Departament d’Estadística i I.O., Universitat de València.
Facultat de Matemàtiques, 46100–Burjassot, València, Spain.
Tel. 34.6.363.6048, Fax 34.6.363.6048 (direct), 34.6.386.4735 (office)Internet: bernardo@uv.es, Web: http://www.uv.es/˜bernardo/
Printed on December 5, 1996
To appear in J. Statist. Planning and Inference

Noninformative Priors Do Not Exist:
A Discussion with José M. Bernardo
PREAMBLE
José Bernardo, who is the originator and, with James Berger, a driving force behind the use ofwhat are known as “reference priors”, shared a ride with Telba Irony and Nozer Singpurwalla
from Pittsburgh, Pa. to Washington, D.C., sometime during the latter part of 1995. After theusual gossip, which statisticians of all persuasions tend to relish, the discussion turned to the
topic of “dishonest priors”, and their increasing encroachment on the Empire of Chance. The
discussion evolved in a Socratic tradition, with José playing the role of Socrates (albeit answering
the questions in this particular dialog), and his driving partners, the pupils; fortunately José was
not driving. This question and answer format of discussion turned out to be very fruitful,and even more enjoyable than gossip, because José’s knowledge of the topic and its historic
evolution, not to mention his passion for statistics (among other things), provided a usefulperspective on a topic of much controversy. José was then asked to write up this perspective asa contribution for the Discussion Forum of JSPI, to which his suggestion was that we reproduce
the dialogue and write it up as such. We thought that this suggestion was a great idea and so,when José visited Washington D.C. in May 1996, the dialogue was reconstructed. Given below
is an account of this reconstruction; we feel that those of us who are not cognoscenti about
dishonest priors may benefit from this conversational overview.
Telba Z. Irony
Nozer D. Singpurwalla
Question 1: It is often said that noninformative priors do not exist and yet under this label,plus many others, such as conventional, default, flat, formal, neutral, non-subjective, objective
and reference priors, they do get used. Does this imply that the users of noninformative priors
are really not honest Bayesians?
Answer: I would not say they that are dishonest but, too often, they are not precisely
aware of the implications of its use. By definition, “non-subjective prior distributions” are notintended to describe personal beliefs, and in most cases, they are not even proper probability
distributions in that they often do not integrate one. Technically, they are only positive functions
to be formally used in Bayes theorem to obtain “non-subjective posteriors” which, for a given
model, are supposed to describe whatever the data “have to say” about some particular quantityJosé M. Bernardo is Professor of Statistics at the University of Valencia. Research partially funded with grantPB93-1204 of the DGICYT, Madrid, Spain. The author gratefully acknowledges stimulating comments by Jim
Berger, Gauri Datta, Phil Dawid and Dennis Lindley to an earlier version of this manuscript.

A discussion on noninformative priors

2of interest, either a function of the parameters in the model, or a function of future observations.
Whether or not they achieve this goal or, indeed, whether or not this goal is at all achievable, isoften a matter of debate.
Question 2: Are non-subjective posteriors always proper?
Answer: They should be! Indeed, a proper posterior for any minimum-size sample shouldbe the first property to be required from any method of deriving non-subjective priors. But
you should realise that a naïve use of “standard” non-subjective priors may lead to improper
posteriors. Casella (1996) describes how, in a components of variance problem, this has lead tounaware users of Gibbs sampling to obtaining nice pictures of posterior distributions that do notexist! Actually, Dennis Lindley told me that he knew of this components of variance example
since the 50’s, and it is mentioned in Berger (1985, p. 187), but people seem to take for granted
a crucial property, —the propriety of the posterior— which must however be carefully checked
whenever an improper prior is used.
Question 3: Are there any non-subjective priors that are proper?
Answer: There are indeed. The simplest example is the Beta distribution Be(θ|1/2, 1/2),
widely regarded as the appropriate non-subjective prior to make inferences about the parameterθ of a binomial model. In fact, proper non-subjective priors are usually found whenever theparameter space is bounded, although this is not a general rule: the non-subjective prior typically
recommended for the parameter θ of a negative binomial model is π(θ) = θ−1 (1−θ)−1/2 , which
is improper although the parameter space is bounded, while a sensible non-subjective prior to
make inferences about the ratio of two multinomial parameters turns out to be proper eventhough the parameter space ]0, ∞[ is not bounded. Actually, one could always work withproper non-subjective priors if the parameter spaces were taken to be appropriately chosenbounded sets. For example, the standard non-subjective prior for a real location parameter isuniform on , which is improper; however, if given some experimental measures, inferences
are made on the true value of some physical quantity a more realistic parameter space would be[0, c], for some context-dependent constant c, and the presumably appropriate non-subjective
prior, a uniform on [0, c], would indeed be proper. Similarly, in the negative binomial setting
where the probability of success must be strictly positive, a parameter space of the form [, 1],
for some  > 0 would lead to a proper non-subjective prior.
Question 4: How do you interpret probability?
Answer: Naturally, as a measure of personal belief. Of course, this does not mean that
I would systematically be prepared to bet in terms of non-subjective posterior distributions,because my personal beliefs may well be not closely approximated by any particular nonsubjective prior.
Question 5: I agree that betting quotients may not reflect true belief, but do you subscribe
to the axioms of probability?
Answer: Yes, I do. I am a strong believer in foundational arguments: the intellectual
strength of the Bayesian argument comes directly from the fact that mathematical logic requires one to express all uncertainties by means of probability measures. In fact there are two
independent arguments to support this claim:(i) Coherent decision theory: If you try to guarantee that your decision making criteria aresensible —in that they meet some intuitively appealing axioms— or if you want to avoid“inadmissible” decisions, then you must express all your uncertainties by probabilities. For

