The Annals of Statistics 2009,Vol.37,No.2.905-938 D0:10.1214/07-A0S587 Institute of Mathematical Statistics,2009 THE FORMAL DEFINITION OF REFERENCE PRIORS BY JAMES O.BERGER.1 JOSE M.BERNARDO2 AND DONGCHU SUN3 Duke University,Universitat de Valencia and University of Missouri-Columbia Reference analysis produces objective Bayesian inference,in the sense that inferential statements depend only on the assumed model and the available data,and the prior distribution used to make an inference is least informative in a certain information-theoretic sense. Reference priors have been rigorously defined in specific contexts and heuristically defined in general,but a rigorous general definition has been lacking.We produce a rigorous general definition here and then show how an explicit expression for the reference prior can be ob- tained under very weak regularity conditions.The explicit expression can be used to derive new reference priors both analytically and nu- merically. 1.Introduction and notation. IA9S10'060:A!XIe 1.1.Background and goals.There is a considerable body of conceptual and theoretical literature devoted to identifying appropriate procedures for the formulation of objective priors;for relevant pointers see Section 5.6 in Bernardo and Smith [13],Datta and Mukerjee [20],Bernardo [11],Berger [3],Ghosh,Delampady and Samanta 23]and references therein.Refer- ence analysis,introduced by Bernardo [10]and further developed by Berger and Bernardo [4,5,6,7],and Sun and Berger [42],has been one of the most utilized approaches to developing objective priors;see the references in Bernardo [11]. Reference analysis uses information-theoretical concepts to make precise the idea of an objective prior which should be maximally dominated by the Received March 2007:revised December 2007. Supported by NSF Grant DMS-01-03265. 2Supported by Grant MTM2006-07801. 3Supported by NSF Grants SES-0351523 and SES-0720229. AMS 2000 subject classifications.Primary 62F15;secondary 62A01,62B10. Key words and phrases.Amount of information,Bayesian asymptotics,consensus pri- ors,Fisher information,Jeffreys priors,noninformative priors,objective priors,reference priors. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2009,Vol.37,No.2,905-938.This reprint differs from the original in pagination and typographic detail
arXiv:0904.0156v1 [math.ST] 1 Apr 2009 The Annals of Statistics 2009, Vol. 37, No. 2, 905–938 DOI: 10.1214/07-AOS587 c Institute of Mathematical Statistics, 2009 THE FORMAL DEFINITION OF REFERENCE PRIORS By James O. Berger,1 Jos´e M. Bernardo2 and Dongchu Sun3 Duke University, Universitat de Val`encia and University of Missouri-Columbia Reference analysis produces objective Bayesian inference, in the sense that inferential statements depend only on the assumed model and the available data, and the prior distribution used to make an inference is least informative in a certain information-theoretic sense. Reference priors have been rigorously defined in specific contexts and heuristically defined in general, but a rigorous general definition has been lacking. We produce a rigorous general definition here and then show how an explicit expression for the reference prior can be obtained under very weak regularity conditions. The explicit expression can be used to derive new reference priors both analytically and numerically. 1. Introduction and notation. 1.1. Background and goals. There is a considerable body of conceptual and theoretical literature devoted to identifying appropriate procedures for the formulation of objective priors; for relevant pointers see Section 5.6 in Bernardo and Smith [13], Datta and Mukerjee [20], Bernardo [11], Berger [3], Ghosh, Delampady and Samanta [23] and references therein. Reference analysis, introduced by Bernardo [10] and further developed by Berger and Bernardo [4, 5, 6, 7], and Sun and Berger [42], has been one of the most utilized approaches to developing objective priors; see the references in Bernardo [11]. Reference analysis uses information-theoretical concepts to make precise the idea of an objective prior which should be maximally dominated by the Received March 2007; revised December 2007. 1Supported by NSF Grant DMS-01-03265. 2 Supported by Grant MTM2006-07801. 3Supported by NSF Grants SES-0351523 and SES-0720229. AMS 2000 subject classifications. Primary 62F15; secondary 62A01, 62B10. Key words and phrases. Amount of information, Bayesian asymptotics, consensus priors, Fisher information, Jeffreys priors, noninformative priors, objective priors, reference priors. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Statistics, 2009, Vol. 37, No. 2, 905–938. This reprint differs from the original in pagination and typographic detail. 1
