of the form ( ϕ, ν ) → ( ϕ, λ ), where λ is a function of both ϕ and ν . Then the reference posterior for ϕ is unchanged by the transformation. Reference priors on unbounded parameter spaces are usually improper, which invalidates the application of Bayes’ theorem. To circumvent this problem one introduces a nested sequence of compact subsets Θ$_{1}$ ⊂ Θ$_{2}$ ⊂ . . . of the parameter space Θ, such that Θ$_{ℓ}$ → Θ as ℓ → ∞ . Given an improper prior π ( θ ), its restriction to Θ$_{ℓ}$ will be proper, so that Bayes’ theorem can be applied to construct the corresponding restricted posterior π$_{ℓ}$ ( θ | x ). The unrestricted posterior for the entire parameter space is then defined by the limit of the π$_{ℓ}$ ( θ | x ) as ℓ → ∞ . The practical justification for this procedure is that one often knows the shape, but not the size, of the physical region of parameter space where the prior has nonzero weight. As this size is typically very large, the limiting posterior can be viewed as an approximation to the posterior on the physical region. Interestingly, the limiting posterior can also be obtained by direct, formal application of Bayes’ theorem to the improper prior π ( θ ), provided the marginal distribution of the data, is finite. It can then be shown that the restricted posteriors π$_{ℓ}$ ( θ | x ) converge logarithmically to their limit π ( θ | x ): where is the Kullback-Leibler divergence between p ( θ ) and q ( θ ). This divergence is a parametrization-independent, non-negative measure of the separation between two densi- ties; it is zero if and only if the densities are identical. Unfortunately, pointwise logarithmic convergence is not enough to avoid inferential inconsistency in some special cases [16], so that a stronger form of convergence is needed, expected logarithmic convergence: where the expectation is taken with respect to the marginal density m ( x ). The above discussion motivates the following terminology [16]. Given a statistical model on a parameter space Θ, a standard prior is a strictly positive and continuous function on Θ 7 lim ℓ →∞ D [ π$_{ℓ}$ ( θ | x ) , π ( θ | x )] = 0 , (7) D [ p ( θ ) , q ( θ )] ≡ ∫ q ( θ ) log q ( θ ) p ( θ ) dθ (8) lim ℓ →∞ E$_{m}$ { D [ π$_{ℓ}$ ( θ | x ) , π ( θ | x )] } = 0 , (9) m ( x ) ≡ ∫ p ( x | θ ) π ( θ ) dθ, (6)