exchangeable belief structures

Exchangeable Belief StructuresAuthor(s): Michael GoldsteinSource: Journal of the American Statistical Association, Vol. 81, No. 396 (Dec., 1986), pp. 971-976Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2289070 .

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Exchangeable Belief Structures MICHAEL GOLDSTEIN*

The static features of beliefs, namely those structures that relate to one's beliefs at a particular moment, are described. It is argued that expectation (or prevision) should be the fundamental quantification of individual statements of uncertainty and that inner product spaces (or belief structures) should be the fundamental organizing structure for collections of such statements. Objections are given to the usual Bayesian justification of relative frequency-based statistical models via exchangeability. I offer an alternative approach, via exchangeable and co-exchangeable belief structures, and derive the representation theorem, and thus the implied statistical models, for these structures. KEY WORDS: Linear methods; Prevision; Relative frequency; Repre- sentation theorems.

1. INTRODUCTION

Now that Bayesian ideas have become part of the statistical orthodoxy, it is reasonable to ask whether we can find better (simpler, more efficient, easier to program, better justified, of wider application) methods for achiev- ing the same apparent objectives.

In this article, I consider the static features of beliefs, namely those structures that relate to beliefs at a particular moment. I consider that beliefs are worthy of logical study in their own right (as well as providing the basis for various practical methodologies). Thus the purpose of the article is to provide a general system by which an individual can coherently specify and revise beliefs. This system is not restricted to simple idealized versions of the world or to hypothetical perfectly rational individuals but relates to the actual, structured belief statements that the individual chooses to make. In particular, I pay close attention to the interpretation of "statistical models" through various notions of exchangeability.

2. PREVISION AND BELIEF STRUCTURES

In this section, the objects of interest in the theory are established. The guiding principle for the constructions is that we should respect as closely as we can the limits on our ability to specify beliefs.

2.1 Prevision

The basic ingredients of "probabilistic" reasoning are probabilities and expectations. There are great advantages in considering expectation to be the fundamental quantity, as we can make any probability statement as a single expectation statement for the indicator function of the event, whereas if probability is fundamental, then a single expectation statement may require probability specifications over an "infinite" partition of possible values that the quantity can take. In problems with a realistic level of com- plexity, we will be unwilling and unable to do this.

I define expectation as a primitive notion, following the notation and development of de Finetti (1974), where the expectation of a random (i.e., unknown) quantity, X, is

* Michael Goldstein is Reader in Statistics, Department of Statistics, University of Hull, Hull, HUG 7RX, N. Humberside, England.

formalized by the concept of the prevision of X, P(X). [De Finetti provided a full operational development of prevision, emphasizing the fundamental coherence property P(X + Y) = P(X) + P(Y) for any X, Y.]

The prevision of X does not "replace" the probability distribution of X. Probabilities over limiting partitions, however, are a much richer set of previsions than you would typically wish, need, or be able to specify. I further suggest that, in most cases of substance, the arguments that you make depend fundamentally on expectation properties. It is not possible to rephrase these arguments in terms of probability, as they involve random quantities whose expectation properties are an immediate conse- quence of your specifications, but whose probabilistic specification would be, at best, extremely laborious.

2.2 Belief Structures

As subjectivist theory addresses the relationships between coherent statements of belief, the object of study here is organized collections of belief statements. Our choice of organizing framework must have enough structure to exhibit all of the fundamental properties of the theory while not having any extraneous restrictions, imposed simply for convenience in a few applications. I now describe such a framework, which I term a belief structure. This is a minimal organization of beliefs (for purposes of "general reasoning"). [A related construction is described in de Finetti (1974, sec. 4.17). De Finetti briefly exhibited the construction to give simple geometric illustrations of general ideas.]

The structure is built in two stages. First, we make explicit the linear structure of prevision. Thus suppose that you have specified your prevision for each of a collection X, ...., Xk of random quantities. This specification uniquely fixes your prevision for all linear combinations a1X1 + ... + akXk (and no other general functions).

