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This paper contains an analysis of the performance of Bayes conditional-mean parameter estimators. The main result is that on a finite parameter space such estimates exhibit a mean-square error that diminishes exponentially with the number of observations, the observations being assumed to be independent. Two situations are discussed: true parameter included in the parameter space and true parameter not included in the parameter space. In the former instance only very general assumptions are required to demonstrate the exponential convergence rate. In the latter case the existence of an information function must be invoked. Comments on the continuous-parameter-space realization of the estimator and a discussion of the convergence mechanism are also included.