Title: Analysis of a nonlinear importance sampling scheme for Bayesian parameter estimation in state-space models,
Abstract: The Bayesian estimation of the unknown parameters of state-space (dynamical)
systems has received considerable attention over the past decade, with a
handful of powerful algorithms being introduced. In this paper we tackle the
theoretical analysis of the recently proposed {\it nonlinear} population Monte
Carlo (NPMC). This is an iterative importance sampling scheme whose key
features, compared to conventional importance samplers, are (i) the approximate
computation of the importance weights (IWs) assigned to the Monte Carlo samples
and (ii) the nonlinear transformation of these IWs in order to prevent the
degeneracy problem that flaws the performance of conventional importance
samplers. The contribution of the present paper is a rigorous proof of
convergence of the nonlinear IS (NIS) scheme as the number of Monte Carlo
samples, $M$, increases. Our analysis reveals that the NIS approximation errors
converge to 0 almost surely and with the optimal Monte Carlo rate of
$M^{-\frac{1}{2}}$. Moreover, we prove that this is achieved even when the mean
estimation error of the IWs remains constant, a property that has been termed
{\it exact approximation} in the Markov chain Monte Carlo literature. We
illustrate these theoretical results by means of a computer simulation example
involving the estimation of the parameters of a state-space model typically
used for target tracking.