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Suppose $L$ simultaneous independent stochastic systems generateobservations, where the observations from each system depend on the underlyingparameter of that system. The observations are unlabeled (anonymized), in thesense that an analyst does not know which observation came from whichstochastic system. How can the analyst estimate the underlying parameters ofthe $L$ systems? Since the anonymized observations at each time are anunordered set of L measurements (rather than a vector), classical stochasticgradient algorithms cannot be directly used. By using symmetric polynomials, weformulate a symmetric measurement equation that maps the observation set to aunique vector. By exploiting that fact that the algebraic ring ofmulti-variable polynomials is a unique factorization domain over the ring ofone-variable polynomials, we construct an adaptive filtering algorithm thatyields a statistically consistent estimate of the underlying parameters. Weanalyze the asymptotic covariance of these estimates to quantify the effect ofanonymization. Finally, we characterize the anonymity of the observations interms of the error probability of the maximum aposteriori Bayesian estimator.Using Blackwell dominance of mean preserving spreads, we construct a partialordering of the noise densities which relates the anonymity of the observationsto the asymptotic covariance of the adaptive filtering algorithm.

Adaptive Filtering Algorithms for Set-Valued Observations -- Symmetric Measurement Approach to Unlabeled and Anonymized Data