Bayesian State Estimation Using Generalized Coordinates Bhashyam Balaji a , and Karl Friston b a Radar Systems Section, Defence Research and Development Canada–Ottawa, 3701 Carling Avenue, Ottawa, ON, Canada K1A 0Z4 b The Wellcome Trust Centre for Neuroimaging, Institute of Neurology, UCL, 12 Queen Square, London, WC1N 3BG, UK ABSTRACT This paper reviews a simple solution to the continuous-discrete Bayesian nonlinear state estimation problem that has been proposed recently. The key ideas are analytic noise processes, variational Bayes, and the formulation of the problem in terms of generalized coordinates of motion. Some of the algorithms, specifically dynamic expectation maximization and variational filtering, have been shown to outperform existing approaches like extended Kalman filtering and particle filtering. A pedagogical review of the theoretical formulation is presented, with an emphasis on concepts that are not as widely known in the filtering literature. We illustrate the appliction of these concepts using a numerical example. Keywords: Variational Filtering, Continuous-Discrete Filtering, Kolmogorov equation, Fokker-Planck equation, Dynamical Causal Modelling, Hierarchical dynamical models 1. INTRODUCTION The continuous-discrete Bayesian filtering problem is to estimate some state given the measurements, where the state is assumed to evolve according to a continuous-time stochastic process and the measurements are samples of a discrete-time stochastic process. 1 The conditional probability density function provides a complete probabilistic solution to the problem, and can be used to compute state estimators such as the conditional mean. Several approaches have been proposed in the literature. The standard extended Kalman filter is the bench- mark nonlinear filtering algorithm. It is based on the application of the linear Kalman filter to the model obtained via linearization of the nonlinear state and measurement models. Another related approach is the unscented Kalman filter. 2 Such approaches often work well for practical problems. However, they are not general solutions; for instance, they cannot model formally multi-modal posterior distributions. A more general solution is provided by particle filters. 3–5 They are based on sequential importance sampling based Monte-Carlo approximations based on point mass, or particle, representation of the probability densities. In principle, they provide a more general solution than the EKF or the UKF; for instance, they can describe multi-modal densities. However, the basic particle filter often requires too many particles, i.e., it succumbs to the “curse of dimensionality”, even for relatively benign models (e.g., linear model with unstable plant noise that is easily tackled using the KF). 6 Another approach proposed to tackling the continuous-discrete and continuous-continuous filtering problems is based on Feynman path integral methods that are used in quantum field theory. 7–9 The simplest path- integral approximation, the Dirac-Feynman approximation, has been shown to be sufficiently accurate for solving challenging problems. In this paper, we provide a pedagogical review of a novel Bayesian state estimation scheme recently proposed by Friston and collaborators. 10–12 The key concepts underlying this approach rests on an analytic noise process (rather than Wiener process), variational Bayes, 13, 14 and the formulation of the problem in terms of generalized coordinates. It has been shown to be very versatile and robust, and has also been successfully applied to simulated models, as well as real data in the problem of deconvolving hemodynamic states and neuronal activity Further author information: (Send correspondence to Bhashyam Balaji) E-mail: Bhashyam.Balaji@drdc-rddc.gc.ca, Telephone: 1 613 998 2215 Signal Processing, Sensor Fusion, and Target Recognition XX, edited by Ivan Kadar, Proc. of SPIE Vol. 8050, 80501Y · © 2011 SPIE · CCC code: 0277-786X/11/$18 · doi: 10.1117/12.883513 Proc. of SPIE Vol. 8050 80501Y-1 Downloaded from SPIE Digital Library on 11 Nov 2011 to 193.62.66.144. Terms of Use: http://spiedl.org/terms