ABSTRACT
In the mathematical modeling of physical and biological systems, situations often arise where some facet of the underlying dynamics (in the form of a parameter) is not constant but rather is distributed probabilistically within the system or across the population under study. While traditional inverse problems (as discussed in Chapter 3) involve the estimation, given a set of data/observations, of a fixed set of parameters contained within some finite dimensional admissible set, models with distributed parameters require the estimation of a probability measure or distribution over the set of admissible parameters. This problem is well known to both applied mathematicians and statisticians and both schools have developed their own set of tools for the estimation of a measure or distribution over a set. Not surprisingly, the methods developed by the two schools are best suited to the particular types of problems most frequently encountered in their respective fields. As such, the techniques for measure estimation (along with their theoretical justifications) are widely scattered throughout the literature in applied mathematics and statistics, often with few cross references to related ideas; a more complete review is given in [25]. In this chapter we present two approaches to these techniques, paying par-
ticular attention to the theoretical underpinnings of the various methods as well as to computational issues which result from the theory. Throughout this chapter, we will focus on the nonparametric estimation of a probability measure. That is, we want to establish a meaningful estimation problem while simultaneously placing a minimal set of restrictions on the underlying measure. The fully parametric case (that is, when the form of the underlying measure is assumed to be known and determined uniquely by a small number of parameters – for example, its mean and covariance matrix) can be considered as a standard parameter estimation problem over Euclidean space for which computational strategies are well known and readily available. Some of these were discussed in Chapter 3.