A discussion on noninformative priors

3specific details, you can look at Savage (1954), Fishburn (1981, 1986), or Bernardo andSmith (1994, Ch. 2), and references therein.
(ii) Representation theorems: If you accept a probabilistic description of the behaviour of observables —as all statisticians presumably do—, and you want to describe mathematicallythe idea that some observations are “similar” in some sense —and hence some type ofprediction is possible—, then the general representation theorem tells you, that these “exchangeable” observations are a random sample from some underlying model, indexed bysome parameter defined as the limit of some function of the observations, and there exists
a prior probability distribution over such parameter. Key references are Finetti (1937),
Hewwit and Savage (1955), Smith (1981) and Diaconis (1988); you can get an overview
from Bernardo and Smith (1994, Ch. 4), and references therein;
These are proven existence results; they imply that the common sentence “a prior does
not exist” is a mathematical fallacy: for mathematical consistency one must be a Bayesian.
However, these are only existence results; they leave open the question of specifying a particularprior in each problem.
Question 6: Don’t the axioms imply that the probability of a tautology should be 1 and,thus, that priors have to be proper?
Answer: The natural axioms do not: they only lead to finite additivity, which is compatible
with improper measures; however, the further natural assumption of conglomerability leads to
σ-additivity and, hence, to proper measures; some signposts to this debate are Renyi (1962),
Heath and Sudderth (1978, 1989), Hartigan (1983), Cifarelli and Regazzini (1987), Consonni
and Veronese (1989) and Lindley (1996). Nevertheless, it must be stressed that what reallymatters is the posterior of the quantity of interest —which under σ-additivity must certainly beproper—, because this is what you have to use either in inference or in decision making.
Question 7: But those proper non-subjective posteriors are often derived from improper
priors; how should one interpret the improper priors?
Answer: One should not interpret any non-subjective prior as a probability distribution.
Non-subjective priors are merely technical devices to produce non-subjective posteriors byformal use of Bayes theorem, and “sensible” non-subjective posteriors are always proper.
Question 8: What do you mean by a “sensible” non-subjective posterior?
Answer: One that, —after careful scrutiny of its properties—, you would be prepared touse for scientific communication. Of course, there is no way to give a formal definition for
this; indeed, an important part of the discussion on methods for deriving non-subjective priors
is based on the analysis of the statistical properties of the posteriors they produce in specific,“test-case” examples.
Question 9: If the prior to posterior conversion describes the change in one’s betting
behaviour (and this is a debatable issue), should that prior also not be proper?
Answer: Non-subjective priors are limits. Any sensible non-subjective prior may be seen
as some appropriately defined limit of a sequence of proper priors. In fact, as I have mentioned
before, they are only improper because of the use of convenient, typically unbounded parameter
spaces. If we tried to be more realistic, and worked with appropriately chosen bounded parameter
spaces, non-subjective priors would always be proper; if one prefers to work with conventionalparameter spaces, increasing sequences of bounded approximations to those parameter spaces
may be used to provide sequences of proper priors which converge to the corresponding improper

A discussion on noninformative priors

4non-subjective priors in the precise sense that, for any data set, the sequence of posteriors thus
produced converges to the corresponding non-subjective posterior.
Question 10: Why then have improper priors entered our lives? Can we not do away withthem by concentrating only on compact spaces?
Answer: We could indeed, but this would probably be mathematically inconvenient. But,
again, the main question is not whether non-subjective priors are proper or improper, but whether
or not they lead to sensible non-subjective posteriors. Actually, if an improper prior leads
to a posterior with undesirable properties, the posterior which would result from a proper
approximation to that prior (like that obtained by truncation), will typically have the same
undesirable
 2 properties: for instance the posterior of the sum of the squares of normal meansφ =µi based on a joint uniform prior on the means π(µ1 , . . . , µk ) ∝ 1 is extremely
unsatisfactory as a non-subjective posteriorfor φ (Stein, 1959), but so it is that based on theproper multinormal prior π(µ1 , . . . , µk ) ∝ i N(µi |0, σ), for large σ. Proper or improper, weneed non-subjective priors which appropriately represent a lack of relevant prior knowledge
about the quantity of interest relative to that provided by the data.
Question 11: What do you mean by a prior describing a lack of knowledge? Given myutility function, the amount I am willing to bet signals the extent of my knowledge.
Answer: The contribution of the data in constructing the posterior of interest shouldbe “dominant”. Note that this does not mean that a non-subjective prior is a mathematicaldescription of “ignorance”. Any prior reflects some form of knowledge. Non-subjective priors
try to make precise the type of prior knowledge which, for a given model and for a particularinference problem within this model, would make data dominant.
Question 12: We now see your point: namely, that non-subjective priors are those forwhich the contribution of the data are posterior dominant for the quantity of interest. This issensible, but to construct non-subjective priors does one need to consider data?
Answer: Non-subjective priors do not typically depend on the data, but on the probabilistic
model which is assumed to have produced them: as one would expect, the prior which makes thedata posterior dominant for the quantity of interest usually depends both on the model assumed
for the data and on the quantity of interest.
Question 13: It appears that, in using the prior as a mere technical device in a formal
use of Bayes’ law, and with the aim of making the posterior data dominant, one is proposing aparadigm for inference which is not in the spirit of Bayes (and Laplace) nor is it in the spirit ofFisher, or Neyman-Pearson-Wald. If so, may one conclude that a non-subjective Bayesian is asensible pseudo-Bayesian?
Answer: No, I think he is totally Bayesian. Non-subjective Bayesian analysis is just apart, —an important part, I believe—, of a healthy sensitivity analysis to the prior choice: it
provides an answer a very important question in scientific communication, namely, what could
one conclude from the data if prior beliefs were such that the posterior distribution of the quantityof interest were dominated by the data.
Question 14: What are the historical precedents of non-subjective priors?
Answer: The pioneers were Bayes (1763), who considered a uniform prior in a binomial
setting, and Laplace (1812), who used an improper uniform prior in the normal case. Nobody
considered using priors that were different from the uniform in those days when, by the way, alarge part of statistical inference was based on inverse probability calculations.

