Recursive computation of the score and observed information matrix in hidden Markov models

Hidden Markov Models (henceforth abbreviated to HMMs), taken in their most general acception, that is, including models in which the state space of the hidden chain is continuous, have become a widely used class of statistical models with applications in diverse areas such as communications, engineering, bioinformatics, econometrics and many more. This contribution focus on the computation of derivatives of the log-likelihood and proposes a (comparatively!) simple and general framework, based on the use of Fisher and Louis identities, to obtain recursive equations for computing the score and observed information matrix. This approach is thought to be simpler than (although equivalent to) the solution provided by the so-called sensitivity equations. It is based on the original remark that recursive smoothers for HMMs are also available for some functionals of the hidden states which do not reduce to sum functionals. This view of the problem also suggests ways in which these exact equations could be approximated using sequential Monte Carlo methods.