Event-triggered smoothing for hidden Markov models: Risk-sensitive and MMSE results

An event-triggered smoothing problem for hidden Markov models (HMMs) is investigated in this paper. The transmission of the measurements is jointly determined by a stochastic event-triggering condition and a Gilbert–Elliott communication channel. Firstly, the event-triggered risk-sensitive smoothed estimate is characterized by constructing an augmented processing of the smoothed information state, which is given by the product of the forward recursive information state and the backward recursive information state under a reference measure. Secondly, the risk-neutral smoothed estimate (namely, the MMSE smoother) is proved to be a special case of the obtained risk-sensitive one when the risk-sensitive parameter approaches zero. The implementation issues of the obtained results are discussed by introducing an alternative smoothing algorithm that is numerically equivalent to the original algorithm. The effectiveness of the proposed results is evaluated through a numerical example and comparative simulations with a naive risk-sensitive smoother that treats unreceived information as packet dropout.