A discussion on noninformative priors

5

Question 15: So Laplace did use uniform priors that were improper?
Answer: Yes, although he seemed to be aware that this was only a sensible approximation
to using a proper uniform prior in the bounded parameter space which would more closelyreflect the underlying physical problem.
Question 16: How does this connect with Laplace’s “principle of insufficient reason”?
Answer: The “rationale” for using a uniform prior was that any other prior would reflectspecific knowledge. Of course, as I stated before, any prior reflects some knowledge; what
happens here is that in any location problem, the uniform prior is precisely that which makes
the data posterior dominant.
Question 17: When did problems with uniform priors surface?
Answer: By the early 20’s it was widely realised that the universal use of a uniform
prior did not make sense. Since most statisticians were not prepared to use personal priors inscientific work, alternative “objective” methods of statistical inference were produced, which
only depended on the assumed probability model; this gave rise to both fiducial and frequentist
inference. It was not until the 40’s that Jeffreys (1946) produced an alternative to using theuniform as a non-subjective prior; however, Jeffreys was a physicist, barely integrated in theworld of academic statistics, and his work did not achieve the impact it deserved.
Question 18: What did Jeffreys do?
Answer: He was motivated by invariance requirements and suggested, using differential
geometry arguments, a solution which provides a non-subjective prior known as Jeffreys’ prior.
He then proceeded to a detailed investigation of the consequences of such an approach (Jeffreys,1961).
Question 19: What are these invariance requirements? Are they really crucial?
Answer: They are invariance under one-to-one transformations, and invariance undersufficient statistics. I certainly believe that those properties are crucial for “sensible” nonsubjective distributions, to the point that one should not seriously consider a proposal for
non-subjective Bayesian inference which does not satisfy them:(i) Invariance under one-to-one transformations. If θ = θ(φ) is a one-to-one function of φ,then π(φ|x)is logically equivalent to π(θ|x) = π(φ|x)|dφ/dθ|; hence, the non-subjective
posterior for φ directly obtained from p(x|φ) and the non-subjective posterior for θ obtained
from the same model reparametrized in terms of θ must be related by that equation.
(ii) Invariance under sufficient statistics. If t is a sufficient statistic for the model π(φ|x), then
the non-subjective posterior π(φ|x) obtained from model p(x|φ) must be the same as thenon-subjective posterior π(φ|t) obtained from p(t|φ).
Some pointers to the literature on the role of invariance in the selection of non-subjective priors
are Hartigan (1964), Jaynes (1968) and Dawid (1983) and Yang (1995).
Question 20: Can you give an intuitive explanation of Jeffreys’ prior?
Answer: His invariance requirements are easy to understand, but he did not offer an
intuitively convincing explanation for his particular choice; however, a modern description by
Kass (1989) offers a heuristic explanation based on the idea that “natural” volume elements,
defined in terms of Fisher’s matrix, should have equal prior probability. Moreover, whenapplicable (continuity and appropriate regularity conditions) the one-dimensional version ofJeffreys’ prior has been justified from many different viewpoints: these include Perks (1947),

A discussion on noninformative priors

6

Lindley (1961), Welch and Peers (1963), Hartigan (1965), Good (1969), Kashyap (1971), Box
and Tiao (1973, Sec. 1.3), Bernardo (1979), Kass (1990), Wasserman (1991) and Clarke andBarron (1994). The derivation of the one-dimensional Jeffreys’ prior by Welch and Peers (1963),
as that prior for which the coverage probabilities of one-sided posterior credible intervals are
asymptotically as close as possible to their posterior probabilities, may be specially appealing tofrequentist trained statisticians: this means that under Jeffreys’ one-parameter prior and for large
sample sizes, an interval with posterior probability 1 − α may be approximately be interpreted
as a confidence interval, in the Neyman-Pearson sense, with significance level α.
Question 21: You have just mentioned that, from a technical point of view, Jeffreys prioris related to Fisher’s information matrix. Is Fisher’s information matrix related to the notion of“lack of information”?
Answer: Not directly. The connection comes from the role of Fisher’s matrix in asymptotics. By the way, this should be simply called “Fisher’s matrix”, not “Fisher’s information
matrix”: it is only directly related to general information measures under normal assumptions
or in regular asymptotic conditions.
Question 22: Is Jeffreys’ prior a flat prior?
Answer: I guess you mean “nearly uniform”. Then no, generally it is not, unless theparameter is a location parameter. I will use your question to stress that “flat” priors, —typicallyuniform or log-uniform—, which are too often used as being synonymous with “noninformative”
priors, may be very informative on non-location parameters. For instance, as I have mentioned
before, a “flat” prior on the means of a multivariate normal implies strong knowledge about
their sum of squares, thus producing Stein’s (1959) paradox.
Question 23: Then in what sense is Jeffreys prior neutral; i.e., it reflects a lack of knowledge?
Answer: It does not reflect “lack of knowledge”, but it may be argued that —with one
parameter and under regularity conditions— Jeffreys’ prior describes the type of prior knowledge
which would make the data as posterior dominant as possible. Thus, the corresponding posterior
distribution may be argued to provide a benchmark, a “reference”, for the class of the posterior
distributions which may be obtained from other, possibly subjective, priors.
Question 24: What is the problem with Jeffreys’ prior and in which cases it is appropriate?
Answer: Jeffreys himself realized that his proposal only works, and then under regularity
conditions, in one-parameter continuous problems. He suggested a collection of ad hoc rules todeal with multiparameter problems, with mixed results. Moreover, he seemed to be convincedthat a unique appropriate non-subjective prior could be defined for any given model, whatever
the quantity of interest. This was later seen to be not true.
Question 25: What were the developments after Jeffreys’?
Answer: For a while, it was thought that clever analysis of multiparameter problems using
some combination of Jeffreys’ proposals would produce appropriate non-subjective priors forany problem. Lindley’s (1965) book was an explicit attempt is this sense, and it does provethat most standard “textbook” inference problems have a non-subjective Bayesian solution
within this framework, and one which produces credible intervals which are often numericallyeither identical or very close to their frequentist counterparts. But then, in the early 70’s, themarginalization paradoxes emerged.
Question 26: What are the marginalization paradoxes?