2 J.O.BERGER.J.M.BERNARDO AND D.SUN data,in the sense of maximizing the missing information (to be precisely defined later)about the parameter.The original formulation of reference priors in the paper by Bernardo [10]was largely informal.In continuous one parameter problems,heuristic arguments were given to justify an explicit expression in terms of the expectation under sampling of the logarithm of the asymptotic posterior density,which reduced to Jeffreys prior (Jeffreys [31,32])under asymptotic posterior normality.In multiparameter problems it was argued that one should not maximize the joint missing information but proceed sequentially,thus avoiding known problems such as marginal- ization paradoxes.Berger and Bernardo [7]gave more precise definitions of this sequential reference process,but restricted consideration to continuous multiparameter problems under asymptotic posterior normality.Clarke and Barron [17]established regularity conditions under which joint maximization of the missing information leads to Jeffreys multivariate priors.Ghosal and Samanta [27]and Ghosal [26]provided explicit results for reference priors in some types of nonregular models. This paper has three goals. GOAL 1.Make precise the definition of the reference prior.This has two different aspects. Applying Bayes theorem to improper priors is not obviously justifiable. Formalizing when this is legitimate is desirable,and is considered in Sec- tion 2. Previous attempts at a general definition of reference priors have had heuristic features,especially in situations in which the reference prior is improper.Replacing the heuristics with a formal definition is desirable, and is done in Section 3. GOAL 2.Present a simple constructive formula for a reference prior. Indeed,for a model described by density p(x6),where x is the complete data vector and 0 is a continuous unknown parameter,the formula for the reference prior,()will be shown to be π(0)=lim fk(0) k-→oof(00) f(0)= expp()logi"x where 0o is an interior point of the parameter space ,x()={x1,.. stands for k conditionally independent replications of x,and *(x()) is the posterior distribution corresponding to some fixed,largely arbitrary prior *()
2 J. O. BERGER, J. M. BERNARDO AND D. SUN data, in the sense of maximizing the missing information (to be precisely defined later) about the parameter. The original formulation of reference priors in the paper by Bernardo [10] was largely informal. In continuous one parameter problems, heuristic arguments were given to justify an explicit expression in terms of the expectation under sampling of the logarithm of the asymptotic posterior density, which reduced to Jeffreys prior (Jeffreys [31, 32]) under asymptotic posterior normality. In multiparameter problems it was argued that one should not maximize the joint missing information but proceed sequentially, thus avoiding known problems such as marginalization paradoxes. Berger and Bernardo [7] gave more precise definitions of this sequential reference process, but restricted consideration to continuous multiparameter problems under asymptotic posterior normality. Clarke and Barron [17] established regularity conditions under which joint maximization of the missing information leads to Jeffreys multivariate priors. Ghosal and Samanta [27] and Ghosal [26] provided explicit results for reference priors in some types of nonregular models. This paper has three goals. Goal 1. Make precise the definition of the reference prior. This has two different aspects. • Applying Bayes theorem to improper priors is not obviously justifiable. Formalizing when this is legitimate is desirable, and is considered in Section 2. • Previous attempts at a general definition of reference priors have had heuristic features, especially in situations in which the reference prior is improper. Replacing the heuristics with a formal definition is desirable, and is done in Section 3. Goal 2. Present a simple constructive formula for a reference prior. Indeed, for a model described by density p(x | θ), where x is the complete data vector and θ is a continuous unknown parameter, the formula for the reference prior, π(θ), will be shown to be π(θ) = lim k→∞ fk(θ) fk(θ0) , fk(θ) = expZ p(x (k) | θ)log[π ∗ (θ | x (k) )] dx (k) , where θ0 is an interior point of the parameter space Θ, x (k) = {x1,...,xk} stands for k conditionally independent replications of x, and π ∗ (θ | x (k) ) is the posterior distribution corresponding to some fixed, largely arbitrary prior π ∗ (θ)
