This paper considers a number of related inverse filtering problems for hidden Markov models (HMMs). In particular, given a sequence of state posteriors and the system dynamics; i) estimate the corresponding sequence of observations, ii) estimate the observation likelihoods, and iii) jointly estimate the observation likelihoods and the observation sequence. We show how to avoid a computationally expensive mixed integer linear program (MILP) by exploiting the algebraic structure of the HMM filter using simple linear algebra operations, and provide conditions for when the quantities can be uniquely reconstructed. We also propose a solution to the more general case where the posteriors are noisily observed. Finally, the proposed inverse filtering algorithms are evaluated on real-world polysomnographic data used for automatic sleep segmentation.

The paper addresses recovery of the observation sequence given known posterior state estimates, but unknown observations and/or sensor model and also in an extension, noise-corrupted measurements. There is a nice progression of the problem through IP, LP, and MILP followed by a more careful analytical derivation of the answers in the noise-free case, and a seemingly approximate though empirically effective approach (cf. Honestly, most of the motivations seem to be unrealistic, especially the cyber-physical security setting where one does not observe posteriors, but simply an action based on a presumed argmax w.r.t. The EEG application (while somewhat narrow) seems to be the best motivation, however, the sole example is to compare resconstructed observations to a redundant method of sensing -- is this really a compelling application? Is it actually used in practice?

This paper considers a number of related inverse filtering problems for hidden Markov models (HMMs). In particular, given a sequence of state posteriors and the system dynamics; i) estimate the corresponding sequence of observations, ii) estimate the observation likelihoods, and iii) jointly estimate the observation likelihoods and the observation sequence. We show how to avoid a computationally expensive mixed integer linear program (MILP) by exploiting the algebraic structure of the HMM filter using simple linear algebra operations, and provide conditions for when the quantities can be uniquely reconstructed. We also propose a solution to the more general case where the posteriors are noisily observed. Finally, the proposed inverse filtering algorithms are evaluated on real-world polysomnographic data used for automatic sleep segmentation.