We add the unit constant X0 (i.e., X0 is identically 1) and denote by L the vector space in which each Xi is represented as a vector and linear combinations of vectors are the corresponding linear combinations of random quantities. Note that, for example, X and X2 are different random quantities (as they are not in general linearly related), although they depend on the same (mathematical) vari- able. Thus they will correspond to different vectors Xi, Xj, if you declare previsions for both X and X2. You may choose to declare either, neither, or both.

Having fixed the linear structure, we superimpose the multiplicative, geometric structure as follows.

Definition. A belief structure is an inner product space, A, constructed as follows.

1. Choose a specified collection of random quantities,

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December 1986, Vol. 81, No. 396, Theory and Methods

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972 Joumal of the American Statistical Association, December 1986

C = [X0, Xl,. . . . Xm] (not necessarily finite and always including the unit constant). Call C the base of the belief structure A, denoted by C = b(A).

2. Construct the linear space L of all finite linear combinations of elements of C.

3. Construct the inner product and norm over L defined for each X, Y by (X, Y) = P(XY), IIX1I = (X, X)1"2.

[We restrict C to contain elements X for which P(X2) is finite. We identify "equivalence classes" of vectors whose differences "coincide with 0," that is, whose differences D satisfy P(IDI ? c) = 0, for every positive c.]

The base of the structure summarizes those aspects of your beliefs about which you are both able and willing to make explicit previsions to sufficient level of detail. Note that we can add two belief structures E and F by constructing the combined space E + F spanned by all of the vectors in E and F This involves the additional specification of P(ef) for each e in b(E) and f in b(F).

Compare Bayes specification, which concerns a probability space W, and a prior probability measure P over W. The implied belief structure A, in this case, is the Hilbert space of all square integrable functions over W with respect to P, with inner product (f, g) = ff(w)g(w) dP(w). The general belief structure can be considered to be any closed subspace of such a structure. Thus the specification of a belief structure must be at least as simple as the corresponding Bayes specification, and we are not concerned with "second-order structure" but rather any (linear) slice of a general joint measure. Our development is more general, however, in that we do not require the existence of full measures.

Consider an example. A doctor is about to examine a patient. He will take a series of measurements, such as blood pressure, temperature, and so forth. He explicitly declares those aspects of his beliefs that he will formally express by constructing a list, C, of quantities in the base of his belief structure A. He then specifies his prevision for each element and each cross-product in C to form A. He must decide whether to include X, log(X), the indicator for X positive, the product of X and Y, and so forth (so he has the option of not including almost all of these quantities).

Under Bayes specification requirements, the doctor would have to list all possible combinations that the quantities could take and then specify his probability for each possible outcome. This is an impossible task. There is no reason to impose this arbitrary burden on the theory. All of the features of the usual Bayesian analysis can be viewed simply as special cases of the corresponding analysis of a general belief structure. (Aspects of this will be demonstrated subsequently; the remainder will be reported elsewhere.) Probability specifications are better viewed as simple, easy- to-manipulate "models" of belief structures. [Further de- tails are given in Goldstein (1984).]

3. EXCHANGEABILITY AND RELATIVE FREQUENCY Relative frequency ideas form the basis for almost all

probabilistic and statistical training, in all quantitative dis- ciplines. It is impossible to find a precise statement, how-

ever, as to what relative frequency means. The most careful definition of relative frequency to date uses exchangeability, as described in de Finetti (1936; translated in Kyburg and Smokler 1964). Consider the simplest example: tossing coins. Compare sampling with replacement from an urn with a proportion p of items black and the rest white with the coin tossing problem. Traditional statistical theory em- phasizes the similarity between these two cases. There are also, however, basic differences. You can open the urn and physically determine p, but there is no corresponding op- eration on the coin yielding a value p. De Finetti (1936; translated in Kyburg and Smokler 1964) wrote The difference between these two cases is essential, and it cannot be neglected; one cannot "by analogy" recover in the second case the reasoning which was valid in the first case, for this reasoning no longer applies in the second case. (p. 142)

