Dutch Book against some ‘Objective’ Priors
byMorris L. Eaton1 and David A. Freedman2
1. School of Statistics, University of Minnesota, 313 Ford Hall, 224 Church St. SE,
Minneapolis, MN 55455, USA. E-mail: eaton@stat.umn.edu
2. Department of Statistics, University of California, Berkeley, CA 94705-3860.
Abstract‘Dutch book’ and ‘strong inconsistency’ are generally equivalent: there is asystem of bets that makes money for the gambler, whatever the state of nature may
be. As de Finetti showed, an odds-maker who is not a Bayesian is subject to a dutch
book, under certain highly stylized rules of play – a fact often used as an argument
against frequentists. However, so-called ‘objective’ or ‘uninformative’ priors may
also be subject to a dutch book. This note explains, in a relatively simple andself-contained way, how to make dutch book against a frequently-recommended
uninformative prior for covariance matrices.
Keywords: Dutch book, incoherence, Jeffreys prior, multivariate analysis, objective prior, prediction, strong inconsistency, uninformative prior
1. Introduction
Our aim here is to sketch a relatively simple and self-contained argument toshow that dutch book can be made against certain ‘objective’ priors, namely, invariant measures that have infinite mass and are used as ‘priors’ in formal Bayesiancalculations. Such improper priors are often said to be ‘uninformative’. Our main
example involves a prior that was recommended by Jeffreys – and adopted by many
Bayesians – for use in multivariate normal distributions where the covariance matrix is unknown.
To fix ideas, we begin with de Finetti’s dutch-book argument against the frequentists. Let  be a finite set. A bookie has to post finite, positive odds on
every subset of , apart from the empty set and  itself, accepting bets on each
set at those odds. Bets can be laid in any amount (positive or negative) on any
combination of sets. The Bayesian bookie will have a prior probability π on ,1and will post odds π(A)/[1 − π(A)] on A. The non-Bayesian bookie posts odds
compatible with no π. In this context, ‘dutch book’ means a system of bets that –no matter what ω is chosen from  – yields a positive payoff to the bettor.
Plainly, dutch book cannot be made against a Bayesian bookie, since the expected payoff is 0. On the other hand, as de Finetti showed, dutch book can be madeagainst any non-Bayesian bookie. The relevance to applied statistical work is notentirely clear, since few statisticians place bets when doing data analysis, and few
bookies follow de Finetti’s rules for accepting bets. However, the argument hasoften been deployed against the frequentists: for discussion, see Freedman (1995).
The possibility of a dutch book is sometimes referred to as a ‘money pump’: ifyou can win a dollar, you make the poor bookie play the game over and over again,
pumping money from him to you.
De Finetti’s example can be viewed as a prediction problem: Nature will chooseω from , and the odds are a stylized way of describing opinions about the future.
Freedman and Purves (1969) modified the argument to cover a two-stage processwith conditional bets. The pair (x, z) is chosen at random from Pθ , where θ is an
unknown parameter; x is observed first, z second. (The parameter space and theobservation space are required to be finite, as in de Finetti’s work.) The gambler
is allowed to bet on z, and bets are allowed to depend on x. The concept of dutch
book must be extended slightly: the clever gambler can arrange to have a positive
expected payoff from a non-Bayesian bookie, simultaneously for all θ – but may
have to take a loss for some combinations of θ, x, and z. Some observers mayview the passage from unconditional to conditional bets as a small variation on de
Finetti’s setup; others consider this generalization to be a major – and subversive –idea.
Our main example involves prediction, in a setting like that of Freedman and
Purves (1969) – although the spaces are infinite. There are n independent observations from a common multivariate normal distribution, having mean 0 and(unknown) positive definite covariance matrix . The observations are denoted
X1 , . . . , Xn ; they are used to predict an n + 1st observation, denoted Z. Indeed,
having observed X1 , . . . , Xn , the statistician is required to produce a ‘predictive
distribution’ for Z. If the Jeffreys prior is used to generate this predictive distribution, the statistician is exposed to a dutch book.
A similar example can be constructed for estimation (Section 2.2), but the argument is a little harder. Stone (1976) has the concept of ‘strong inconsistency’,
defined below. In Section 2, we show that the Jeffreys prior leads to strong inconsistency. Section 3 demonstrates that strong inconsistency is equivalent to a dutch
book. Section 4 reviews the literature.
22. The Main Example
Let X1 , . . . , Xn be independent random p×1 vectors, with a common Np (0, )
distribution; the covariance matrix  is p × p and positive definite. Here, n ≥p > 1. An (n + 1)st observation Z will be drawn independently from Np (0, ).
How can the data X = (X1 , . . . , Xn ) be used to predict Z? Since the problem isinvariant under multiplication by a p × p non-singular matrix A,
Xi → AXi , Z → AZ,  → AA ,an invariant ‘prior distribution’might suggest itself. (Quote marks are used because
the prior is improper, with infinite total mass.) The invariant prior is d/||(p+1)/2 .
It√ is unique up to a positive constant, and can be recognized as the Jeffreys prior|I |d, where I is the Fisher information matrix and |M| is the determinant of M.
A predictive distribution for Z can be computed from the Jeffreys prior, by aformal application of Bayes rule. In more detail, let φ(x, z|) be the multivariate
normal density of (X, Z) given ; similarly, φ(x|) is the multivariate normal
density of X given . Here, x ∈ (R p )n and z ∈ R p . Formally, the ‘predictive
density’ for X, Z – in advance of data collection – is obtained by integrating against the Jeffreys prior:φ(x, z) = φ(x, z|)d/||(p+1)/2 .
(1)
Likewise, the ‘predictive density’ for X is
φ(x) = φ(x|)d/||(p+1)/2 .

