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A discrete-time nonlinear filtering lower bound algorithm is given for evaluating the error in a desired function of the state vector. The algorithm is based on a rate distortion bound derived previously by the author. The problem is formulated in terms of Monte Carlo analysis. The theory of backward Markovian models is used to evaluate the conditional expectation appearing in the Bucy representation for the ease of Gauss-Markov signal models. An approximation procedure is given for the case of nonlinear signal models. In comparison with the author's previous bound the bound algorithm obtained here is tighter and does not require the difficult computation of the entropy of the state vector.