case, the Bayes risk is not even well-defined, nor is there any well-defined distribution over . However, the posterior —and hence the expected loss—may be well-defined for each , so that it is still possible to define a generalized Bayes rule. Admissibility of (generalized) Bayes rules According to the complete class theorems, under mild conditions every admissible rule is a (generalized) Bayes rule (with respect to some prior —possibly an improper one—that favors distributions where that rule achieves low risk). Thus, in frequentist decision theory it is sufficient to consider only (generalized) Bayes rules. Conversely, while Bayes rules with respect to proper priors are virtually always admissible, generalized Bayes rules corresponding to improper priors need not yield admissible procedures. Stein's example is one such famous situation. Examples The James–Stein estimator is a nonlinear estimator of the mean of Gaussian random vectors which can be shown to dominate, or outperform, the ordinary least squares technique with respect to a mean-square error loss function. Thus least squares estimation is not an admissible estimation procedure in this context. Some others of the standard estimates associated with the normal distribution are also inadmissible: for example, the sample estimate of the variance when the population mean and variance are unknown. Notes References Bayesian statistics Optimal decisions