Exchangeability attempts to reconcile the two cases as follows. Your joint probability measure over an infinite exchangeable sequence can be represented as a mixture of iid sequences from probability distributions, F, under some mixing measure, P, over F Essentially, the representation theorem claims to give meaning to the "long-run frequency" of tossing heads by creating an "urn" for the coin tosses, this urn being the actual future tosses of the coin. How should we assess this claim? In rejecting relative frequency as a fundamental concept, I simply take a skeptical view of vaguely defined, totally unobservable quantities, preferring to express concrete arguments about observable quantities. The raw materials that the exchangeability argument uses are your probabilities for every possible sequence of outcomes that the coin might display. To apply the argument you must specify all such probabilities for all sequence lengths. You still require an infinite sequence of tosses of the coin (because you can only attach probabilities to "real" quantities). But we have now imposed an extra ingredient, namely an infinite number of probability as- sessments that you will never have the inclination or ability to make.

This is how foundations become irrelevant! Who uses exchangeability to construct models? Who even knows what exchangeability is? (I was not aware quite what a cult topic this was until I gave seminars on exchangeability, only to be informed routinely that very few members of the au- dience had even heard of the term.) Can we build models directly from actual belief specifications? This depends on our approach to exchangeability.

Long-Run Frequency

To motivate our formal development, consider the "true but unknown" probability p that a coin will land heads. The Bayesian approach denies the frequentist notion of a physical limit for the outcomes but constructs p from an infinite sequence of (hypothetical) probability judgments. A probability judgment, however, cannot have the status of a "4real but unknown" quantity. Either it is specified or it is not.

It may be that certain actual beliefs that you express about the sequence are not sensitive to the ordering. Con- sider the minimal number of such statements that you could



Goldstein: Exchangeable Belief Structures 973

make. First, you might consider your probability, q,, that the rth toss is a head. If the sequence is to be exchangeable, then the values q, will take some constant value, Pi, for all r. Next you might consider the probability, qij, that both the ith and the jth toss are heads. Again, exchangeability demands that qij is a constant value, P2, for i # j. You can hardly avoid making these evaluations if you want to ana- lyze the sequence. They are enough, however, to generate a representation of the "unknown" probability p, as follows.

Call P,, the proportion of heads in the first n tosses. Write Pn = (I, + ... + In)In, where Ij = 1 if the jth toss is heads, 0 otherwise. You have assigned EIj = Pi for each j, E(IjIk) = P2, for j # k. Your specifications imply that for any n < m you must specify E(Pn - pm)2 = ((1/n) - (lIm))(p1 - P2). Thus Pn is a Cauchy sequence in mean square, so if the sequence is in principle of infinite length, your stated beliefs are consistent with the existence of a further random quantity P for which limnE(Pn - p)2 = 0.

Thus your actual exchangeability specifications lead directly to a further quantity P similar to the quantity p in the usual representation theorem. That is, P is a hypothetical random quantity consistent with your stated beliefs, whose construction simplifies and clarifies your analysis.

The conditions imposed on P by the mean squared con- vergence of Pn imply the following representation. For each r, I, = P + Jr, where (i) E(P) = Pl, var(P) = P2 - 14, (ii) E(Jr) = 0, for all r, (iii) var(Jr) = Pl - P2, for all r, (iv) the sequence, J1, J2, . .. , is uncorrelated, and (v) P is uncorrelated with each Jr.

Compare the usual representation theorem, in which your (fully specified) beliefs are consistent with p, conditional on which your beliefs are independent and identical for each toss. In our construction "given P" your beliefs over tosses are uncorrelated with identical mean and vari- ance for each toss. This is the difference between hypothetical idealized conclusions and those following from stated beliefs.

