We consider fixed-interval smoothing problems for counter-adversarial autonomous systems. An adversary deploys an autonomous filtering and control system that i) measures our current state via a noisy sensor, ii) computes a posterior estimate (belief) and iii) takes an action that we can observe. Based on such observed actions and our knowledge of our state sequence, we aim to estimate the adversary's past and current beliefs - this forms a foundation for predicting, and counteracting against, future actions. We derive the optimal smoother for the adversary's beliefs (we treat the problem in a Bayesian framework). Moreover, we demonstrate how the smoother can be computed for discrete systems even though the corresponding backward variables do not admit a finite-dimensional characterization. Finally, we illustrate our results in numerical simulations.

Can the parameters of a hidden Markov model (HMM) be estimated from a single sweep through the observations – and additionally, without being trapped at a local optimum in the likelihood surface? That is the premise of recent method of moments algorithms devised for HMMs. In these, correlations between consecutive pair- or triplet-wise observations are empirically estimated and used to compute estimates of the HMM parameters. Albeit computationally very attractive, the main drawback is that by restricting to only low-order correlations in the data, information is being neglected which results in a loss of accuracy (compared to standard maximum likelihood schemes). In this paper, we propose extending these methods (both pair- and triplet-based) by also including non-consecutive correlations in a way which does not significantly increase the computational cost (which scales linearly with the number of additional lags included). We prove strong consistency of the new methods, and demonstrate an improved performance in numerical experiments on both synthetic and real-world financial time-series datasets.

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.

We consider estimating the transition probability matrix of a finite-state finite-observation alphabet hidden Markov model with known observation probabilities. We propose a two-step algorithm: a method of moments estimator (formulated as a convex optimization problem) followed by a single iteration of a Newton-Raphson maximum-likelihood estimator. The two-fold contribution of this letter is, first, to theoretically show that the proposed estimator is consistent and asymptotically efficient, and second, to numerically show that the method is computationally less demanding than conventional methods-in particular for large datasets.