(2)

According to ‘Bayes rule’, the ‘predictive density’ for Z when X = x isφ(z|x) = φ(x, z)/φ(x).

(3)

We have quote marks because the prior is improper, so
φ(x, z)dxdz = φ(x)dx = ∞.
On the other hand, φ(z|x) is a proper density for Z, because φ(x, z|)dz =φ(x|), so φ(x, z)dz = φ(x), and φ(z|x)dz = 1.
The main result of this section can now be stated: the predictive density φ(z|x)
is strongly inconsistent, and dutch book can be made against a statistician who
uses it. (Strong inconsistency is defined below.)3

Theorem 1. Let n ≥ p > 1. Suppose X1 , . . . , Xn , Z are independent Np (0, ).
The predictive distribution for Z given X1 , . . . , Xn , computed from the Jeffreys
prior on the covariance matrix , is strongly inconsistent.
Anof the proof comes next, with details in Sections 2.1 and 3. Let
outlinen
S = i=1 Xi Xi . We take Xi and Z to be p × 1 column vectors, so S is a p × p
matrix. By eliminating a null set, we can take S to be positive definite. Write Z1
for the first coordinate of Z, and S11 for the (1,1) elementof S. Then, as is almost√
obvious, the sampling distribution D0 of T = Z1 / S11 does not depend on :√
nT ∼ tn .
(4)Here, ‘∼’ means ‘is distributed as’, and tn is t with n degrees of freedom.
When X = x, the predictive distribution Q(dz|x) for the (n + 1)st observation
Z, computed formally by Bayes rule from the Jeffreys prior, has a density on R pgiven by
(5)
φ(z|x) = C/ |s|(1 + z s −1 z)n+1 ,nwhere C is a constant, and s = i=1 xi xi is the value of S at the observed X = x.
As before, we take xi and z to be p × 1, so x is p × n and s is p × p. The proofof (5) is ‘just’ calculus (Section 2.1). The constant C depends on n and p, noton x or z.
√
Next, the predictive distribution D1 of T = Z1 / S11 when X = x does notdepend on x; indeed,n − p + 1 T ∼ tn−p+1 ,(6)
as will also be proved in Section 2.1. By (4) and (6),D1 = D0 .

(7)

Inequality (7) is the key point, and strong inconsistency will soon follow. (If p = 1
then D1 = D0 ; that is why we assumed p > 1.)
Let E denote expectation relative to our sampling model for X1 , . . . , Xn , Z,
and recall that Q(dz|x) is the predictive distribution for Z given X = x. ‘Strong
inconsistency’ means there is a bounded measurable function f and an $ > 0 withf (x, z)Q(dz|x) + $ ≤ E {f (X, Z)}(8)
for all x and . Section 3 discusses the definition in a more general framework,
but here is the point. The left hand side of (8) depends on x not ; the right hand4

side, on  not x. Thus,
α = sup f (x, z)Q(dz|x) < inf E {f (X, Z)} = β.
xConsider a ‘lottery’ that pays f (x, z) when X = x and Z = z. The expected valueof the lottery, in advance of data collection, is at least β. Thus, a statistician (of thekind envisaged by de Finetti) should pay β to buy the lottery. On the other hand,
after seeing X, a statistician who uses the predictive distribution Q should happily
sell the lottery for α, no matter what X proves to be. The gap between α and β
reflects an inconsistency in the pricing, and leaves room for dutch book against Q.
Strong inconsistency is easily demonstrated, by constructing f and $. In viewof (7), there is a bounded measurable function h and $ > 0 with ∞
 ∞h(v)D1 (dv) + $ ≤h(v)D0 (dv).
(9)
−∞

