Joseph S. J. Peri

Dempster-Shafer Theory, Bayesian Theory and Measure Theory

We use measure theoretic methods to describe the relationship between the Dempster Shafer (DS) theory and Bayesian (i.e. probability) theory. Within this framework, we demonstrated the relationships among Shafers belief and plausibility, Dempsters lower and upper probabilities and inner and outer measures. Dempsters multivalued mapping is an example of a random set, a generalization of the concept of the random variable. Dempsters rule of combination is the product measure on the Cartesian product of measure spaces. The independence assumption of Dempsters rule arises from the nature of the problem in which one has knowledge of the marginal distributions but wants to calculate the joint distribution. We present an engineering example to clarify the concepts.

Dempster-Shafer theory and connections to information theory

The Dempster-Shafer theory is founded on probability theory. The entire machinery of probability theory, and that of measure theory, is at one’s disposal for the understanding and the extension of the Dempster-Shafer theory. It is well known that information theory is also founded on probability theory. Claude Shannon developed, in the 1940’s, the basic concepts of the theory and demonstrated their utility in communications and coding. Shannonian information theory is not, however, the only type of information theory. In the 1960’s and 1970’s, further developments in this field were made by French and Italian mathematicians. They developed information theory axiomatically, and discovered not only the Wiener- Shannon composition law, but also the hyperbolic law and the Inf-law. The objective of this paper is to demonstrate the mathematical connections between the Dempster Shafer theory and the various types of information theory. A simple engineering example will be used to demonstrate the utility of the concepts.

Fundamentals of the Dempster-Shafer theory

In this paper, I discuss the basic notions of the Dempster Shafer theory. Using a simple engineering example, I highlight sources of confusion in the Dempster Shafer literature, and some questions that arise in the course of applying the Dempster Shafer algorithm. Finally, I discuss the measure theoretic foundation that reveals the intimate connections between the Dempster Shafer theory and Probability theory.

Dempster-Shafer information measures in category theory

In the Dempster Shafer context, one can construct new types of information measures based on belief and plausibility functions. These measures differ from those in Shannon’s theory because, in his theory, information measures are based on probability functions. Other types of information measures were discovered by Kampe de Feriet and his colleagues in the French and Italian schools of mathematics. The objective of this paper is to construct a new category of information. I use category theory to construct a general setting in which the various types of information measures are special cases.

Categorification of the Dempster Shafer theory

This paper contains preliminary steps in demonstrating how the Dempster Shafer theory can be placed into the framework of category theory. In the Dempster Shafer setting, the elements of the base set of a probability space are, typically, subsets of some set. Consequently, the elements of the corresponding sigma algebra are not subsets of a set, but rather, subsets of subsets of a set. A probability function, in this case, no longer has the classical meaning. This situation lends itself to the more general notions of inner and outer measures, which Shafer calls belief and plausibility, respectively. The categorical approach attempts to unify classical and non-classical concepts into a setting, so that, depending on the nature of the stochastic problem at hand, a general framework may be specialized appropriately to attack the problem.

Dempster-Shafer theory and connections to Choquet's theory of capacities and information theory

The axiomatic development of information theory, during the 1960s, led to the discovery of various composition laws. The Wiener-Shannon law is well understood, but the Inf law holds particular interest because it creates a connection with the Dempster-Shafer theory. Proceeding along these lines, in a previous paper, I demonstrated the connection between the Dempster-Shafer theory and Information theory. In 1954, Gustave Choquet developed the theory of capacities in connection with potential theory. The basic concepts of capacity theory arise from electrostatics, but a capacity is a generalization of the concept of measure in Analysis. It is well known that Belief and Plausibility in the Dempster-Shafer theory are Choquet capacities. However, it is not well known that the inverse of an information measure is a Choquet capacity. The objective of this paper is to demonstrate the connections among the Dempster- Shafer theory, Information theory and Choquets theory of capacities.

Lidar characteristics for detecting and tracking high-speed bullets

In this paper, we discuss the possible use of a light-weight lidar system to detect and track a snipers high-speed bullet. The analysis includes the calculation of the beam waist, the irradiance per pulse, average irradiance, the maximum time between pulses and the minimum pulse repetition frequency, all as functions of range, beam diameter and beam quality (M2). We discuss, briefly, the possible cueing of such a lidar system by an IR system. The measurement of the BRDF of a bullet is briefly described. Finally, we report on the detection range, based on SNR calculations, as a function of energy per pulse, beam diameter and M2.