A nested particle filter for online Bayesian parameter estimation in state-space systems

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Description:	Miguez, J (Universidad Carlos III de Madrid) Wednesday 30 April 2014, 11:30-12:30


Created:	2014-05-07 09:37
Collection:	Advanced Monte Carlo Methods for Complex Inference Problems
Publisher:	Isaac Newton Institute
Copyright:	Miguez, J
Language:	eng (English)


Abstract:	We address the problem of approximating the probability measure of the fixed parameters of a state-space dynamic system using a sequential Monte Carlo method (SMC). The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability law of the static parameters and the dynamic variables of the system of interest, in the vein of the recent SMC^2 algorithm. However, different from the SMC^2 scheme, the proposed algorithm operates in a purely recursive manner and the scheme for the rejuvenation of the particles in the parameter space is simpler. We show analytical results on the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters computed via the proposed scheme. For a finite time horizon and under mild assumptions, we prove that the approximation errors vanish with the usual 1/?N rate, where N is the number of particles in the parameter space. Under a set of stronger assumptions related to (i) the stability of the optimal filter for the model, (ii) the compactness of the parameter and state spaces and (iii) certain bounds on the family of likelihood functions, we prove that the convergence of the approximation errors is uniform over time, and provide an explicit rate function. The uniform convergence result has some relevant consequences. One of them is that the proposed scheme can asymptotically identify the parameter values for a class of state-space models. A subset of the assumptions that yield uniform convergence also lead to a positive lower bound, uniform over time and the number of particles, on the normalized effective sample size the filter. We conclude with a simple numerical example that illustrates some of the theoretical findings

Abstract:

We address the problem of approximating the probability measure of the fixed parameters of a state-space dynamic system using a sequential Monte Carlo method (SMC). The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability law of the static parameters and the dynamic variables of the system of interest, in the vein of the recent SMC^2 algorithm. However, different from the SMC^2 scheme, the proposed algorithm operates in a purely recursive manner and the scheme for the rejuvenation of the particles in the parameter space is simpler. We show analytical results on the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters computed via the proposed scheme. For a finite time horizon and under mild assumptions, we prove that the approximation errors vanish with the usual 1/?N rate, where N is the number of particles in the parameter space. Under a set of stronger assumptions related to (i) the stability of the optimal filter for the model, (ii) the compactness of the parameter and state spaces and (iii) certain bounds on the family of likelihood functions, we prove that the convergence of the approximation errors is uniform over time, and provide an explicit rate function. The uniform convergence result has some relevant consequences. One of them is that the proposed scheme can asymptotically identify the parameter values for a class of state-space models. A subset of the assumptions that yield uniform convergence also lead to a positive lower bound, uniform over time and the number of particles, on the normalized effective sample size the filter. We conclude with a simple numerical example that illustrates some of the theoretical findings

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