−∞

√
Let f (x, z) = h(z1 / s11 ), where z1 is the first coordinate ofp × 1 column
thenvector z, and s11 is the (1,1) element of the p × p matrix s = i=1 xi xi . Now (9)boils down to (8), by the change-of-variables formula for integrals. On the left hand√side, D1 is the Q(dz|x)-distribution√ of z1 / s11 . On the right hand side, D0 is the
sampling distribution of T = Z1 / S11 . With our f , both sides of (8) are constant,by (4) and (6). This completes a sketch of the argument for strong inconsistency.
Some details are given next, and dutch book is discussed in Section 3.
2.1 Some Details
We begin with (5). Step 1 is showing that the predictive density φ(x) of
X = (X1 , . . . , Xn ), computed from the model and the Jeffreys
nprior in advance ofn/2
data collection, is Cn,p /|s| , where Cn,p is a constant, s = i=1 xi xi is positive
definite (a.e.), and |s| is the determinant of s. As in (2),
φ(x) = φ(x|)µ(d),where φ(x|) is the multivariate normal density,
φ(x|) = (2π)−np/2  −n/2 exp − 21 ni=1 xi  −1 xi ,and µ(d) = d/||(p+1)/2 is the Jeffreys prior. As before, x = (x1 , . . . , xn ),with xi a p × 1 column vector, and  is a p × p positive definite matrix. Since5trace(LM) = trace(ML) when both products are defined,n

xi  −1 xi = trace( −1 s) = trace(s 1/2  −1 s 1/2 ).

(10)

i=1

Since |LM| = |ML| for p × p matrices, −n/2 = s −n/2 s −1/2 s −1/2 −n/2 .
So

(11)


φ(x) =

φ(x|)µ(d)

−n/2
= s
ψ(s|)µ(d)

−n/2
= s
ψ(Ip×p |)µ(d),(12)where Ip×p is the p × p identity matrix,
 1
ψ(s|) = Dn,p (s) exp − trace(s 1/2  −1 s 1/2 ) ,2Dn,p (s) = (2π )−np/2 s −1/2 s −1/2 −n/2 .
The first line in (12) is just (2). The second line holds by (10) and (11), with a
bit of algebraic juggling for the constants. The third line holds because the Jeffreys
prior µ is invariant. Equation (12) is the required formula for the predictive density
φ(x) of X = (X1 , . . . , Xn ): the last integral in (12) is Cn,p , the mystery constantin that density. (Computing the integral is a task not lightly to be undertaken.) This
completes Step 1 in proving (5).
Step 2. The predictive density for Z when X = x is obtained from (3), as thequotient of (1) and (2). In view of (12),φ(z|x) = (Cn+1,p /Cn,p ) s n/2 s + zz −(n+1)/2 .

(13)

This simplifies to (5), because of the identity
Ip×p + ww = 1 + w w,
6

(14)

