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In this paper, we consider the problem of actively providing an estimate of the state of a stochastic dynamic system over a (possibly long) finite time horizon. The active estimation problem (AEP) is formulated as a stochastic optimal control one, in which the minimization of a suitable uncertainty measure is carried out. Toward this end, the use of the Renyi entropy as an information measure is proposed and motivated. A neural control scheme, based on the application of the extended Ritz method (ERIM) and on the use of a Gaussian sum filter (GSF), is then presented. Simulation results show the effectiveness of the proposed approach.