Paper:
Inverse Filtering for Hidden Markov Models
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.

Review:
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. Fig 2) using spherical K-means as a subroutine in the noisy case.
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. posteriors.  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?  
Some additional details are provided on page 8, but this is not enough for me to fully understand the rationale for the approach.  One is given posterior state estimates of sleep stage for a new patient and the goal is to provide the corresponding mean EEG frequency observations?  I don't see the point.
Minor terminological question: I've always viewed the HMM to be a type of sequential graphical model structure allowing any type of random variable (discrete or continuous) of which discrete HMMs and Kalman filters (continuous Gaussian) are special cases.  Does the terminology "HMM" really assume discrete random variables?
Overall, though the motivation for the work is not very strong for me, the results in this paper address a novel problem, and make technically innovative and honestly surprising insights in terms of the tractability and uniqueness of the recovery problems addressed in the paper.  I see this paper as a very technically interesting solution that may find future application in use cases the authors have not considered.