DEFINITION OF REFERENCE PRIORS 3 The interesting thing about this expression is that it holds (under mild conditions)for any type of continuous parameter model,regardless of the asymptotic nature of the posterior.This formula is established in Section 4.1,and various illustrations of its use are given. A second use of the expression is that it allows straightforward compu- tation of the reference prior numerically.This is illustrated in Section 4.2 for a difficult nonregular problem and for a problem for which analytical determination of the reference prior seems very difficult. GOAL 3.To make precise the most common practical rationale for use of improper objective priors,which proceeds as follows: In reality,we are always dealing with bounded parameters so that the real parameter space should,say,be some compact set o. It is often only known that the bounds are quite large,in which case it is difficult to accurately ascertain which o to use. This difficulty can be surmounted if we can pass to the unbounded space and show that the analysis on this space would yield essentially the same answer as the analysis on any very large compact Oo Establishing that the analysis on is a good approximation from the refer- ence theory viewpoint requires establishing two facts: 1.The reference prior distribution on e,when restricted to eo,is the ref- erence prior on 0o. 2.The reference posterior distribution on e is an appropriate limit of the reference posterior distributions on an increasing sequence of compact sets f:converging to e. Indicating how these two facts can be verified is the third goal of the paper. 1.2.Notation.Attention here is limited mostly to one parameter prob- lems with a continuous parameter,but the ideas are extendable to the mul- tiparameter case through the sequential scheme of Berger and Bernardo [7]. It is assumed that probability distributions may be described through probability density functions,either in respect to Lebesgue measure or count- ing measure.No distinction is made between a random quantity and the particular values that it may take.Bold italic roman fonts are used for observable random vectors (typically data)and italic greek fonts for un- observable random quantities (typically parameters);lower case is used for variables and upper case calligraphic for their domain sets.Moreover,the standard mathematical convention of referring to functions,say fx and gx of xE,respectively by f(x)and g(x),will be used throughout.Thus,the conditional probability density of data x E&given 6 will be represented
DEFINITION OF REFERENCE PRIORS 3 The interesting thing about this expression is that it holds (under mild conditions) for any type of continuous parameter model, regardless of the asymptotic nature of the posterior. This formula is established in Section 4.1, and various illustrations of its use are given. A second use of the expression is that it allows straightforward computation of the reference prior numerically. This is illustrated in Section 4.2 for a difficult nonregular problem and for a problem for which analytical determination of the reference prior seems very difficult. Goal 3. To make precise the most common practical rationale for use of improper objective priors, which proceeds as follows: • In reality, we are always dealing with bounded parameters so that the real parameter space should, say, be some compact set Θ0. • It is often only known that the bounds are quite large, in which case it is difficult to accurately ascertain which Θ0 to use. • This difficulty can be surmounted if we can pass to the unbounded space Θ and show that the analysis on this space would yield essentially the same answer as the analysis on any very large compact Θ0. Establishing that the analysis on Θ is a good approximation from the reference theory viewpoint requires establishing two facts: 1. The reference prior distribution on Θ, when restricted to Θ0, is the reference prior on Θ0. 2. The reference posterior distribution on Θ is an appropriate limit of the reference posterior distributions on an increasing sequence of compact sets {Θi}∞ i=1 converging to Θ. Indicating how these two facts can be verified is the third goal of the paper. 1.2. Notation. Attention here is limited mostly to one parameter problems with a continuous parameter, but the ideas are extendable to the multiparameter case through the sequential scheme of Berger and Bernardo [7]. It is assumed that probability distributions may be described through probability density functions, either in respect to Lebesgue measure or counting measure. No distinction is made between a random quantity and the particular values that it may take. Bold italic roman fonts are used for observable random vectors (typically data) and italic greek fonts for unobservable random quantities (typically parameters); lower case is used for variables and upper case calligraphic for their domain sets. Moreover, the standard mathematical convention of referring to functions, say fx and gx of x ∈ X , respectively by f(x) and g(x), will be used throughout. Thus, the conditional probability density of data x ∈ X given θ will be represented
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)d0
4 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 posterior 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 denoted by appropriate names. Thus, if x is an observable random quantity with a normal distribution of mean µ and variance σ 2 , its probability density 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 function 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 restrict attention to strictly positive functions because any believably objective prior would need to have strictly positive density, and this restriction eliminates 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 posterior density π(θ | x) = p(x | θ)π(θ) R Θ p(x | θ)π(θ) dθ (2.1)
