Deterministic and Stochastic Particle Filters in State-Space Models

Optimal or Bayesian filtering in state-space models is a question of computing series of linked numerical integrals where output from one is input to the other (Bucy and Senne 1971). Particle filtering can be regarded as comprising techniques for solving these integrals by replacing the complicated posterior densities involved by discrete approximations, based on particles (Kitagawa 1996). There is evidence that the numerical errors as the process is iterated often stabilise, or at least do not accumulate sharply (see section 5.2.5). Filters of this type can be constructed in many ways. Most of the contributions to this volume employ Monte Carlo designs (see also (Doucet 1998) and the references therein). Particles are then random drawings of state vectors under the current posterior. This amounts to Monte Carlo evaluations of integrals. Numerically inaccurate, but often practical and easy to implement, general methods to run the sampling have been developed. Alternatively, particles can be laid out through a deterministic plan, using more sophisticated and more accurate numerical integration techniques. Such an approach has been discussed in (Kitagawa 1987), (Pole and West 1988) and (Pole and West 1990), but recently most work has been based on Monte Carlo methods. To some extent, Monte Carlo and deterministic particle filters are complementary approaches, and one may also wonder whether they may be usefully combined (see (Monahan and Genz 1997) for such a combination in a non-dynamic setting). Emphasis in this paper is on deterministic filtering. A general framework can be found in the above-mentioned references and in (West and Harrison 1997)[Section 13.5]. We shall present a common perspective in the next section, where our contribution will be on design issues.