A discussion on noninformative priors

7

Answer: They may collectively be seen as a proof that the original idea of a unique nonsubjective prior for each model is untenable: we may only agree on a unique non-subjective
prior for each quantity of interest within a model. A simple example of marginalization paradox
is provided by the standarized mean φ = µ/σ of a normal distribution: Stone and Dawid (1972)
showed that the posterior distribution of φ only depends on the data through some statistics t,
whose sampling distribution only depends on φ. Hence, one would expect that inferences derived
from the model p(t|φ) would match those derived from the full model N (x|µ, σ), but they provedthat this is not possible if one uses the “standard” non-subjective prior π(µ, σ) = 1/σ, which
everybody agrees it is the appropriate non-subjective prior to make inferences about either µor σ. It was immediately seen (Dawid, Stone and Zidek, 1973) that marginalization paradoxes
are ubiquitous in multiparameter problems: any future development of non-subjective Bayesiananalysis would have to come to terms with them.
Question 27: Was your own work on reference distributions a reaction to this?
Answer: It was not a direct reaction, but I was certainly influenced by these results. In themid 70’s, as a part of my Ph.D. work on Bayesian design of experiments, I become interested in
non-subjective, “noninformative” priors. The marginalization paradoxes made obvious to me
that some new work in that area was necessary: reference analysis was the result.
Question 28: Can you explain what do you mean by reference analysis?
Answer: Reference analysis may be described as a method to derive model-based, nonsubjective posteriors, based on information-theoretical ideas, and intended to describe the inferential content of the data for scientific communication. It is, to the best of my knowledge,
the only general method available which has the required invariance properties and successfully
deals with the marginalization paradoxes.
Question 29: This is a very convincing statement in favour of your paradigm, but what doyou mean by the inferential content of the data? How may this be quantified?
Answer: In a sense, each possible answer to that pair of related questions may be the basis
for a method to derive non-subjective posteriors. I personally believe that the inferential content
of the data is appropriately measured by the amount of information they provide on the quantityof interest, where the word “information” is used in the technical sense of Shannon (1948) and
Lindley (1956). This was the starting point for the definition of a reference posterior (Bernardo,1979).
Question 30: Can you be more explicit on the relationship between reference distributions
and information theoretic ideas?
Answer: Sure. The amount of information to be expected from the data is naturally afunction of the prior knowledge, as described by the prior distribution: the more prior information available, the less information may one expect from the data. With only one real-valued
parameter, one may unambiguously define a limit functional which measures, in terms of theprior distribution, the amount of missing information about the parameter which data from agiven model could possibly be expected to provide; the reference prior is that which maximizes the missing information. The multiparameter case is handled by using recursively the
one-parameter solution.
Question 31: How reference priors differ from other proposals? In particular, how do theydiffer from Jeffreys’ priors?

8

A discussion on noninformative priors

Answer: The reference prior approach is totally general and, as far as I am aware, it includes
within a single framework all generally accepted non-subjective solutions to specific cases. In
one-parameter problems, the reference prior reduces to Jaynes (1968) maximum entropy prior
if the parameter space has a finite number of points, and it reduces to Jeffreys’ prior in thecontinuous regular case. In regular continuous multiparameter problems, one often obtains thesolutions which Jeffreys suggested using ad hoc arguments rather than his general multivariate
rule. Moreover, reference analysis can deal with non-regular cases which cause problems forother methods.
Question 32: Can you give some examples of reference priors in the one-parameter continuous case?
Answer: As I have just mentioned, under regularity conditions to guarantee asymptotic
normality, the reference prior is simply Jeffreys’ prior, namely
1/2
 2

d
,
π(θ) ∝ E x|θ − log p(x|θ)
dθbut I will give you a couple of non-regular examples:(i) Uniform distribution on [θ − a, θ + a]. The reference prior is then uniform on , and thereference posterior is uniform over the set of θ values which remain feasible after the data
have been observed; (Bernardo and Smith, 1994, p. 311)(ii) Uniform distribution on [0, θ]. The reference prior is then π(θ) ∝ θ−1 , and the reference
posterior is a Pareto distribution; (Bernardo and Smith, 1994, p. 438).
Question 33: You have sketched the derivation of reference posteriors associated to models
with only one real-valued parameter, and stated that those are invariant under reparametrization,
but how do you deal with nuisance parameters?
Answer: Recursively: the idea is very simple, although there are delicate technical issues
involved. Consider the simplest case; suppose that you are interested in the reference posterior
distribution π(φ|x1 , . . . , xn ) of some quantity φ given a random sample from a model p(x|φ, λ),which contains one real-valued nuisance parameter λ ∈ Λ ⊂ . Working conditionally on φ,this is a one-parameter problem, and hence the one-parameter solution may be used to provide
a conditional reference prior π(λ|φ). If this is proper, then it may be used to integrate out thenuisance parameter λ and obtain a model with one real-valued parameter p(x|φ) to which the
one-parameter solution is applied again to derive the marginal reference prior π(φ); the desiredreference posterior is then simplyn{p(xi |φ, λ)}π(λ|φ)dλ.
π(φ|x1 , . . . , xn ) ∝ π(φ)Λ

i=1If π(λ|φ) is not proper, the procedure is performed within an increasing sequence of bounded
approximations {Λj , j = 1, 2, . . .} to the nuisance parameter space Λ, chosen such that π(λ|φ)
is integrable within each of them; the reference posterior is then the limit of the resulting
sequence {πj (φ|x1 , . . . , xn ), j = 1, 2, . . .} of posterior distributions (Berger and Bernardo,
1989, 1992b).
Question 34: Does this mean that, within a single model, you may have as many reference
priors as possible parameters of interest?
Answer: It does indeed. Given a model, say p(x | θ1 , θ2 ), the reference algorithm provides
a reference posterior distribution for each parameter of interest φ = φ(θ1 , θ2 ), and those mayA discussion on noninformative priors