4. EXCHANGEABLE BELIEF STRUCTURES

We now give the general formulation. There is a general collection of situations (such as a doctor examining patients) that is, in a sense to be defined, exchangeable. We must establish a "frequency model" from a general representation theorem, which is formulated as an intrinsic property of the formal structure of Section 2, rather than, as in the preceding section, depending on some implicitly defined and illusory underlying probability measure.

4.1 Definitions

Consider a doctor examining patients. For each patient the doctor will make a series of observations. Suppose that he wishes to establish a structured representation of certain aspects of his knowledge related to these examinations. First, he constructs one belief structure for each patient. Thus he constructs a base for each structure. He is inter- ested in features common to patients and so constructs the

same base for each individual. (For example, if he considers certain aspects of blood pressure for one individual, then he considers the same aspects for each other individual.) We express this by constructing a pure base for the system: a collection C = [X(O), X(1), X(2), . . .], finite or infinite, where each X(i) is a real valued function of the quantities such as blood pressure, age, weight, and so forth that he will observe on each patient. [By our previous convention X(o) = 1.] For example, if height and weight are observed, then one element of C might be log(weight/ height), if this was a meaningful index to the doctor.

From C we generate a base for each case. For case i, the base Ci is [Xoi, X1i, X2i, . . .], where Xrs is the value of quantity X(,) for case s [e.g., if X(t) = log(weight/height), then Xrs = log(weight of patient s/height of patient s)]. C, is a set of observable random quantities that forms the base of a belief structure Ai, relating to case i. We form a com- posite belief structure A* containing each Ai as a subspace from the base C*, the union of all of the elements in all of the bases Ci, where i ranges over some indexing set of cases. We refer to C*, A * as the full base and the full belief structure. Exchangeability applies to beliefs about sequences that are unaffected by permuting the order of the sequence. This imposes two sets of constraints. (i) You express similar beliefs about each case, so (Xis, Xj) = P(XisXjs) = [X(i), X(j)] (a constant for all s). (ii) You express similar beliefs about each pair of situations, so (Xir, Xjs) = P(XirXjs) = (X(i), X(j)) (a constant for all r # s).

If conditions (i) and (ii) hold, then the full belief structure A* is uniquely determined by the specification of the belief structure A1 + A2. Thus whatever the number of previsions involved in the specification of A*, what is ac- tually required is the consideration of two cases, with all other values following from the perceived "symmetries" between cases. Therefore, we make this our formal definition.

Definition. The pure base, C, is exchangeable over the full base C*, or equivalently, A* is composed of exchangeable situations Aj, if the previsions assessed for each pair of elements of C* satisfy the aforementioned conditions (i) and (ii).

As an illustration, consider again the coin tossing example of Section 3. Here, a case is a toss of the coin. The pure base C is the constant and I, the indicator function for heads. So the base Cj is [XO, Ij], where Ij is the indicator for heads on toss j. Conditions (i) and (ii) reduce to those imposed in our earlier discussion of coin tossing, for example, (XO, I,) = (Ij, I) = P(I) = Pi, a constant for all j, and (Ir Is) = P(IrIs) = P2, a constant for all r $0 s.

The defining features of a case and the elements of the base are chosen to express whatever features are judged to be important. (For example, we might take the exam- ination of two patients as forming the structure A1 to express beliefs about the relationships between pairs of patients.)

4.2 The Representation Theorem The usual representation for exchangeable sequences

summarizes all of the probabilistic relationships between



974 Journal of the American Statistical Association, December 1986

the infinite collection of quantities by constructing a further random entity, F, conditional on which the sequence is iid from F We summarize all of the relationships expressed between the structures Aj by constructing a further belief structure, B, so that all of the relationships between the various Aj may be represented by the common relationship between each Aj and B. The construction is as for the coin tossing example. For every element of the pure base we form the average of the corresponding elements in the first n structures Aj. These averages are Cauchy sequences in A * and so have limit points in the closure of A *. These limit points form the base of the structure B.

