Robust Sequential Analysis
Final Report Abstract
A fast inference which is reliable even if the underlying model is not completely known is important in most signal processing applications. Fast inference naturally calls for sequential methods, whereas a reliable inference under model uncertainty requires robust methods. Over the past decades, there has been a lot of research in both fields, sequential inference and robust inference. However, until now, there are only a few results on the combination of both fields, i.e., on robust and sequential inference procedures. The aim of this project was to develop a unified framework for analyzing and implementing robust sequential procedures that covers robust sequential hypothesis tests as well as robust procedures for the problem of simultaneous hypothesis testing and parameter estimation. During this project, we have developed a framework for minimax robust sequential hypothesis tests for multiple simple hypotheses. We have shown that least favorable distributions, i.e., distributions that maximize the expected number of used samples and the error probabilities, are given by the minimizers of an f -dissimilarity. The optimal testing policy has been characterized by a non-linear integral equation that induces this f -dissimilarity. Based on these results, a set of sufficient conditions for minimax optimal sequential hypothesis tests has been derived. In a variety of applications, hypothesis testing and parameter estimation occur in a coupled way and both are of primary interest. We have investigated this problem in a sequential framework. That is, we have proposed a framework to design a sequential procedure that simultaneously infers the true hypothesis as well as the true parameter by using on average as few samples as possible and keeping the detection and estimation errors below predefined levels. The optimal procedure has been characterized by a system of non-linear and recursively defined cost functions. We have further developed algorithms to obtain the coefficients that parameterize the cost functions such that the constraints on the detection and estimation errors are fulfilled and the resulting procedure uses indeed on average a minimum number of samples. Deriving the optimal procedure for the problem of sequential joint detection and estimation requires the evaluation of the recursively defined cost function and, hence, can be computationally tedious for more complex scenarios. Therefore, we have developed an asymptotically pointwise optimal (APO) procedure for the problem of sequential joint detection and estimation. That is, a procedure that becomes optimal when the constraints detection and estimation errors are sufficiently small. The proposed APO stopping rule can be simply implemented by comparing the instantaneous cost to a time-varying threshold, whereas the strictly optimal stopping rule is induced by the recursively defined cost functions. This easy implementation enables the application to real-world problems. We have further shown that even for moderate constraints on the detection and estimation errors, the strictly optimal procedure requires only slightly less samples than the APO procedure.
Publications
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“Bayesian Sequential Joint Detection and Estimation.” In: Sequential Analysis 37.04 (2018), pp. 530–570
D. Reinhard, M. Fauß, and A. M. Zoubir
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“Minimax optimal sequential hypothesis tests for Markov processes.” In: Ann. Statist. 48.5 (Oct. 2020), pp. 2599–2621
M. Fauß, A. M. Zoubir, and H. V. Poor
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“Minimax Robust Detection: Classic Results and Recent Advances.” In: IEEE Transactions on Signal Processing (2021), pp. 1–1
M. Fauss, A. M. Zoubir, and H. V. Poor