DEFINITION OF REFERENCE PRIORS 5 must be justified,even when fep(x )()de<oo so that t(x)is a proper density. The most convincing justifications revolve around showing that r(x) is a suitable limit of posteriors obtained from proper priors.A variety of versions of such arguments exist;cf.Stone [40,41]and Heath and Sudderth 28].Here,we consider approximations based on restricting the prior to an increasing sequence of compact sets and using logarithmic convergence to define the limiting process.The main motivation is,as mentioned in the introduction,that objective priors are often viewed as being priors that will yield a good approximation to the analysis on the "true but difficult to specify"large bounded parameter space. DEFINITION 3(Approximating compact sequence).Consider a paramet- ric model M={p(x|f),x∈，0∈O}and a strictly positive continuous functionπ(0)，0∈Θ，such that,.for all x∈X,fep(x|0)π(0)d0<o.An approximating compact sequence of parameter spaces is an increasing se- quence of compact subsets ofΘ，{ei}≌1，converging to e.The correspond- ing sequence of posteriors with support on ei,defined as [i(x)1,with Ti(6|x)xp(x|8)π(0)，Ti()=cπ(e)le,andc=Jaπ(e)d,is called the approximating sequence of posteriors to the formal posterior r(x). Notice that the renormalized restrictions mi()of (0)to the O;are proper [because the e;are compact and r(0)is continuous].The following theorem shows that the posteriors resulting from these proper priors do converge,in the sense of logarithmic convergence,to the posterior r(x). THEOREM 1.Consider model M={p(x|0),x∈X,0∈Θ}and a strictly positive continuous function n(0),such that fep(x0)(0)de<oo,for all xE X.For any approrimating compact sequence of parameter spaces,the corresponding approrimating sequence of posteriors converges logarithmi- cally to the formal posterior (x)p(x0)n(0) PROOF.To prove that k{(x)|i(x)}converges to zero,define the predictive densities pi(x)=Je,p(x)ni(0)do and p(x)=Jep(x)(0)do (which has been assumed to be finite).Using for the posteriors the expres- sions provided by Bayes theorem yields π(0|x)log le: 01风d0=e p(x)π：(0 _ao π(0|x) π(0|x)1og P(x)π(0 =。(01x)log p(x) d0 i(x)ci =log p(x) pi(x)ci 二log Jep(x0)(0)do Je.p(x0)(0)do
DEFINITION OF REFERENCE PRIORS 5 must be justified, even when R Θ p(x | θ)π(θ) dθ < ∞ so that π(θ | x) is a proper density. The most convincing justifications revolve around showing that π(θ | x) is a suitable limit of posteriors obtained from proper priors. A variety of versions of such arguments exist; cf. Stone [40, 41] and Heath and Sudderth [28]. Here, we consider approximations based on restricting the prior to an increasing sequence of compact sets and using logarithmic convergence to define the limiting process. The main motivation is, as mentioned in the introduction, that objective priors are often viewed as being priors that will yield a good approximation to the analysis on the “true but difficult to specify” large bounded parameter space. Definition 3 (Approximating compact sequence). Consider a parametric model M = {p(x | θ),x ∈ X ,θ ∈ Θ} and a strictly positive continuous function π(θ), θ ∈ Θ, such that, for all x ∈ X , R Θ p(x | θ)π(θ) dθ < ∞. An approximating compact sequence of parameter spaces is an increasing sequence of compact subsets of Θ, {Θi}∞ i=1, converging to Θ. The corresponding sequence of posteriors with support on Θi , defined as {πi(θ | x)}∞ i=1, with πi(θ | x) ∝ p(x | θ)πi(θ), πi(θ) = c −1 i π(θ)1Θi and ci = R Θi π(θ) dθ, is called the approximating sequence of posteriors to the formal posterior π(θ | x). Notice that the renormalized restrictions πi(θ) of π(θ) to the Θi are proper [because the Θi are compact and π(θ) is continuous]. The following theorem shows that the posteriors resulting from these proper priors do converge, in the sense of logarithmic convergence, to the posterior π(θ | x). Theorem 1. Consider model M = {p(x | θ),x ∈ X ,θ ∈ Θ} and a strictly positive continuous function π(θ), such that R Θ p(x | θ)π(θ) dθ < ∞, for all x ∈ X . For any approximating compact sequence of parameter spaces, the corresponding approximating sequence of posteriors converges logarithmically to the formal posterior π(θ | x) ∝ p(x | θ)π(θ). Proof. To prove that κ{π(· | x) | πi(· | x)} converges to zero, define the predictive densities pi(x) = R Θi p(x | θ)πi(θ) dθ and p(x) = R Θ p(x | θ)π(θ) dθ (which has been assumed to be finite). Using for the posteriors the expressions provided by Bayes theorem yields Z Θi πi(θ | x) log πi(θ | x) π(θ | x) dθ = Z Θi πi(θ | x) log p(x)πi(θ) pi(x)π(θ) dθ = Z Θi πi(θ | x) log p(x) pi(x)ci dθ = log p(x) pi(x)ci = log R Θ p(x | θ)π(θ) dθ R Θi p(x | θ)π(θ) dθ