takes the limit of π$_{k}$ ( θ ) as k → ∞ . Additional care is required when the parameter space Θ is unbounded, since in that case the prior that maximizes I$_{k}$ is often improper, and I$_{k}$ is undefined for improper priors. The solution is to define reference priors via their restrictions on arbitrary compact subsets Θ$_{ℓ}$ of Θ. Thus one is led to the formal definition of a reference prior for θ as any permissible prior π$_{R}$ ( θ ) that satisfies the so-called maximizing missing information (MMI) property, namely that for any compact set Θ$_{ℓ}$ and candidate prior π ( θ ), where π$_{R,ℓ}$ and π$_{ℓ}$ are the renormalized restrictions of π$_{R}$ and π to Θ$_{ℓ}$. A candidate prior is a standard prior that incorporates any prior knowledge about θ . lim k →∞ [ I$_{k}$ { π$_{R,ℓ}$ } − I$_{k}$ { π$_{ℓ}$ } ] ≥ 0 (13) ∫ p ( x$_{(}$$_{k}$$_{)}$ θ ) h ( θ ) dθ where θ$_{0}$ is an arbitrary fixed point in Θ, h ( θ ) is any continuous, strictly positive function, such as h ( θ ) = 1, and p ( x$_{(}$$_{k}$$_{)}$ | θ ) = ∏ k i =1 p ( x$_{i}$ | θ ) is the probability model for a sample of k independent observations. We emphasize that this constructive definition only guarantees that the MMI property (13) is satisfied. The permissibility part of the reference prior definition must be separately verified. However, the proponents of reference priors view the MMI property as considerably more important than permissibility [16], and also believe that it would be highly unusual for a prior satisfying the MMI property to fail permissibility [11] (counter-examples are known, but they are rather exotic). π$_{R}$ ( θ ) = lim k →∞ π$_{k}$ ( θ ) π$_{k}$ ( θ$_{0}$ ) , with π$_{k}$ ( θ ) = exp {∫ p ( x$_{(}$$_{k}$$_{)}$ | θ ) ln [ p ( x$_{(}$$_{k}$$_{)}$ | θ ) h ( θ ) p ( x$_{(}$$_{k}$$_{)}$ θ ) h ( θ ) dθ ] dx$_{(}$$_{k}$$_{)}$ } , (14) A key result is the following constructive definition of the reference prior π$_{R}$ ( θ ) [16], A further useful result that we shall exploit is that, when certain regularity conditions are met — essentially those that guarantee asymptotic normality of the posterior — the reference prior for models with one continuous parameter reduces to the well-known Jeffreys prior [4], π$_{R}$ ( θ ) = √ E [ − d 2 dθ 2 ln p ( x | θ ) ] , (15) where the expectation is taken with respect to the sampling model p ( x | θ ). In general the analytical derivation of reference priors can be extremely challenging. However, Eq. (14) is amenable to numerical integration [16]. 9