Optimal Smoothing for Finite State Hidden Reciprocal Processes

This technical note addresses modelling and estimation of a class of finite state random processes called hidden reciprocal chains (HRC). A hidden reciprocal chain consists of a finite state reciprocal process, together with an observation process conditioned on the reciprocal process much as in the case of a hidden Markov model (HMM). The key difference between Markov models and reciprocal models is that reciprocal models are non-causal. The technical note presents a characterization of a HRC by a finite set of hidden Markov bridges, which are HMMs with the final state fixed. The technical note then uses this characterization to derive the optimal fixed interval smoother for a HRC. Performance of linear and optimal smoothers derived for both HMM and HRC are compared (using simulations) for a class of HRC derived from underlying Markov transitions. These experiments suggest that, not surprisingly, the performance of the optimal HMM and HRC smoothers are signifcantly better than their linear counterparts, and that some performance improvement is obtained using the HRC smoothers compared to the HMM smoothers. The technical note concludes by mentioning some ongoing and future work which exploits this new Markov bridge characterization of a HRC.