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

Suppose
$\boldsymbol{L}$
simultaneous independent stochastic systems generate observations, where the observations from each system depend on the underlying model parameter of that system. The observations are unlabeled (anonymized), in the sense that an analyst does not know which observation came from which stochastic system. How can the analyst estimate the underlying model parameters of the
$\boldsymbol{L}$
systems? Since the anonymized observations at each time are an unordered set of
$\boldsymbol{L}$
measurements (rather than a vector), classical stochastic gradient algorithms cannot be directly used. By using symmetric polynomials, we formulate a symmetric measurement equation that maps the observation set to a unique vector. By exploiting the fact that the algebraic ring of multi-variable polynomials is a unique factorization domain over the ring of one-variable polynomials, we construct an adaptive filtering algorithm that yields a statistically consistent estimate of the underlying parameters. We analyze the asymptotic covariance of these estimates to quantify the effect of anonymization. Finally, we characterize the anonymity of the observations in terms of the error probability of the maximum aposteriori Bayesian estimator. Specifically using Blackwell dominance of mean preserving spreads, we construct a partial ordering of the noise densities which relates the anonymity of the observations to the asymptotic covariance of the adaptive filtering algorithm.