Biometrika {I9$l),68, 3, pp. 696-8 6 9 5 Printed in Great Britain

This note investigates possible extensions of Fisher's measure of information to the case where there is prior knowledge. Some key u»nfe:Cram6r-Rao inequality; Fisher information. Let {p(x 18); 8 e 0} denote a family of densities depending on a single parameter 6. Let X be the sample space which is assumed to be a Euclidean space and let 0 be the parameter space. We assume that 0 = (a, b), a real open interval, finite or not. Further, we assume that a proper prior density £(d) exists and we denote by £(d \ x) oc p(x \ 8) £(0) the corresponding posterior density function. The expectations with respect to the likelihood, prior and posterior distributions are denoted by Ex\e, E6 and Eg\x, respectively. The marginal density of the data is denoted by q(x). Further, in the following we require some regularity conditions, basically differentiability of p(x \ 8) and £(0) with respect to 8 and convergence and interchangeability of some integrals and limits. If 8(x) is a point estimate of 8 and if f(x, 6) = p(x \ 8) £(6), then it follows that = [Exle{(d-0)£(d)}]ba-[[{6-8)^{\ogf)fd8dx. (1) Now, if the bias of § is denoted by B(6) = Ex]e(d-d) and i