Particle filter based entropy

For many problems in the field of tracking or even the wider area of filtering the a posteriori description of the uncertainty can oftentimes not be described by a simple Gaussian density function. In such situations the characterization of the uncertainty by a mean and a covariance does not capture the true extent of the uncertainty at hand. For example, when the posterior is multi-modal with well separated narrow modes. Such descriptions naturally occur in applications like target tracking with terrain constraints or tracking of closely spaced multiple objects, where one cannot keep track of the objects identities. In such situations a co-variance measure as a description of the uncertainty is not appropriate anymore. In this paper we look at the use of entropy as an uncertainty description. We show how to calculate the entropy based on a running particle filter. We will verify the particle based approximation of the entropy numerically. We we also discuss theoretical convergence properties and provide some motivating examples.