9well correspond to different priors, because beliefs which maximize the missing information
about φ = φ(θ1 , θ2 ) will generally differ from those which maximize the missing information
about η = η(θ1 , θ2 ), unless η happens to be a one-to-one function of φ.
Note also that, as I mentioned before, using different priors for different parameters of interest is the only way to have non-subjective priors which avoid the marginalization paradoxes. For
instance, in a normal model, N(x|µ, σ), the reference posterior for µ is the Student distribution
St(µ|x, (n − 1)−1/2 s, n − 1), obtained from the “conventional” improper prior π(σ|µ)π(µ) =σ −1 , while the reference posterior for φ = µ/σ is obtained from π(σ|φ)π(φ) = (2+φ2 )−1/2 σ −1 ,a different improper prior, producing a reference posterior for φ which avoids the marginalization
paradox that you would get if you used again the conventional prior (Bernardo, 1979).
Question 35: We now see how to deal with a single nuisance parameter, but how do you
proceed when there are more than one?
Answer: The algorithm I have just described may easily be extended to any number{λ1 , . . . , λm } of ordered nuisance parameters: get the one-parameter conditional reference
prior π(λm |φ, λ1 , . . . , λm−1 ) and use this to integrate out λm ; get π(λm−1 |φ, λ1 , . . . , λm−2 )and use this to integrate out λm−1 ; continue until you get π(φ); then use
π(λm |φ, λ1 . . . , λm−1 )π(λm−1 |φ, λ1 . . . , λm−2 ) × . . . × π(λ1 |φ)π(φ)
in Bayes theorem, and marginalize to obtain the desired reference posterior π(φ|x1 , . . . , xn ).
The result might possibly depend on the order in which the nuisance parameters are considered which, in that case, should reflect their order of importance in the problem analysed,
the least important being integrated out first. We have found however that this is usually not thecase: in most problems, the reference posterior of the quantity of interest is independent of theorder in which the nuisance parameters are considered.
Question 36: Can you give some examples of this?
Answer: Sure. In a multinomial model, Mu(r1 , . . . , rm |θ1 , . . . , θm , n), the reference
posterior for, say, θ1 , is the Beta distribution Be(θ1 |r1 +1/2, n−r1 +1/2) and this is independentof the order in which the other θi ’s are considered (Berger and Bernardo, 1992a); Note, by theway, that this does depends not on the irrelevant number of categories m as the posterior fromJeffreys’ multivariate prior does; thus, the reference algorithm avoids the agglomeration paradox
typically present in other proposals. Similarly, within the same model, the reference posterior
for φ = θ1 /θ2 is
φr1 −1/2,π(φ|r1 , . . . , rm , n) ∝(1 + φ)r1 +r2 +1(which, again, does not depend on m, but corresponds to a different prior), and this is independentof the order in which the nuisance parameters are considered (Bernardo and Ramón, 1996).
Many more examples are referenced in Yang and Berger (1996).
Question 37: What has now happened to the invariance properties on which you insisted
before?
Answer: They are still there. The reference posterior of any quantity of interest φ does notdepend on whether one uses the full model or the joint sampling distribution of a set of sufficient
statistics. Moreover, for any model p(x | φ, λ1 , . . . , λm ), the reference posterior π(φ | x) does
not depend on the particular parametrization chosen for each of the nuisance parameters and,besides, if θ = θ(φ) is a one-to-one transformation of φ, then π(θ|x) = π(φ|x)|dφ/dθ|. Datta

A discussion on noninformative priors

10and Ghosh (1996) have recently shown that these invariance properties are often not shared byother proposed methods to derive non-subjective posteriors.
Question 38: How do you compute in practice reference distributions?
Answer: Reference priors only depend on the model through its asymptotic behaviour;
essentially, if you know the asymptotics of your model, than you may easily find its associatedreference priors. Under regularity conditions for asymptotic normality, any reference prior may
be obtained from a relatively simple algorithm in terms of Fisher’s matrix (Berger and Bernardo,
1992b). However, the derivation of reference priors in non-regular or complex models may be
a difficult mathematical problem.
Of course, once you have obtained the appropriate reference prior for some quantity ofinterest, you simply use Bayes theorem and marginalize to derive the required reference posterior. It turns out that, within the exponential family, reference priors often correspond to some
limiting form of the corresponding natural conjugate family and, in that case, the correspondingreference posteriors may often be obtained in closed form. When this is not the case, numericalreference posteriors may be efficiently obtained using MCMC sampling-resampling techniques,
as described by Stephens and Smith (1992).
Question 39: We now have a procedure to derive reference posterior distributions whenno prior information is available about the parameter of interest; however, even for scientificcommunication, one may sometimes want to use some partial information (possibly intersubjectively agreed). Can reference analysis deal with this situation?
Answer: It surely can; you define the reference prior under partial information as thatwhich maximizes the missing information subject to whatever constraints are imposed by the
information assumed. If the restrictions take the form of expected values, then explicit forms forthe corresponding restricted reference priors are readily obtained (Bernardo and Smith, 1994, pp
316–320). Note that restricted reference analysis typically leads to proper priors; for instance,
in a location model, the reference prior which corresponds to the partial information provided by
the first two moments of the unknown parameter is the normal distribution with those moments.
Question 40: Please, talk about the new developments on reference priors.
Answer: I can see several directions in which further research is needed:(i) Bounded approximations: I have mentioned before that to implement the reference algorithm in multiparameter problems when the conditional reference priors are not proper, abounded approximation to the parameter space in which the conditional reference priors
are integrable is required. It may be seen (Berger and Bernardo, 1989) that the result may
depend on the bounded approximation chosen. Although in specific applications it is usually clear what the “natural” bounded approximation is, —and this should be the same forall parameters of interest within the same model—, a general definition of the appropriate
bounded approximation is needed.
(ii) Grouping: With many parameters, one may apply the reference prior algorithm by lumping
the parameters in just two groups (parameters of interest and nuisance parameters), or one
may lump them into any number of groups and proceed sequentially (Berger and Bernardo,
1992a, 1992b, 1992c). There is evidence to suggest, however, that one should not group theparameters but proceed recursively using the one-parameter solution as I have described to
you before; this seems to guarantee both admissible coverage properties and the absence ofmarginalization paradoxes, but further research is needed to substantiate this point.

