4 J.O.BERGER.J.M.BERNARDO AND D.SUN by p(x),with p(x)>0 and fap(x)dx =1,and the reference pos- terior distribution of 0ee given x will be represented by r(x),with π(0|x)≥0 and Jeπ(e|x)d0=l.This admittedly imprecise notation will greatly simplify the exposition.If the random vectors are discrete,these functions naturally become probability mass functions,and integrals over their values become sums.Density functions of specific distributions are de- noted by appropriate names.Thus,if z is an observable random quantity with a normal distribution of mean u and variance o2,its probability den- sity function will be denoted N(xu,o2);if the posterior distribution of A is Gamma with mean a/b and variance a/b2,its probability density func- tion will be denoted Ga(a,b).The indicator function on a set C will be denoted by 1c. Reference prior theory is based on the use of logarithmic divergence,often called the Kullback-Leibler divergence. DEFINITION 1.The logarithmic divergence of a probability density p() of the random vector 6ee from its true probability density p(),denoted by kp p,is p0d0, p()log0) provided the integral (or the sum)is finite. The properties of pp}have been extensively studied;pioneering works include Gibbs 22],Shannon 38],Good 24,25],Kullback and Leibler [35], Chernoff [15],Jaynes [29,30],Kullback [34]and Csiszar [18,19] DEFINITION 2 (Logarithmic convergence).A sequence of probability density functions [pi converges logarithmically to a probability density p if,and only if,limi(p pi)=0. 2.Improper and permissible priors. 2.1.Justifying posteriors from improper priors.Consider a model M= {p(x|e),x∈X,e∈Θ}and a strictly positive prior function m(e).(Were strict attention to strictly positive functions because any believably objective prior would need to have strictly positive density,and this restriction elim- inates many technical details.)When r()is improper,so that Je r()do diverges,Bayes theorem no longer applies,and the use of the formal poste- rior density (2.1) π(0|x)= p(x|0)π(0) J∫ep(x|0)π(0)d04 J. O. BERGER, J. M. BERNARDO AND D. SUN by p(x | θ), with p(x | θ) ≥ 0 and R X p(x | θ) dx = 1, and the reference pos￾terior distribution of θ ∈ Θ given x will be represented by π(θ | x), with π(θ | x) ≥ 0 and R Θ π(θ | x) dθ = 1. This admittedly imprecise notation will greatly simplify the exposition. If the random vectors are discrete, these functions naturally become probability mass functions, and integrals over their values become sums. Density functions of specific distributions are de￾noted by appropriate names. Thus, if x is an observable random quantity with a normal distribution of mean µ and variance σ 2 , its probability den￾sity function will be denoted N(x | µ,σ2 ); if the posterior distribution of λ is Gamma with mean a/b and variance a/b2 , its probability density func￾tion will be denoted Ga(λ | a,b). The indicator function on a set C will be denoted by 1C. Reference prior theory is based on the use of logarithmic divergence, often called the Kullback–Leibler divergence. Definition 1. The logarithmic divergence of a probability density ˜p(θ) of the random vector θ ∈ Θ from its true probability density p(θ), denoted by κ{p˜ | p}, is κ{p˜ | p} = Z Θ p(θ) log p(θ) p˜(θ) dθ, provided the integral (or the sum) is finite. The properties of κ{p˜ | p} have been extensively studied; pioneering works include Gibbs [22], Shannon [38], Good [24, 25], Kullback and Leibler [35], Chernoff [15], Jaynes [29, 30], Kullback [34] and Csiszar [18, 19]. Definition 2 (Logarithmic convergence). A sequence of probability density functions {pi}∞ i=1 converges logarithmically to a probability density p if, and only if, limi→∞ κ(p | pi) = 0. 2. Improper and permissible priors. 2.1. Justifying posteriors from improper priors. Consider a model M = {p(x | θ),x ∈ X ,θ ∈ Θ} and a strictly positive prior function π(θ). (We re￾strict attention to strictly positive functions because any believably objective prior would need to have strictly positive density, and this restriction elim￾inates many technical details.) When π(θ) is improper, so that R Θ π(θ) dθ diverges, Bayes theorem no longer applies, and the use of the formal poste￾rior density π(θ | x) = p(x | θ)π(θ) R Θ p(x | θ)π(θ) dθ (2.1)