with w = s −1/2 z, a p × 1 column vector. To verify (14), let L rotate w into
(a, 0, . . . , 0) , so LL = Ip×p and a 2 = w w. The left hand side of (14) equals
LL + Lww L = Ip×p + Lww L = 1 + a 2 .
This proves (14), completing the argument for (5).
We turn now to (6). Let zj be the j th coordinate of the p × 1 column vector z;
recall that s11 is the (1,1) entry of the matrix s. What is the Q(dz|x)-distribution√of z1 / s11 ? When s = Ip×p , the density is proportional to 1/(1 + z1 2 )(n−p+2)/2 ,as one shows by integrating out z2 , . . . , zp . To do the integral, setzj = 1 + z1 2 wj
(15)
for j = 2, . . . , p. This proves (6) when s = Ip×p .
Next, Q(dz|x) is the distribution qs on R p whose density isφ(z|x) = C/ |s|(1 + z s −1 z)n+1 .
(16)Dependence on x is only through s = ni=1 xi xi : see (5). Abbreviate I = Ip×p .
Then qs is the qI -law of s 1/2 z. What remains to be seen is that√(17)
the qI -law of (s 1/2 z)1 / s11 does not depend on s,
where (s 1/2 z)1 is the first coordinate of the p × 1 column vector s 1/2 z. Claim (17)√is immediate from the invariance of qI under rotation: s11 is the 12 norm of thefirst row of s 1/2 . (Bear in mind that s 1/2 is symmetric.) Since qs is the predictive
distribution for Z when X = x, the argument for (6) is complete, and with it, theproof of Theorem 1.
2.2 Estimation
We have established strong inconsistency when the Jeffreys prior is used in a
multivariate normal prediction problem. A parallel – but more technical – argument
yields the same conclusion in an estimation context. Here is a brief sketch; readers
can skip to Section 3 without loss of continuity. Let X1 , . . . , Xn be independent
random p×1 vectors, with a common Np (0, ) distribution. Consider the Jeffreys
prior distribution µ(d) = d/||(p+1)/2 . Then the formal posterior distributionfor θ =  −1 given X1 , . . . , Xn is Wishart:θ ∼ W (S −1 , p, n),
7

(18)


with S = ni=1 Xi Xi . This is non-trivial, but can be verified starting from thefact that the Jeffreys prior µ is invariant under the transformation  →  −1 , by
Jacobian trickery. Given S, the posterior density of θ =  −1 with respect to µ is
proportional to1|θS|n/2 exp[− trace(θS)],
2−1
which is the density of W (S , p, n), as in Eaton (1983, p. 240). One analog ofthe statistic T is
θ11
U = 11 ,S
where S 11 is the (1,1) element of S −1 . When – given the data – θ has the distribution (18), thenU ∼ χn2(19)
for each S, as 
is immediate from the definition of the Wishart: if ζi are iid
Np (0, K), then ni=1 ζi ζi ∼ W (K, p, n). However, under the sampling model,
S is W (, p, n), and2U ∼ χn−p+1(20)
for any , by Proposition (8.7) in Eaton (1983). The fact that (19) and (20) are
different for p > 1 leads to strong inconsistency, as before.
3. Strong Inconsistency is Equivalent to a Dutch Book
Our purpose here is to show that strong inconsistency is equivalent to a dutch
book. A ‘measurable prediction problem’ consists of: (i) X ∈ X ; (ii) Z ∈ Z,
which is to be predicted from X; (iii) a set of parametric models {P (dx, dz|θ) :θ ∈ 7} specifying the joint distribution of X and Z. Here, X and Z are measurable
spaces, while X and Z are measurable functions on some underlying probability
space. A ‘predictive distribution’ Q(dz|x) is a distribution for Z that depends
on the observed value X = x. If x → Q(A|x) is measurable on X for every
measurable A ⊂ Z, we will say that Q is measurable.
One way to evaluate Q involves gambling scenarios, as follows. Consider ameasurable subset C ⊂ X × Z and let Cx = {z : (x, z) ∈ C} be the x-sectionof C. Then a ‘simple payoff function’ isψC (x, z) = IC (x, z) − Q(Cx |x).

(21)

If Q is measurable, then (x, z) → ψC (x, z) is measurable by the usual argument,
starting from measurable rectangles.
8By way of interpretation, ψC (x, z) is the net payoff to a gambler who puts
down Q(Cx |x) dollars to get a dollar if Z ∈ Cx . This net payoff is 1 − Q(Cx |x)
if Z ∈ Cx and −Q(Cx |x) if Z ∈/ Cx . No money changes hands if Cx = ∅ or Z.
(In particular, there are no interesting bets on x.)
Now, consider measurable subsets C1 , . . . , Ck of X × Z. After X = x isobserved, allow the gambler to pay bi (x)Q(Ci,x |x) in order to get bi (x) dollars if
Z ∈ Ci,x . The gambler is allowed to use any bounded measurable bi ; this is viewed
as encouraging honesty on the part of the odds-maker. Bets are settled separately,
and then summed. The net payoff to a gambler who uses the sets {C1 , . . . , Ck } andthe betting functions {bi : i = 1, . . . , k} isψ(x, z) =kbi (x) ICi (x, z) − Q(Ci,x |x) .