A discussion on noninformative priors

11(iii) Prediction and hierarchical models: A reference prior is technically defined for an orderedparametrization suggested by the problem of interest. What are the appropriate orderedparametrizations to use in prediction and hierarchical model problems? Again, althoughsome answers are available in specific cases, the general strategy is not clear.
(iii) Model choice: Reference distributions are not directly applicable to model choice between
models of different dimensionalities: indeed, reference distributions are typically only
defined up to proportionality constant, and those constants become relevant in this case.
Nice results are available however (Bernardo, 1996) by posing the question as a decision
problem, and working with the reference posterior of the quantity of interest implied by thecorresponding utility function.
(iv) Numerical reference analysis: The derivation of reference priors may sometimes be adifficult mathematical problem, but numerical reference posteriors may be obtained, inprinciple, by simulation methods. This is easily done in one or two parameters, but thegeneral problem is not trivial (Efstathiou, 1996, Ch. 5), as computational explosion has tobe avoided in higher dimensions.
Question 41: Is there anything to be said about the long-term frequentist properties of
non-subjective posteriors? I realize that good Bayesians should not be raising this type of aquestion but, politically speaking, it may be wise to raise the issue.
Answer: Politics aside, this is a very interesting issue, and one that is central to discussions
on comparative statistical inference. Interest on the frequentist coverage probabilities of credibleintervals derived from non-subjective posteriors has a long history; key references include thepioneering work by Welch and Peers (1963) that I have already mentioned, Peers (1965),
Hartigan (1966), Tibshirani (1989), Ghosh and Mukerjee (1992), Mukerjee and Day (1993),
Nicolau (1993), and Datta and Ghosh (1995). The coverage probabilities of credible regions hasoften been an important element in arguing among competing non-subjective posteriors, as inBerger and Bernardo (1989) or Ye and Berger (1991) and, indeed, discussions on the coverage
properties of non-subjective priors were prominent in the workshop on “default” Bayesian
methodology recently held in the States (Purdue University, November 1–3, 1996). Reference
posteriors have consistently been found to have attractive coverage properties, —what may beseen as a form of calibration— but, as far as I am aware, no general results have been established.
Question 42: Is there any other issue on comparative inference over which non-subjective
priors may have a bearing?
Answer: A very important one is the procedure used to eliminate nuisance parameters. The
marginalization paradox examples may be used to demonstrate than non-Bayesian methods toeliminate nuisance parameters (plug-in estimates, profile likelihood, naïve integrated likelihood
and the like) are often inconsistent within their own paradigms in that the resulting ‘marginal’
likelihood may differ from the simplified likelihood. For example, the sampling distributionof the sample coefficient of correlation r in a bivariate normal model only depends on the
population coefficient of correlation ρ, so that non-Bayesian statisticians would presumably
consider p(r | ρ) to be an ‘exact’ ’marginal’ likelihood from which inferences about ρ could be
made. Yet, the more sophisticated non-Bayesian techniques to eliminate nuisance parameters
fail to derive p(r | ρ) (Efron, 1993), while integration of the nuisance parameters with theconditional reference priors (taken in any order) easily produces the ‘exact’ marginal likelihood(Bayarri, 1981; Lindley, 1965, pp. 215–219, Bernardo and Smith, 1994, pp. 363–364).
Question 43: What about admissibility?

A discussion on noninformative priors

12

Answer: Non-subjective priors are sometimes criticized on the grounds that, since they areoften improper, they may lead, for instance, to inadmissible estimates. We have seen, however,that sensible non-subjective priors are, in an appropriate sense, limits of proper priors; regarded
as a “baseline” for admissible inferences, non-subjective posteriors need not be themselves
admissible, but only arbitrarily close to admissible posteriors. That said, admissibility is notreally the relevant concept: truncating the parameter space may lead to technically admissible butvery unsatisfactory posteriors, as in the sum of squares of normal means example I described
before. Besides, admissibility crucially depends on the loss function; thus, if one is reallyinterested in estimation, one should explicitly work in terms of the corresponding decision
problem, with an appropriate, context dependent, loss function. To deal with those problems,
reference analysis may be extended to define reference decisions, as those optimal under the priorwhich maximizes the missing utility (Bernardo, 1981, Bernardo and Smith, 1994, Sec. 5.4.1).
This is, by the way, another very promising area for future research.
Question 44: Could you talk about general criticisms to non-subjective priors?
Answer: The major criticism usually comes from subjectivist Bayesians: the prior should
be an honest expression of the analyst’s prior knowledge, not a function of the model, specially
if this involves integration over the sample space and hence violates the likelihood principle. I
believe there are two complementary answers to this:(i) Foundational: A non-subjective posterior is the answer to a what if question, namely what
could be said about the quantity of interest given the data, if one’s prior knowledge wasdominated by the data; if the experiment is changed, or a different quantity of interest
is considered, the non-subjective prior may be expected to change correspondingly. If
subjective prior information is specified, the corresponding subjective posterior could becompared with the non-subjective posterior in order to assess the relative importance on theinitial opinions in the final inference.
(ii) Pragmatic: In the complex multiparameter models which are now systematically used asa consequence of the availability of numerical MCMC methods, there is little hope for a
detailed assessment of a huge personal multivariate prior; the naïve use of some “tractable”
prior may then hide important unwarranted assumptions which may easily dominate theanalysis (see e.g., Casella, 1996, and references therein). Careful, responsible choice of anon-subjective prior is possibly the best available alternative.
Question 45: Could we end with some signposts for those interested in pursuing this
discussion at a more technical level?
Answer: The classic books by Jeffreys (1961), Lindley (1965) and Box and Tiao (1973)
are a must for anyone interested in non-subjective Bayesian inference; other relevant books are
Zellner (1971) and Geisser (1993).
The construction of non-subjective priors has a very interesting history, which dates backto Laplace (1812), and includes Jeffreys (1946, 1961), Perks (1947), Lindley (1961), Geisser
and Cornfield (1963), Welch and Peers (1963), Hartigan (1964, 1965), Novick and Hall (1965),
Jaynes (1968, 1971), Good (1969), Villegas (1971, 1977), Box and Tiao (1973, Sec. 1.3), Zellner(1977, 1986), Bernardo (1979), Rissanen (1983), Tibshirani (1989) and Berger and Bernardo(1989, 1992c) as some of the more influential contributions.
For a general overview of the subject, see Bernardo and Smith (1994, Sec. 5.6.2), Kassand Wasserman (1996), and references therein. Yang and Berger (1996) is a problem-specific(partial) catalog of the many non-subjective priors which have been proposed in the literature.