Recall that any inner product space S can be embedded in the minimal closure of the space, S*. Essentially this is done by adding, for each Cauchy sequence [s,] of elements of S for which the limit point of the sequence does not exist in 5, a new element s* whose inner product with each element s in S is (s, s*) = limj(s, sj).

We have the following representation.

Theorem (the representation for infinite exchangeable belief structures).

1. Suppose that the full belief structure A* is composed of an infinite number of exchangeable situations Ai. There exist, within A** (the closure of A*), a series of mutually orthogonal belief structures B, R1, R2, . . ., so that for each j, Ai is a subspace of B + R1.

2. The space B is constructed as follows. If the pure base for the system is [X(O), X(1), X(2), . . .], then the base for B is [XO, X1, X2, ... .], where each Xj is the Cauchy limit of the sequence of partial means Sjn = (Xjl + ... + Xjn)/n.

3. The inner product over B is given by (Xi, Xj) = (X(i), X(j)) [the common value assigned to each P(XirXjs) for all r :A s]. Further, the inner product between A * and B is given by (Xj, Xrs) = (X(j), X(t)), for all j, r, s.

4. The base for Rr is [Vir, V2r, . . ], where VIr =Xi -

Xj, for each j. All elements of each Rr have zero prevision, and the inner product over each Ri is essentially equivalent; that is, for all i, j, r, s (Vir, Vjr) = (Vis Vjs) = [X(i) X(1j)]- (X(i), X(j)).

Proof. For each X(j) in the pure base, b(M), let Sjn = (XjI + - + Xj,)In. For any n < m, we have

jSj, - Sjpjf2 = ((1/n) - (11m))([X(j), X(j)] - (X(j), X(j))),

where [X(j), X(j)] is the common value of P(Xjr). Thus each sequence Sjn in A* is a Cauchy sequence in

A*, so Xj = limnSjn exists in A**, and for each Y in A**, (Xi, Y) = limn(Sin, Y)-

Let B be the subspace of A * * generated by the collection of all of the quantities of form Xj, for X(j) in b(M). For any two such quantities Xi, Xj, and any r, s, we have-

+ (Xi1n Xrs))Iln = (X(j), X(r)),

(Xi, Xj) = limn((Xil, Xjs) + *'

+ (XYn X)))n = (X(Y-, X()).

Thus for any X(i), X(j) in b(M), and any r # s, we have

(Xir, Xjs) = (Xir, Xj) = (Xi, Xj) SO (Vir, XA) = (Vir,

Vi.) = 0. From these relations, it is immediate that the spaces B,

R1, R2, ... . , described in the statement of the theorem are mutually orthogonal, with inner product as stated. As each Xjr = Vjr + Xj, it is immediate that each Aj is a subspace of B + R1, and the theorem follows.

We call the derived spaces B, Rj in the previous representation theorem the pure belief structure for the infinite exchangeable system and the residual belief structure for case j. The pure belief structure B summarizes A* as follows. As each Ai is a subspace of B + Ri, and the spaces B, R1, R2, . . ., are mutually orthogonal, the relationship between any two quantities Xir, Xjs (r $A s) is entirely determined by the relationship between the orthogonal pro- jections of Xir, Xjs into B. The projection of Xir into B is Xi, so the relationship is that, for r :# s, (Xir, Xjs) = (Xi, X1). Note also the following relations:

I.X1112 = (X(i) .X(i)), ljI.A12 = I.X112 + IIVqj12.

As in our coin tossing example, our construction pro- duces a belief structure given which the residual belief structures are orthogonal (i.e., noncorrelated). I will demonstrate elsewhere the effect of these properties on the revision of beliefs. (Intuitively, large samples eliminate uncertainty over the pure structure, but have no effect on residual uncertainties.)

