Continuous Time: Smoothing Algorithms

Abstract

In this chapter, we describe various algorithms for the smoothing problem in continuous time. We begin, in Section 7.1, by describing the Kalman–Bucy smoother , which applies in the case of linear dynamics when the initial conditions and the observational noise are Gaussian; the explicit Kalman–Bucy formulas are useful for the building of intuition. In Section 7.2, we discuss MCMC methods to sample from the smoothing distributions of interest. However, as in the discrete-time case, sampling the posterior can be prohibitively expensive. For this reason, it is of interest to identify the point that maximizes probability, using techniques from optimization , rather than explore the entire probability distribution—the variational method . This optimization approach is discussed in Section 7.3. Section 7.4 is devoted to numerical illustrations of the methods introduced in the previous sections. The chapter concludes with bibliographic notes in Section 7.5 and exercises in Section 7.6.