Abstract The Bernoulli filter is a very general, computationally feasible Bayes‐optimal approach for tracking a single disappearing and reappearing target, using a single sensor whose observations are corrupted by missed detections and a known, general point‐clutter process. This paper shows how to generalise it to the dyadic labelled random finite set (DLRFS) filter—that is, a very general, computationally feasible Bayes‐optimal approach for tracking two disappearing and reappearing and possibly correlated targets, using a single sensor whose observations are corrupted by missed detections and a known, general clutter process. It is further shown that, like the Bernoulli filter, the DLRFS filter is an exact special case of the labelled multitarget recursive Bayes filter (LMRBF)—and thus that, given the target and sensor models, there cannot be a theoretically better tracking filter. The paper also describes the relationship between the DLRFS filter and the (unlabelled) Gauss–Poisson filter of Singh, Vo, Baddeley, and Zuyev.

In this paper, we extend the notion of Cauchy-Schwarz divergence to point processes and establish that the Cauchy-Schwarz divergence between the probability densities of two Poisson point processes is half the squared L2-distance between their intensity functions. Extension of this result to mixtures of Poisson point processes and, in the case where the intensity functions are Gaussian mixtures, closed form expressions for the Cauchy-Schwarz divergence are presented. Our result also implies that the Bhattacharyya distance between the probability distributions of two Poisson point processes is equal to the square of the Hellinger distance between their intensity measures. We illustrate the result via a sensor management application where the system states are modeled as point processes.

The PHD filter propagates a multitarget statistical first moment, the probability hypothesis density (PHD), in place of the full multitarget posterior distribution. It has been the basis of a systematic approach to multisensor, multitarget sensor management based on the posterior expected number of targets (PENT) objective function. The PHD filter has since been generalized to the cardinalized PHD (CPHD) filter, which propagates not only the PHD but also the full probability distribution on target number. The CPHD filter provides more accurate estimates of target number and target states. This paper shows how PENT-based objective functions can be naturally extended for use with the CPHD filter.

The concept of miss distance-Euclidean, Mahalanobis, etc.-is a fundamental, far-reaching, and taken-for-granted element of the engineering theory and practice of single-target systems. In this paper we introduce a comprehensive L/sub p/-type theory of distance metrics for multitarget (and, more generally, multiobject) systems. We show that this theory extends, and provides a rigorous theoretical basis for, an intuitively appealing optimal-assignment approach proposed by Drummond for evaluating the performance of multitarget tracking algorithms. We describe tractable computational approaches for computing such metrics based on standard optimal assignment or convex optimization techniques. We describe the potentially far-reaching implications of these metrics for applications such as performance evaluation and sensor management. In the former case, we demonstrate the application of multitarget miss-distance metrics as measures of effectiveness (MoEs) for multitarget tracking algorithms.

This paper describes "modified Dempster-Shafer" (MDS), an approach to object identification which incorporates Bayesian prior distributions into an altered Dempster-Shafer rule of combination. The MDS combination rule reduces, under strong independence assumptions, to a special case of Bayes' rule. We show that MDS has rigorous probabilistic foundations in the theory of random sets. We also demonstrate close relationships between MDS and Smets' "pignistic" probabilities (1990), which in the MDS framework become true posterior distributions. We describe the application of MDS to a practical classification algorithm which uses an information-theoretic technique to limit the combinatorial explosion of evidence. We also define a non-ad hoc, MDS-based classification "miss distance" metric used to measure the performance of this algorithm.

I. R. Goodman, H. T. Nguyen, and others have proposed the theory of random sets as a unifying paradigm for knowledge-based systems. The more types of ambiguous evidence that can be brought under the random-set umbrella, the more potentially useful the theory becomes as a systematic methodology for comparing and fusing seemingly incongruent kinds of information. Conditional event algebra provides a general framework for dealing with rule-based evidence in a manner consistent with probability theory. This and a previous companion article bring conditional event logic—and through it, rules and iterated rules—under the random set umbrella. In this article, we derive a new conditional event algebra, denoted GNW2, for rules which are contingent on other rules. We show that this logic is the only one which extends the GNW first-degree logic, admits a simple embedding into random sets, and has GNW-like behavior. We show that GNW events are contained in a Boolean subalgebra GNW* of GNW2. Finally, we show how GNW2 can be used to recursively define a GNW-like Boolean logic for iterated rules of any degree. © 1996 John Wiley & Sons, Inc.

This paper describes conditioned Dempster-Shafer (CDS) theory, a probabilistic calculus for dealing with possibly non-Bayesian evidence when the underlying a priori knowledge base is possibly non-Bayesian. Specifically, we show that the Dempster-Shafer combination operator can be "conditioned" to reflect the influence of any a priori knowledge base which can be modeled by a Dempster-Shafer belief measure. We show that CDS is firmly grounded in probability theory-specifically, in the theory of random sets. We also show that it is a generalization of the Bayesian theory to the case when both evidence and a priori knowledge are ambiguous. We derive the algebraic properties of the theory when a priori knowledge is assumed fixed. Under this assumption, we also derive the form of CDS in the special case when fixed a priori knowledge is Bayesian.