As an illustration, consider the coin tossing example. The pure belief structure, B, for this system has base [XO, PI, where P is the Cauchy limit of the sequence S", in this case the proportion of heads in the first n tosses. The inner product over B is given by (XO, P) = pI, (P, P) = P2. The inner product between P and A* is fixed by (P, I,) = P2, for all j. The base for Rj is [Ij - P]. The mutual orthogonality between the spaces B, RI, R2, . . ., corresponds to the representation that was given where each Ir = P + Jr, where P, JI, J2, . .. , are mutually uncorrelated. The other properties of the representation given in (3.2) follow similarly. For example, with I as the indicator for heads in the pure base, var(Jr) = IIJrII2 = [I, I] - (I, I) = Pl -P2-

I have emphasized that our specifications are based on explicit construction of A1 + A2 and implicit extension, by symmetry, to all other cases (such as all other coin tosses). There will be circumstances, however, for which there is a clear upper bound on the number of cases that it is meaningful to consider. If A* is composed of only a finite number, m, of exchangeable structures Ai, that is, A* = A1 + ... + Am, then the representation theorem takes the following form. If the pure base of A* is [XO, X(1), X(2), ... I, then the base of the pure belief structure, B, is [XO, X1, X2,.. .1, where Xi = (Xi, + * + Xim)/m. The base of the rth residual structure, Rr, is [VOr, Vir, V2r, ... *] where Vi, = Xi, - Xi. Each Ai is contained in B + Ri, and each Ri is orthogonal to B. The residual structures, however, are not orthogonal, but

(Vir, Vjr) = (m - 1)Im([Xi, X1] -(Xi, Xj)),

(Vir, VjS) = - IVir, V,r)/(m - 1), for all r # s.



Goldstein: Exchangeable Belief Structures 975

Thus any inferential arguments based on strict orthogonality between residual spaces will require modification in this case for "orthogonality of order 1/m."

4.3 Co-exchangeable Belief Structures I now describe the relationship between an exchangeable

belief structure and a further belief structure relevant to a particular problem. Suppose that with notation as in (4.1), A* = A1 + A2 + * is an exchangeable system, and that there is another collection of random quantities of interest [Zo, Zl, Z2, . .] which form the base for a further belief structure F We say that F is exchangeable with A* if the inner product on each space (Aj + F) is essentially the same, that is, if for each i, r, j, P(ZiXrj) = air, a constant for all j.

Any such structure inherits the representation theorem for the exchangeable system, in that F is related to each individual belief structure Ai strictly through the pure belief structure for the system.

Theorem. If the belief structure F is exchangeable with the infinite exchangeable belief structure A *, then all of the relationships between F and A* are contained in the relationship between F and B(A *); that is, in the combined belief structure (A** + F), F is orthogonal to each residual belief structure.

Proof. A typical base element of the residual structure R, is Xrs - Xr. A general element Zj in the base of F satisfies

(Zj, Xrs - Xr) = P(ZjXrs)

- limnP(Zj(Xrl + * + Xrn)In) = 0,

by exchangeability, and the result follows.

Often we will construct exchangeable structures that are themselves composed of exchangeable belief structures. For example, a doctor might wish to compare several treatments. Patients within a treatment might be considered exchangeable. Typically, patients would not be exchangeable across treatments, but the structures would be exchangeable as above. Thus I extend our previous definition as follows. Two exchangeable systems, A* = A1l + A12 + ***, and A* =A2 +A22 + , are termed co- exchangeable if each A1l is exchangeable with A and each A2s is exchangeable with A . Taking typical base elements of Alr, A2s as Xiir, X2j,, respectively, this is equivalent to the requirement that for all i, r, j, s, P(XlirX2js) = ai1, a constant for all r, s. The counterpart to the above theorem is that all of the relationships between co-exchangeable belief structures are contained in the relationships between the corresponding pure belief structures.

Theorem. Suppose that A *, A*, A*, ... , are a sequence of co-exchangeable infinite exchangeable systems. For each i, denote the pure belief structure and the sequence of residual structures for system A$' as Bi, Ri1 (i = 1, 2, . . .). In the combined belief structure (Ar * + A2 * + A3 * + ...), all of the residual belief structures, Ri1, are mutually orthogonal and are also orthogonal to all of the pure belief structures, Bi.

