Perturbation analysis for parameter estimation in continuous space HMMs

This paper is concerned with parameter identification in continuous space hidden Markov models. We provide a sensitivity analysis on a filtering density with respect to some model parameters to obtain the gradient of the log-likelihood function. The resulting estimator is combined with an efficient variance reduction technique in a stochastic gradient ascent method to identify the parameter. The main ideas consist in using a non-naive Finite Difference approach and an Infinitesimal Perturbation Analysis by defining an augmented Markov chain which allows to express the gradient of the filtering density as an expectation of a functional along this chain. This provides a general methodology that enables us to transform any algorithm that estimates the filtering density, such as interacting particle methods, into an algorithm that estimates its gradient. We propose therefore two algorithms, which are proven to be consistent and whose computational complexity is linear in the number of particles. Numerical experiments show their respective performances in terms of variance, accuracy, and computational complexity.