces an algebraic flavor into statistical inference. It emphasizes the view that in statistical inference different pieces of information have to be combined and then focussed onto the question of interest. This theory covers Bayesian and Fisher type inference as two extreme cases of a more general theory of inference.

Using propositional logic to compute probabilities in multistate systems

In this paper we start with presentation of a general language for representing subsets of a cartesian product of finite sets. This language is used to represent the set of diagnoses in the general theory of model-based diagnosis presented by Reiter (R. Reiter, Artif. Intell. 32 (1987) 57-95) when the components have more than two possible operating modes. After having established some general results about Boolean algebras, which turn out to be the appropriate mathematical structure to define the language precisely, they are applied in the special case of propositional logic and product spaces, thereby defining a language for the description of events in product spaces. Then we present three different symbloic methods for computing the probability of a formula in the language without explicitly constructing the corresponding system states. The first two methods are based on the algorithm of Abraham (J.A. Abraham, IEEE Transactions on Reliablity 28 (1979) 58-61) whereas the last method is based on the algorithm of Bertschy-Monney (R. Bertschy, P.A. Monney, J. Comput. Appl. Math. 76 (1996) 55-76). All these methods transform the original formula into an equivalent formula for which it is very simple to compute the probability. The problem of computing the probability of a logical formula appears for example in model-based diagnostics when we need to compute the conditional probability of a diagnosis given the observations made on the system.

Propagating belief functions through constraint systems

Abstract Constraint systems as used in temporal or spatial reasoning usually describe uncertainty by constraining variables into given sets. Viewing belief functions as random or uncertain sets, uncertainty in such models is quite naturally and more generally described by belief functions. Here a special class of constraint systems arising especially in numerical models underlying temporal and spatial reasoning is introduced. The computations are, as usual, plagued by combinatorial explosion in the general case. Structural properties of the knowledge base must therefore be exploited. It is shown that there are topological properties of a graph representing the model, which can be used to reduce computational complexity. Series-parallel graphs prove to be particularly simple with respect to computations. They play a role analogous to that of qualitative Markov trees in multivariate models. Moreover, the idea of reference elements leads to a natural hierarchical structuring of the knowledge base that permits computational simplifications.

Temporal and Spatial Reasoning

In this section a certain number of examples illustrating the type of problems to be considered in this chapter will be presented. They are all concerned with spatial or temporal reasoning under uncertainty. Then the examples are generalized and computational techniques are presented. To a large extent, this chapter reproduces results given in Monney (1991) and Kohlas, Monney (1991).

A belief function classifier based on information provided by noisy and dependent features

A model and method are proposed for dealing with noisy and dependent features in classification problems. The knowledge base consists of uncertain logical rules forming a probabilistic argumentation system. Assumption-based reasoning is the inference mechanism that is used to derive information about the correct class of the object. Given a hypothesis regarding the correct class, the system provides a symbolic expression of the arguments for that hypothesis as a logical disjunctive normal form. These arguments turn into degrees of support for the hypothesis when numerical weights are assigned to them, thereby creating a support function on the set of possible classes. Since a support function is a belief function, the pignistic transformation is then applied to the support function and the object is placed into the class with maximal pignistic probability.

Model-Based Diagnostics Using Hints

It is often possible to describe the correct functioning of a system by a mathematical model. As long as observations or measurements correspond to the predictions made by the model, the system may be assumed to be functioning correctly. When, however, a discrepancy arises between the observations and the model-based predictions, then an explanation for this fact has to be found. The foundation of this approach to diagnostics has been laid by Reiter (1987). The explanations generated by his method, called diagnoses, are not unique in general. In addition, they are not weighed by a likelihood measure which would make it possible to compare them. We propose here the theory of hints — an interpretation of the Dempster-Shafer Theory of Evidence — as a very natural and general method for model-based diagnostics (for an introduction to the theory of hints, see (Kohlas & Monney, 1995)). Note that (Peng & Reggia, 1990) and (DeKleer & Williams, 1987) also discuss probabilistic approaches to diagnostic problems.