13

A discussion on noninformative priors

For someone specifically interested in reference priors, the original paper, Bernardo (1979)
is easily read and it is followed by a very lively discussion; Berger and Bernardo (1989, 1992b)
contain crucial extensions; Bernardo and Smith (1994, Sec. 5.4) provide a description of reference analysis at a textbook level; Bernardo and Ramón (1996) offer a modern elementary
introduction.
REFERENCES
Bayarri, M. J. (1981). Inferencia Bayesiana sobre el coeficiente de correlación de una población normal bivariante.
Trab. Estadist. 32, 18–31.
Berger, J. O. and Bernardo, J. M. (1989). Estimating a product of means: Bayesian analysis with reference priors.
J. Amer. Statist. Assoc. 84, 200–207.
Berger, J. O. and Bernardo, J. M. (1992a). Ordered group reference priors with applications to a multinomial
problem. Biometrika 79, 25–37.
Berger, J. O. and Bernardo, J. M. (1992b). On the development of reference priors. Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 35–60 (with discussion).
Berger, J. O. and Bernardo, J. M. (1992c). Reference priors in a variance components problem. Bayesian Analysis
in Statistics and Econometrics (P. K. Goel and N. S. Iyengar, eds.). Berlin: Springer, 323–340.
Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference. J. Roy. Statist. Soc. B 41, 113–147(with discussion). Reprinted in Bayesian Inference (N. G. Polson and G. C. Tiao, eds.), Brookfield, VT: Edward
Elgar, (1995), 229–263.
Bernardo, J. M. (1981). Reference decisions. Symposia Mathematica 25, 85–94.
Bernardo, J. M. (1996). Bayesian hypothesis testing: a reference analysis. Workshop on Default Bayesian Statistical
Methodology. Purdue University, November 1–3, 1996.
Bernardo, J. M. and Ramón J. M. (1996). An elementary introduction to Bayesian reference analysis. Tech.
Rep., Universitat de València, Spain.
Bernardo, J. M. and Smith, A. F. M. (1994). Bayesian Theory. Chichester: Wiley.
Box, G. E. P. and Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Reading, MA: Addison-Wesley.
Casella, G. (1996). Statistical inference and Monte Carlo algorithms. Test 5, 249–340, (with discussion).
Clarke, B. and Barron, A. R. (1994). Jeffreys’ prior is asymptotically least favorable under entropy risk. J. Statist.
Planning and Inference 41, 37–60.
Cifarelli, D. M. and Regazzini, E. (1987). Priors for exponential families which maximize the association between
past and future observations. Probability and Bayesian Statistics (R. Viertl, ed.). London: Plenum, 83–95.
Consonni, G. and Veronese, P. (1989). Some remarks on the use of improper priors for the analysis of exponential
regression problems. Biometrika 76, 101–106.
Datta, G. S. and Ghosh, J. K. (1995). On priors providing a frequentist validity for Bayesian inference. Biometrika 82,
37–45.
Datta, G. S. and Ghosh, M. (1996). On the invariance of noninformative priors. Ann. Statist. 24, 141–159.
Dawid, A. P. (1983). Invariant prior distributions. Encyclopedia of Statistical Sciences 4 (S. Kotz, N. L. Johnson
and C. B. Read, eds.). New York: Wiley, 228–236.
Dawid, A. P., Stone, M. and Zidek, J. V. (1973). Marginalization paradoxes in Bayesian and structural inference.
J. Roy. Statist. Soc. B 35, 189–233 (with discussion).
de Finetti, B. (1937). La prévision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincaré 7, 1–68.
Reprinted in 1980 as ‘Foresight; its logical laws, its subjective sources’ in Studies in Subjective Probability(H. E. Kyburg and H. E Smokler, eds.). New York: Dover, 93–158.
Diaconis, P. (1988a). Recent progress on de Finetti’s notion of exchangeability. Bayesian Statistics 3 (J. M. Bernardo, M. H. DeGroot, D. V. Lindley and A. F. M. Smith, eds.). Oxford: University Press, 111-125 (withdiscussion).
Efron, B. (1993). Bayes and likelihood calculations from confidence intervals. Biometrika 80, 3–26.
Efstathiou, M. (1996). Some Aspects of Approximation and Computation for Bayesian Inference. Ph.D. Thesis, Imperial College London, UK.
Fishburn, P. C. (1981). Subjective expected utility: a review of normative theories. Theory and Decision 13,
139–199.
Fishburn, P. C. (1986). The axioms of subjective probability. Statist. Sci. 1, 335–358 (with discussion).

