In previous work (Fertig and Breese, 1989; Fertig and Breese, 1990) we defined a mechanism for performing probabilistic reasoning in influence diagrams using interval rather than point-valued probabilities. In this paper we extend these procedures to incorporate decision nodes and interval-valued value functions in the diagram. We derive the procedures for chance node removal (calculating expected value) and decision node removal (optimization) in influence diagrams where lower bounds on probabilities are stored at each chance node and interval bounds are stored on the value function associated with the diagram's value node. The output of the algorithm are a set of admissible alternatives for each decision variable and a set of bounds on expected value based on the imprecision in the input. The procedure can be viewed as an approximation to a full e-dimensional sensitivity analysis where n are the number of imprecise probability distributions in the input. We show the transformations are optimal and sound. The performance of the algorithm on an influence diagrams is investigated and compared to an exact algorithm.

The most difficult task in probabilistic reasoning may be handling directed cycles in belief networks. To the best knowledge of this author, there is no serious discussion of this problem at all in the literature of probabilistic reasoning so far.

We introduce a new heuristic algorithm for the problem of finding minimum size loop cutsets in multiply connected belief networks. We compare this algorithm to that proposed in [Suemmondt and Cooper, 1988]. We provide lower bounds on the performance of these algorithms with respect to one another and with respect to optimal. We demonstrate that no heuristic algorithm for this problem cam be guaranteed to produce loop cutsets within a constant difference from optimal. We discuss experimental results based on randomly generated networks, and discuss future work and open questions.

In this paper, optimum decomposition of belief networks is discussed. Some methods of decomposition are examined and a new method - the method of Minimum Total Number of States (MTNS) - is proposed. The problem of optimum belief network decomposition under our framework, as under all the other frameworks, is shown to be NP-hard. According to the computational complexity analysis, an algorithm of belief network decomposition is proposed in (Wee, 1990a) based on simulated annealing.

Within diagnostic reasoning there have been a number of proposed definitions of a diagnosis, and thus of the most likely diagnosis, including most probable posterior hypothesis, most probable interpretation, most probable covering hypothesis, etc. Most of these approaches assume that the most likely diagnosis must be computed, and that a definition of what should be computed can be made a priori, independent of what the diagnosis is used for. We argue that the diagnostic problem, as currently posed, is incomplete: it does not consider how the diagnosis is to be used, or the utility associated with the treatment of the abnormalities. In this paper we analyze several well-known definitions of diagnosis, showing that the different definitions of the most likely diagnosis have different qualitative meanings, even given the same input data. We argue that the most appropriate definition of (optimal) diagnosis needs to take into account the utility of outcomes and what the diagnosis is used for.

We show an approach to automated control of machine vision systems based on incremental creation and evaluation of a particular family of influence diagrams that represent hypotheses of imagery interpretation and possible subsequent processing decisions. In our approach, model-based machine vision techniques are integrated with hierarchical Bayesian inference to provide a framework for representing and matching instances of objects and relationships in imagery and for accruing probabilities to rank order conflicting scene interpretations. We extend a result of Tatman and Shachter to show that the sequence of processing decisions derived from evaluating the diagrams at each stage is the same as the sequence that would have been derived by evaluating the final influence diagram that contains all random variables created during the run of the vision system.

Decision tree induction systems are being used for knowledge acquisition in noisy domains. This paper develops a subjective Bayesian interpretation of the task tackled by these systems and the heuristic methods they use. It is argued that decision tree systems implicitly incorporate a prior belief that the simpler (in terms of decision tree complexity) of two hypotheses be preferred, all else being equal, and that they perform a greedy search of the space of decision rules to find one in which there is strong posterior belief. A number of improvements to these systems are then suggested.

In the canonical examples underlying Shafer-Dempster theory, beliefs over the hypotheses of interest are derived from a probability model for a set of auxiliary hypotheses. Beliefs are derived via a compatibility relation connecting the auxiliary hypotheses to subsets of the primary hypotheses. A belief function differs from a Bayesian probability model in that one does not condition on those parts of the evidence for which no probabilities are specified. The significance of this difference in conditioning assumptions is illustrated with two examples giving rise to identical belief functions but different Bayesian probability distributions.

In this paper we provide an overview of a new framework for robot perception, real-world modelling, and navigation that uses a stochastic tesselated representation of spatial information called the Occupancy Grid. The Occupancy Grid is a multi-dimensional random field model that maintains probabilistic estimates of the occupancy state of each cell in a spatial lattice. Bayesian estimation mechanisms employing stochastic sensor models allow incremental updating of the Occupancy Grid using multi-view, multi-sensor data, composition of multiple maps, decision-making, and incorporation of robot and sensor position uncertainty. We present the underlying stochastic formulation of the Occupancy Grid framework, and discuss its application to a variety of robotic tusks. These include range-based mapping, multi-sensor integration, path-planning and obstacle avoidance, handling of robot position uncertainty, incorporation of pre-compiled maps, recovery of geometric representations, and other related problems. The experimental results show that the Occupancy Grid approach generates dense world models, is robust under sensor uncertainty and errors, and allows explicit handling of uncertainty. It supports the development of robust and agile sensor interpretation methods, incremental discovery procedures, and composition of information from multiple sources. Furthermore, the results illustrate that robotic tasks can be addressed through operations performed di- rectly on the Occupancy Grid, and that these operations have strong parallels to operations performed in the image processing domain.

In this paper, we describe a scheme for propagating belief functions in certain kinds of trees using only local computations. This scheme generalizes the computational scheme proposed by Shafer and Logan1 for diagnostic trees of the type studied by Gordon and Shortliffe, and the slightly more general scheme given by Shafer for hierarchical evidence. It also generalizes the scheme proposed by Pearl for Bayesian causal trees (see Shenoy and Shafer). Pearl's causal trees and Gordon and Shortliffe's diagnostic trees are both ways of breaking the evidence that bears on a large problem down into smaller items of evidence that bear on smaller parts of the problem so that these smaller problems can be dealt with one at a time. This localization of effort is often essential in order to make the process of probability judgment feasible, both for the person who is making probability judgments and for the machine that is combining them. The basic structure for our scheme is a type of tree that generalizes both Pearl's and Gordon and Shortliffe's trees. Trees of this general type permit localized computation in Pearl's sense. They are based on qualitative judgments of conditional independence. We believe that the scheme we describe here will prove useful in expert systems. It is now clear that the successful propagation of probabilities or certainty factors in expert systems requires much more structure than can be provided in a pure production-system framework. Bayesian schemes, on the other hand, often make unrealistic demands for structure. The propagation of belief functions in trees and more general networks stands on a middle ground where some sensible and useful things can be done. We would like to emphasize that the basic idea of local computation for propagating probabilities is due to Judea Pearl. It is a very innovative idea; we do not believe that it can be found in the Bayesian literature prior to Pearl's work. We see our contribution as extending the usefulness of Pearl's idea by generalizing it from Bayesian probabilities to belief functions. In the next section, we give a brief introduction to belief functions. The notions of qualitative independence for partitions and a qualitative Markov tree are introduced in Section III. Finally, in Section IV, we describe a scheme for propagating belief functions in qualitative Markov trees.