Construction of nonlinear filter algorithms using the saddlepoint approximation

Abstract

In this thesis we propose the use of the saddlepoint method to construct nonlinear filtering algorithms. To our knowledge, while the saddlepoint approximation has been used very successfully in the statistics literature (as an example the saddlepoint method provides a simple, highly accurate approximation to the density of the maximum likelihood estimator of a non-random parameter given a set of measurements), its potential for use in the dynamic setting of the nonlinear filtering problem has yet to be realized. This is probably because the assumptions on the form of the integrand that is typical in the asymptotic analysis literature do not necessarily hold in the filtering context. We show that the assumptions typical in asymptotic analysis (and which are directly applicable in statistical inference since the statistics applications usually involve estimating the density of a function of a sequence of random variables) can be modified in a way that is still relevant in the nonlinear filtering context while still preserving a property of the saddlepoint approximation that has made it very useful in statistical inference, namely, that the shape of the desired density is accurately approximated. As a result, the approximation can be used to calculate estimates of the mean and confidence intervals and also serves as an excellent choice of proposal density for particle filtering. We will show how to construct filtering algorithms based on the saddle point approximation.