that yields a proper posterior. A permissible prior is a standard prior for which the posterior is the expected logarithmic limit of a sequence of posteriors defined by restriction to compact sets. Reference priors make use of the notion of expected intrinsic information. For one obser- vation from a model p ( x | θ ), the expected intrinsic information about the value of θ when the prior is π ( θ ) is given by the functional The more informative the observation, the greater the expected separation between the posterior and the prior. The larger this separation, the greater the expected intrinsic infor- mation I { π } . Thus, I { π } measures the amount of information about the value of θ that might be expected from one observation when the prior is π ( θ ). Suppose next that we make k independent observations x$_{(}$$_{k}$$_{)}$ = { x$_{1}$, x$_{2}$, . . . , x$_{k}$ } from the model p ( x | θ ). The definition of expected intrinsic information can be generalized to include all k observations: where the expectation is a k -dimensional integral over x$_{(}$$_{k}$$_{)}$ weighted by: As the sample size k grows larger, one expects the amount of information about θ to increase, and in the limit k → ∞ , the true value of θ would become exactly known. In this sense, the limit I ∞ { π } ≡ lim$_{k}$ →∞ I$_{k}$ { π } represents the missing information about θ when π ( θ ) is the prior. This concept of missing information is central to the construction of reference priors. The goal of reference analysis is to contruct a prior that maximizes the missing informa- tion. This maximization cannot be done directly however, because I ∞ typically diverges. To avoid this problem, one first constructs the prior π$_{k}$ ( θ ) that maximizes I$_{k}$ , and then B. Reference Priors for One-Parameter Models 8 m ( x$_{(}$$_{k}$$_{)}$ ) ≡ ∫ p ( x$_{(}$$_{k}$$_{)}$ | θ ) π ( θ ) dθ = ∫ [ k ∏ i =1 p ( x$_{i}$ | θ ) ] π ( θ ) dθ. (12) I$_{k}$ { π } = E$_{m}$ { D [ π ( θ | x$_{(}$$_{k}$$_{)}$ ) , π ( θ )] } , (11) I { π } = E$_{m}$ { D [ π ( θ | x ) , π ( θ )] } . (10) A. The Concept of Missing Information