(22)

i=1

Any such ψ is called a ‘payoff function’. Clearly,ψ(x, z)Q(dz|x) = 0(23)

for all x. Thus, if your predictive distribution for Z – after observing X = x – isQ(dz|x), all these payoff functions seem fair.
Definition 1. Dutch book can be made against the predictive distribution Q(dz|x)
if there is a gambling system that provides a uniformly positive expected payoff to
the gambler: in other words, there exists a payoff function ψ – as defined by (22) –and an $ > 0 such that$ ≤ψ(x, z)P (dx, dz|θ) for all θ ∈ 7.
(24)
To paraphrase Freedman and Purves (1969),
Imagine a Master of Ceremonies who picks some θ ∈ 7 and thendraws (X, Z) from the model P (dx, dz|θ). The value of X = x
is revealed and the statistician announces the predictive distribution
Q(dz|x). The gambler then lays bets with payoff function ψ. When (24)holds, the gambler expects to win at least $ no matter what the value
of θ.

9

Of course, if (24) holds for some positive $, any other positive $ can be obtainedby rescaling the payoff function. We will say that dutch book can be made against
Q; more explicitly, dutch book can made against a bookie who – after seeing thatX = x – sets odds on Z using Q(dz|x). Other language abounds: for instance,
there is a dutch book; or, if Q is computed from an improper prior π, dutch bookcan be made against π.
In this paper, we allow only conditional bets, and obtain results on expected
loss for a non-Bayesian bookie. De Finetti allowed unconditional bets (Section 1),
and obtained results on actual loss. For more discussion, see Freedman and Purves(1969) or Sudderth (1994).
Definition 2. The predictive distribution Q(dz|x) is strongly inconsistent if there
exists a bounded measurable function f (x, z) and an $ > 0 such thatf (x, z)Q(dz|x) + $ ≤f (x, z)P (dx, dz|θ)(25)
for all x ∈ X and θ ∈ 7.
Equation (8) was a special case, with θ = . We now show that Q(dz|x) isstrongly inconsistent iff dutch book can made against Q.
Theorem 2. Let X ∈ X , Z ∈ Z, and {P (dx, dz|θ) : θ ∈ 7} be a measurable
prediction problem. Let Q(dz|x) be a measurable predictive distribution for Z
when X = x. Then Q(dz|x) is strongly inconsistent if and only if dutch book can
be made against Q.
Here is the argument. First, if (24) holds (dutch book), then f (x, z) ≡ ψ(x, z)
is bounded and (25) holds because of (23), proving strong inconsistency. For theconverse, assume (25) holds for some bounded measurable f and $ > 0. The left
side of (25) is a function of x only; the right side, of θ only. Thus,
sup f (x, z)Q(dz|x) + $ ≤ inf
f (x, z)P (dx, dz|θ).
(26)
θ

x

Since f is bounded, it can be uniformly approximated by a simple function f0 ,and
f0 (x, ζ )Q(dζ |x) ≤ α < β ≤f0 (x, z)P (dx, dz|θ)(27)
for suitable real numbers α and β. The inequality holds for all x ∈ X and θ ∈ 7;
using ζ for the variable of integration may be helpful later. Letf0 (x, ζ )Q(dζ |x).
f1 (x, z) = f0 (x, z) −Z

10

Plainly, f1 is a payoff function in the sense of (22). Now 
f1 (x, z)P (dx, dz|θ) = B − A,
Z X

 

where
B=
and

Z X

f0 (x, z)P (dx, dz|θ) ≥ β
 

  
A=

Z X Z

f0 (x, ζ )Q(dζ |x)P (dx, dz|θ) ≤Z X

αP (dx, dz|θ) ≤ α;

