Simulated Pseudo Maximum Likelihood Identification of Nonlinear Models

Nonlinear stochastic parametric models are widely used in various fields. However, for these models, the problem of maximum likelihood identification is very challenging due to the intractability of the likelihood function. Recently, several methods have been developed to approximate the analytically intractable likelihood function and compute either the maximum likelihood or a Bayesian estimator. These methods, albeit asymptotically optimal, are computationally expensive. In this contribution, we present a simulation-based pseudo likelihood estimator for nonlinear stochastic models. It relies only on the first two moments of the model, which are easy to approximate using Monte-Carlo simulations on the model. The resulting estimator is consistent and asymptotically normal. We show that the pseudo maximum likelihood estimator, based on a multivariate normal family, solves a prediction error minimization problem using a parameterized norm and an implicit linear predictor. In the light of this interpretation, we compare with the predictor defined by an ensemble Kalman filter. Although not identical, simulations indicate a close relationship. The performance of the simulated pseudo maximum likelihood method is illustrated in three examples. They include a challenging state-space model of dimension 100 with one output and 2 unknown parameters, as well as an application-motivated model with 5 states, 2 outputs and 5 unknown parameters.