Sequential Monte Carlo methods

In Chapter 2 we introduced the filtering recursion for general state space models (Proposition 2.1). The recursive nature of the algorithm that, from the filtering distribution at time t?1 and the observation yt computes the filtering distribution at time t, makes it ideally suited for a large class of applications in which inference must be made online, before the data collection ends. For those types of applications one must have, at any time, an up-to-date estimate of the current state of the system. Standard examples of such online types of applications include the following: tracking the position and speed of a moving aircraft observed through a radar; monitoring the location and characteristics of a storm based on satellite data; estimating the volatility of the prices of a group of stocks from tick-to-tick data. Unfortunately, for a general state space model the integrations in (2.7) cannot be carried out analytically. DLMs are a special case for which the Kalman filter gives a closed form solution to the filtering problem. However, even in this case, as soon as a DLM contains unknown parameters in its specification, the Kalman filter alone is not enough to compute the filtering distribution and, except in a few simple cases (see Section 4.3.1) one has to resort to numerical techniques.