A discussion on noninformative priors

14

Geisser, S. (1993). Predictive Inference: an Introduction. London: Chapman and Hall.
Ghosh, J. K. and Mukerjee, R. (1992). Non-informative priors. Bayesian Statistics 4 (J. M. Bernardo, J. O. Berger,
A. P. Dawid and A. F. M. Smith, eds.). Oxford: University Press, 195–210 (with discussion).
Good, I. J. (1969). What is the use of a distribution? Multivariate Analysis 2 (P. R. Krishnaiah, ed.). New York:
Academic Press, 183–203.
Hartigan, J. A. (1964). Invariant prior distributions. Ann. Math. Statist. 35, 836–845.
Hartigan, J. A. (1965). The asymptotically unbiased prior distribution. Ann. Math. Statist. 36, 1137–1152.
Hartigan, J. A. (1966). Note on the confidence prior of Welch and Peers. J. Roy. Statist. Soc. B 28, 55-56.
Hartigan, J. A. (1983). Bayes Theory. Berlin: Springer.
Heath, D. L. and Sudderth, W. D. (1978). On finitely additive priors, coherence and extended admissibility. Ann.
Statist. 6, 333–345.
Heath, D. L. and Sudderth, W. D. (1989). Coherent inference from improper priors and from finitely additive priors.
Ann. Statist. 17, 907–919.
Hewitt, E. and Savage, L. J. (1955). Symmetric measures on Cartesian products. Trans. Amer. Math. Soc. 80,
470–501.
Jaynes, E. T. (1968). Prior probabilities. IEEE Trans. Systems, Science and Cybernetics 4, 227–291.
Jaynes, E. T. (1971). The well posed problem. Foundations of Statistical Inference (V. P. Godambe and D. A. Sprott,eds.). Toronto: Holt, Rinehart and Winston, 342–356 (with discussion).
Jeffreys, H. (1946). An invariant form for the prior probability in estimation problems. Proc. Roy. Soc. A 186,
453–461.
Jeffreys, H. (1961). Theory of Probability (3rd ed.). Oxford: University Press.
Kashyap, R. L. (1971). Prior probability and uncertainty. IEEE Trans. Information Theory 14, 641–650.
Kass, R. E. (1989). The geometry of asymptotic inference. Statist. Sci. 4, 188–234.
Kass, R. E. (1990). Data-translated likelihood and Jeffreys’ rule. Biometrika 77, 107–114.
Kass, R. E. and Wasserman, L. (1996). The selection of prior distributions by formal rules. J. Amer. Statist. Assoc. 91,(to appear).
Lindley, D. V. (1956). On a measure of information provided by an experiment. Ann. Math. Statist. 27, 986–1005.
Lindley, D. V. (1961). The use of prior probability distributions in statistical inference and decision. Proc. FourthBerkeley Symp. 1 (J. Neyman and E. L. Scott, eds.). Berkeley: Univ. California Press, 453–468.
Lindley, D. V. (1965). Introduction to Probability and Statistics from a Bayesian Viewpoint. Vol. 2: Statistics.
Cambridge: University Press.
Lindley, D. V. (1996). Some comments on Bayes Factors. J. Statist. Planning and Inference (to appear).
Mukerjee, R. and Dey, D. K. (1993). Frequentist validity of posterior quantiles in the presence of a nuisanceparameter: Higher order asymptotics. Biometrika 80, 499–505.
Nicolau, A. (1993). Bayesian intervals with good frequentist behaviour in the presence of nuisance parameters.
J. Roy. Statist. Soc. B 55, 377–390.
Novick, M. R. and Hall, W. K. (1965). A Bayesian indifference procedure. J. Amer. Statist. Assoc. 60, 1104–1117.
Peers, H. W. (1965). On confidence points and Bayesian probability points in the case of several parameters. J. Roy.
Statist. Soc. B 27, 9–16.
Perks, W. (1947). Some observations on inverse probability, including a new indifference rule. J. Inst. Actuaries 73,
285–334 (with discussion).
Renyi, A. (1962). Wahrscheinlichkeitsrechnung. Berlin: Deutscher Verlag der Wissenschaften. English translationin 1970 as Probability Theory. San Francisco, CA: Holden-Day.
Rissanen, J. (1983). A universal prior for integers and estimation by minimum description length. Ann. Statist. 11,
416–431.
Savage, L. J. (1954). The Foundations of Statistics. New York: Wiley. Second edition in 1972, New York: Dover.
Shannon, C. E. (1948). A mathematical theory of communication. Bell System Tech. J. 27 379–423 and 623–656.
Reprinted in The Mathematical Theory of Communication (Shannon, C. E. and Weaver, W., 1949). Urbana,IL.: Univ. Illinois Press.
Stephens, D. A. and Smith, A. F. M. (1992). Sampling-resampling techniques for the computation of posterior
densities in normal means problems. Test 1, 1–18.
Smith, A. F. M. (1981). On random sequences with centred spherical symmetry. J. Roy. Statist. Soc. B 43, 208–209.
Stein, C. (1959). An example of wide discrepancy between fiducial and confidence intervals. Ann. Math. Statist. 30,
877–880.

A discussion on noninformative priors

15

Stone, M. and Dawid, A. P. (1972). Un-Bayesian implications of improper Bayesian inference in routine statisticalproblems. Biometrika 59, 369–375.
Tibshirani, R. (1989). Noninformative priors for one parameter of many. Biometrika 76, 604–608.
Villegas, C. (1971). On Haar priors. Foundations of Statistical Inference (V. P. Godambe and D. A. Sprott, eds.).
Toronto: Holt, Rinehart and Winston, 409–414 (with discussion).
Villegas, C. (1977). Inner statistical inference. J. Amer. Statist. Assoc. 72, 453–458.
Wasserman L. (1991). An inferential interpretation of default priors. Tech. Rep. 516, Carnegie Mellon University,
USA.
Welch, B. L. and Peers, H. W. (1963). On formulae for confidence points based on intervals of weighted likelihoods.
J. Roy. Statist. Soc. B 25, 318–329.
Yang, R. (1995). Invariance of the reference prior under reparametrization. Test 4, 83–94.
Yang, R. and Berger, J. O. (1996). A catalog of noninformative priors. Tech. Rep., Purdue University, USA.
Ye, K. and Berger, J. O. (1991). Non-informative priors for inferences in exponential regression models. Biometrika 78, 645–656.
Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics. New York: Wiley. Reprinted in 1987,
Melbourne, FL: Krieger.
Zellner, A. (1977). Maximal data information prior distributions. New Developments in the Applications of Bayesian
Methods (A. Aykaç and C. Brumat, eds.). Amsterdam: North-Holland, 211–232.
Zellner, A. (1986). On assessing prior distributions and Bayesian regression analysis with g-prior distributions.
Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti (P. K. Goel and A. Zellner,eds.). Amsterdam: North-Holland, 233–243.