the inequalities hold by (27), establishing the long-sought dutch book, namely,
inequality (24) with $ = β−α and ψ = f1 . This completes the proof of Theorem 2.
The equivalence of strong inconsistency and dutch book also holds for estimation. To see this, just take Z = 7, and let Z be the identity map on 7. Initially,
P (dx|θ) will be defined only on X ; we require some measurable structure on 7
and set P (A × B|θ) = P (A|θ)1B (θ) for measurable A ⊂ X and B ⊂ 7. A predictive distribution is a ‘posterior’ for θ given X = x. Quotes are needed unless
Q is computed from a proper prior – but then, there are no paradoxes to discuss.
4. Literature Review
Ramsey (1926) introduced the idea of betting odds as a means of assessing
probability assignments. De Finetti (1931, 1937) used similar ideas in his discussion of what is now commonly known as coherence.
di un individuo che debba tenere un banco di scommesse su dati eventi,
accettando alle stesse conditizioni qualunque scommessa nell’uno onell’altro senso. Vedremo che egli è costretto allora a rispettare certe
restrizioni, che sono i teoremi del calcolo delle probabilità. Altrimenti
egli pecca di coerenza, e perde sicuramente, purchè l’avversario sappia
sfruttare il suo errore. De Finetti (1931, p. 305).
In free translation,
A person who is obliged to accept bets in any amount, positive or negative, on any finite combination of events, must fix prices according to
the laws of probability theory. Otherwise, this person sins against coherence and loses money with certainty, provided the opponent knows
how to exploit the mistake.
11

Some time afterward, the term ‘dutch book’ entered the lexicon as a synonym forincoherence. The earliest cite we could find was Lehman (1955), although use of‘dutch’ as a pejorative dates back to 17th-century England.
Freedman and Purves (1969) gave the dutch-book idea careful mathematicalexpression, for prediction and estimation, when all the spaces are finite. Rigorous
treatments in a finitely additive setting for the estimation problem can be found in
Heath and Sudderth (1978, 1989). Extensions to the prediction context appear in
Lane and Sudderth (1984). Heath, Lane, and Sudderth allow infinite spaces.
Another foundational idea, ‘strong inconsistency’, was introduced by Stone(1976), and later adapted to the predictive setting by Lane and Sudderth (1984);
also see Eaton and Sudderth (1993, 1999). The equivalence of strong inconsistency and incoherence is discussed in the finitely additive setting by Lane andSudderth (1983). Proof in the countably additive setting is a little different (Section 3). At the risk of the obvious, strong inconsistency is an exact finite-sample
property, rather than an asymptotic large-sample property.
That improper prior distributions can give rise to posterior distributions with
disturbing properties has been known since at least the 1970’s. Stone (1976) and thediscussants of Stone’s paper provide examples, including the Jeffreys prior. Eaton
and Sudderth (1993, 1995, 1998, 1999, 2001, 2002) discuss invariant prediction
problems, and show that in the multivariate linear model, fully invariant predictive distributions are strongly inconsistent: the ‘principle of invariance’ (Berger,
1985, p. 390) therefore leads to dutch book. The Jeffreys prior can be viewed asa prototype where elementary arguments suffice (Section 2). In contrast, much of
the Eaton–Sudderth work relies on separation theorems of the Hahn–Banach type,which makes the results less accessible. Eaton and Sudderth (1999, Section 8)
show that if the transformation group is amenable, there will be an invariant predictive distribution immune to dutch book – although other invariant predictive
distributions will be vulnerable. If the transformation group is not amenable – likethe non-singular linear transformations on R p for p > 1 in Section 2 – all invariant
predictive distributions may be subject to dutch book (Eaton and Sudderth, 1998).
The uniqueness of the invariant prior in Section 2 above is demonstrated by
Eaton (1983, Example 6.19). Use of this prior has been suggested by Jeffreys(1961, pp. 180–81), Box and Tiao (1973, p. 426), Box and Tiao (1992, Section 8.2.2), Geisser (1993, Chapter 9), Schervish (1995, p. 122), Keyes and Levy
(1996). The latter also has a good survey of invariant predictive distributions in
multivariate analysis of variance. For an interesting generalization of (14), see
Eaton (1983, p. 43). Although incoherence is often a synonym for the possibilityof a dutch book, other definitions have been suggested. See Regazzini (1987) as
12