(The proof is as for the preceding theorem.)

As a simple example, suppose that a doctor conducts a double-blind trial between m treatments. Patients are ran- domly assigned to treatments, and the doctor does not know which treatment is which. Patients within a trial are exchangeable. Between treatments, patients are co-exchangeable. The doctor constructs a pure base [X(o), X(1), X(2), . . .1. (Perhaps X(1) is age, X(2) is current severity of disease, X(3) to X(6) are indicator functions for different degrees of recovery, etc.) He then specifies A1l, his belief structure for the first patient on treatment 1. Because of the nature of the trial, he assigns the same inner product over each Aij. He then specifies A1l + A12, his beliefs for the first two patients on treatment 1 incorporating his beliefs as to similarities between patients on the same treatment. Again the inner product will be the same for each Ari + Arj (i # j). This completes specification of each A*. Finally, he constructs A1l + A21, his beliefs for the first patient on each of treatments 1 and 2, representing his beliefs about the degree of similarity between treatments. Again this is the same for each pair An + Asj (r $ s). Thus the full structure, A* = A + A + , is determined.

Applying the representation theorem to each treatment A separates for each patient j the structure Aij as Bi + Rij, the pure and residual structures for each patient. Each Bi has base [XO, Xi,, X12, . . .], with essentially the same inner product expressing beliefs about average treatment effects. Each Rij expresses beliefs about individual differences and has essentially the same inner product. All elements of each Rij have zero prevision. All of the spaces R1j are mutually orthogonal and also orthogonal to each Bi. Each Bi + Bj has the same inner product. As the Bi are finitely exchangeable, we decompose each Bi into B + Ri, where each Ri is orthogonal to B (the "overall treatment effect" structure), although the Ri, expressing differences between treatments, are not orthogonal.

5. REVISING EXCHANGEABLE BELIEF STRUCTURES I have described the "static" features of beliefs. It is the

dynamic features, however, namely how beliefs are mod- ified over time with new information, that determine the system. For example, the Bayesian formalism requires a full specification of probability distributions, because beliefs are updated by conditioning. The approach in this article is likewise guided by the principles that we envisage for the revision of belief structures. In a sequel I will describe the dynamic features of beliefs and demonstrate that for an exchangeable belief structure this revision has a simple, intuitive form and that the pure belief structure for such a system does satisfy the properties that we usually require of a statistical model.

Such an analysis requires a clear set of inferential principles for our system. For a variety of reasons, logical, practical, and philosophical, I find Bayesian conditioning unsatisfactory as the basis for our inferential procedures (though it is a useful special case). We need to construct an inferential logic that respects the limitations of the individual in anticipating and assessing evidence. This will



976 Journal of the American Statistical Association, December 1986

involve the exploitation of various basic notions introduced in Goldstein (1981, 1983, 1985a,b, 1986) and will be reported in detail elsewhere.

[Received February 1984. Revised January 1986.]

REFERENCES

De Finetti, B. (1974), Theory of Probability (Vol. 1), London: John Wiley.

Goldstein, M. (1981), "Revising Previsions: A Geometric Interpreta- tion," Journal of the Royal Statistical Society, Ser. B, 43, 105-130.

(1983), "The Prevision of a Prevision," Journal of the American Statistical Association, 78, 817-819.

(1984), "Belief Structures," Technical Report 2738, University of Wisconsin-Madison, Mathematics Research Center.

(1985a), "Temporal Coherence," in Bayesian Statistics 2, eds. J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, Amsterdam: North-Holland.

(1985b), "Adjusted Belief Structures," Technical Report 2804, University of Wisconsin-Madison, Mathematics Research Center.

(1986), "Separating Beliefs," in Bayesian Inference and Decision Techniques, eds. P. Goel and A. Zellner, Amsterdam: Elsevier.

Kyburg, H. E., and Smokler, H. E. (eds.) (1964), Studies in Subjective Probability, London: John Wiley.



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