Prakash P. Shenoy

Axioms for probability and belief-function propagation

In this paper, we describe an abstract framework and axioms under which exact local computation of marginals is possible. The primitive objects of the framework are variables and valuations. The primitive operators of the framework are combination and marginalization. These operate on valuations. We state three axioms for these operators and we derive the possibility of local computation from the axioms. Next, we describe a propagation scheme for computing marginals of a valuation when we have a factorization of the valuation on a hypertree. Finally we show how the problem of computing marginals of joint probability distributions and joint belief functions fits the general framework.

Valuation-based systems for Bayesian decision analysis

This paper proposes a new method for representing and solving Bayesian decision problems. The representation is called a valuation-based system and has some similarities to influence diagrams. However, unlike influence diagrams which emphasize conditional independence among random variables, valuation-based systems emphasize factorizations of joint probability distributions. Also, whereas influence diagram representation allows only conditional probabilities, valuation-based system representation allows all probabilities. The solution method is a hybrid of local computational methods for the computation of marginals of joint probability distributions and the local computational methods for discrete optimization problems. We briefly compare our representation and solution methods to those of influence diagrams.

A causal mapping approach to constructing Bayesian networks

This paper describes a systematic procedure for constructing Bayesian networks (BNs) from domain knowledge of experts using the causal mapping approach. We outline how causal knowledge of experts can be represented as causal maps, and how the graphical structure of causal maps can be modified to construct Bayes nets. Probability encoding techniques can be used to assess the numerical parameters of the resulting Bayes nets. We illustrate the construction of a Bayes net starting from a causal map of a systems analyst in the context of an information technology application outsourcing decision.

Propagating belief functions in qualitative Markov trees

Abstract This article is concerned with the computational aspects of combining evidence within the theory of belief functions. It shows that by taking advantage of logical or categorical relations among the questions we consider, we can sometimes avoid the computational complexity associated with brute-force application of Dempsters rule. The mathematical setting for this article is the lattice of partitions of a fixed overall frame of discernment. Different questions are represented by different partitions of this frame, and the categorical relations among these questions are represented by relations of qualitative conditional independence or dependence among the partitions. Qualitative conditional independence is a categorical rather than a probabilistic concept, but it is analogous to conditional independence for random variables. We show that efficient implementation of Dempsters rule is possible if the questions or partitions for which we have evidence are arranged in a qualitative Markov tree—a tree in which separations indicate relations of qualitative conditional independence. In this case, Dempsters rule can be implemented by propagating belief functions through the tree.

A Bayesian Network Approach to Making Inferences in Causal Maps

Abstract The main goal of this paper is to describe a new graphical structure called ‘Bayesian causal maps’ to represent and analyze domain knowledge of experts. A Bayesian causal map is a causal map, i.e., a network-based representation of an expert’s cognition. It is also a Bayesian network, i.e., a graphical representation of an expert’s knowledge based on probability theory. Bayesian causal maps enhance the capabilities of causal maps in many ways. We describe how the textual analysis procedure for constructing causal maps can be modified to construct Bayesian causal maps, and we illustrate it using a causal map of a marketing expert in the context of a product development decision.

A valuation-based language for expert systems

Abstract A new language based on valuations is proposed as an alternative to rule-based languages for constructing knowledge-based systems. Valuation-based languages are superior to rule-based languages for maintaining consistency in the knowledge base, for caching inferences, for managing uncertainty, and for nonmonotonic reasoning. An abstract description of a valuation-based language is given. Two specific instances of valuation-based languages are described. The first is designed to represent categorical knowledge. The ability of such a language to maintain consistency and cache inferences is demonstrated with an example. The second is an evidential language—a valuation-based language in which valuations are belief functions. The ability of evidential languages to perform nonmonotonic reasoning and manage uncertainty is demonstrated with an example.

On coalition formation: a game-theoretical approach

This paper deals with the question of coalition formation inn-person cooperative games. Two abstract game models of coalition formation are proposed. We then study the core and the dynamic solution of these abstract games. These models assume that there is a rule governing the allocation of payoffs to each player in each coalition structure called a payoff solution concept. The predictions of these models are characterized for the special case of games with side payments using various payoff solution concepts such as the individually rational payoffs, the core, the Shapley value and the bargaining set M1(i). Some modifications of these models are also discussed.

Binary join trees for computing marginals in the Shenoy-Shafer architecture

Abstract We describe a data structure called binary join trees that is useful in computing multiple marginals efficiently in the Shenoy-Shafer architecture. We define binary join trees, describe their utility, and describe a procedure for constructing them.

On the plausibility transformation method for translating belief function models to probability models

In this paper, we propose the plausibility transformation method for translating Dempster-Shafer (D-S) belief function models to probability models, and describe some of its properties. There are many other transformation methods used in the literature for translating belief function models to probability models. We argue that the plausibility transformation method produces probability models that are consistent with D-S semantics of belief function models, and that, in some examples, the pignistic transformation method produces results that appear to be inconsistent with Dempsters rule of combination.

Using Bayesian networks for bankruptcy prediction : Some methodological issues

Abstract This study provides operational guidance for building naive Bayes Bayesian network (BN) models for bankruptcy prediction. First, we suggest a heuristic method that guides the selection of bankruptcy predictors. Based on the correlations and partial correlations among variables, the method aims at eliminating redundant and less relevant variables. A naive Bayes model is developed using the proposed heuristic method and is found to perform well based on a 10-fold validation analysis. The developed naive Bayes model consists of eight first-order variables, six of which are continuous. We also provide guidance on building a cascaded model by selecting second-order variables to compensate for missing values of first-order variables. Second, we analyze whether the number of states into which the six continuous variables are discretized has an impact on the model’s performance. Our results show that the model’s performance is the best when the number of states for discretization is either two or three. Starting from four states, the performance starts to deteriorate, probably due to over-fitting. Finally, we experiment whether modeling continuous variables with continuous distributions instead of discretizing them can improve the model’s performance. Our finding suggests that this is not true. One possible reason is that continuous distributions tested by the study do not represent well the underlying distributions of empirical data. Finally, the results of this study could also be applicable to business decision-making contexts other than bankruptcy prediction.