well as Berti, Regazzini, and Rigo (1991). For more discussion, see Sudderth(1994, Section 7).
Acknowledgments
We thank Eugenio Regazzini and Philip Stark for helpful comments.
References
Berger, J.O. (1985) Statistical Decision Theory and Bayesian Analysis, 2nd edn.
New York: Springer.
Berti, P., Regazzini, E., and Rigo, P. (1991) Coherent statistical inference and Bayes
theorem. Ann. Statist., 19, 366–381.
Box, G.E.P. and Tiao, G.C. (1973) Bayesian Inference in Statistical Analysis. Reading, Mass.: Addison-Wesley.
Box, G.E.P. and Tiao, G.C. (1992) Bayesian Inference in Statistical Analysis. New
York: Wiley. Reprint of Box and Tiao (1973).
de Finetti, B. (1931) Sul significato soggestivo della probabilità. Fundamenta
Mathematicae, 17, 298–329. English translation in Bruno de Finetti (1993) Probabilita’e Induzione (Induction and probability). Paola Monari and Daniela Cocchi(Editors). Bologna: Biblioteca di Statistica Editrice Clueb.
de Finetti, B. (1937) La prévision: ses lois logiques ses sources subjectives. Ann.
Inst. H. Poincaré, 7, 1–68. English translation in H. E. Kyburg, H. E. Smokler(eds), Studies in Subjective Probability. New York: Wiley, 1964.
Eaton, M. L. (1983) Multivariate Statistics. New York: Wiley.
Eaton, M.L. and Sudderth, W.D. (1993) Prediction in a multivariate normal setting:Coherence and incoherence. Sankhyā, Special Volume 55, Series A, Pt. 3, 481–493.
Eaton, M.L. and Sudderth, W.D. (1995) The formal posterior of a standard flatprior in MANOVA is incoherent. J. Ital. Statist. Soc., 2, 251–270.
Eaton, M.L. and Sudderth, W.D. (1998) A new predictive distribution for normal
multivariate linear models. Sankhyā, Series A, Indian J. Statist., 60, 363–382.
Eaton, M.L. and Sudderth, W.D. (1999) Consistency and strong inconsistency ofgroup-invariant predictive inferences. Bernoulli, 5, 833–854. See esp. p. 849.
Eaton, M. L. and Sudderth, W. D. (2001). Best invariant predictive distributions. In
M.Viana and D. Richards (eds), Contemporary Mathematics Series of the American
Mathematical Society, 287, 49–62. Providence, Rhode Island.

13

Eaton, M.L. and Sudderth, W.D. (2002) Group invariant inference and right Haar
measure. J. Statistical Planning and Inference, 103, 87–99.
Freedman, D.A. (1995) Some issues in the foundations of statistics. Foundationsof Science, 1, 19–83, with discussion. Reprinted in B. van Fraassen (ed), Topics in
the Foundation of Statistics. Dordrecht: Kluwer, 1996.
Freedman, D.A. and Purves, R.A. (1969) Bayes method for bookies. Ann. Math
Statist., 40, 1177–1186.
Geisser, S. (1993) Predictive Inference: An Introduction. NewYork: Chapman Hall.
Heath, D. and Sudderth, W.D. (1978) On finitely additive priors, coherence, andextended admissibility. Ann. Statist., 2, 333–345.
Heath, D. and Sudderth, W.D. (1989) Coherent inference from improper priors andfrom finitely additive priors. Ann. Statist., 17, 907–919.
Jeffreys, H. (1961). Theory of Probability, 3rd edn. Oxford University Press.
Keyes, T.K. and Levy, M.S. (1996) Goodness of prediction fit for multivariate
linear models. J. Amer. Statist. Assoc., 91, 191–197.
Lane, D and Sudderth, W.D. (1983) Coherent and continuous inference. Ann.
Statist., 11, 114–120.
Lane, D. and Sudderth, W.D. (1984) Coherent predictive inference. Sankhyā, Series
A, Indian J. Statist., 46, 166–185.
Lehman, R.S. (1955) On confirmation and rational betting. J. Symbolic Logic, 20,
251–262.
Ramsey, F. (1926) Truth and Probability. Reprinted in H.E. Kyburg and H.E. Smokler (eds), Studies in Subjective Probability, pp. 25–52. Huntington, New York:
Krieger Publishing, 1980.
Regazzini, E. (1987) De Finetti’s coherence and statistical inference. Ann. Statist.,
15, 845–864.
Schervish, M.J. (1995) Theory of Statistics. New York: Springer.
Stone, M. (1976) Strong inconsistency from uniform priors. J. Amer. Statist. Assoc.,
71, 114–125.
Sudderth, W.D. (1994) Coherent inference and prediction in statistics. In D. Prawitz,
B. Skyrms, and D. Westerstáhl (eds), Logic. Methodology and Philosophy of Science IX. Amsterdam: Elsevier Science.

14

15 March 2004
Technical report no. 642
Department of Statistics
University of California
Berkeley, CA 94720-3860
www.stat.berkeley.edu/˜census/642.pdf

15