Newton-based maximum likelihood estimation in nonlinear state space models

Maximum likelihood (ML) estimation using Newton optimization in nonlinear state space models (SSMs) is a challenging problem due to the analytical intractability of the log-likelihood and its gradient and Hessian. We compute the gradient using Fisher's identity in combination with a smoothing algorithm and estimate the Hessian using the output from this procedure. We explore two approximations of the log-likelihood and the solution of the smoothing problem, using linearization and using sampling methods. The linearization approximations are computationally cheap, but the accuracy of the approximations typically varies between models. The sampling approximations can be applied to any SSM, but come with a higher computational cost. We demonstrate our approach for ML parameter estimation on simulated data from two different SSMs with encouraging results.