Approximate inference of nonparametric Hammerstein models

We propose a method for nonparametric identification of Hammerstein models with Gaussian-process models for the impulse response of the linear block and for the input nonlinearity. Interpreting the Gaussian-processes as prior distributions, we can estimate the unknowns using the posterior means given the data. To estimate the hyperparameters we set up an iterative scheme, reminiscent of the expectation-maximization method, where the posterior expectation of the complete likelihood is iteratively maximized. In the Hammerstein case, the posterior density is intractable because, in general, it does not admit a closed form expression. In this work, we propose two approximation approaches to estimate the posterior mean. In the first, we make a particle approximation of the posterior using Markov Chain Monte Carlo. In the second, we use a variational Bayes approach with a mean-field hypothesis. We validate the proposed methods on synthetic datasets of Hammerstein systems.