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The value of information in constrained parametric models

We consider the value of information introduced through constraints on the parameters, by showing the Fisher information gain and the resulting Cramér-Rao bound (CRB) reduction. In doing so, we show that the constrained CRB (CCRB) can be found by a transformation of parameters or a projection onto the tangent of the constraint space. Finally, we determine the characteristics of optimal constraints that minimize the trace of the CCRB, thereby attaining the lowest bound on the sum of mean-square errors of unbiased estimators.