SIMON F R E N C H

ON T H E A X I O M A T I S A T I O N O F

S U B J E C T I V E P R O B A B I L I T I E S

ABSTRACT. This paper essentially makes two remarks that are pertinent to many of the axiomatisations of subjective probability. First, the auxiliary experiment used to quantify qualitative feelings of relative likelihood is essentially distinct from the field of events of actual interest and may be kept so in the axiomatisation. Second, all theories of subjective probability agree that beliefs are conditional on the present state of know- ledge and on the present mood and attitude of the individual concerned. As the indi- vidual moves through time this conditioning set, his knowledge and psychological state, change. This ever present change has implications for the conditions under which Bayes' Theorem may be invoked to prescribe how the individual should rationally update his beliefs in the light of a particular observation.

1. I N T R O D U C T I O N

In truth, I believe that I have very little new to say in the following. Rather I

would see it as a commentary on axiomatisations of subjective probabil i ty, one

which emphasised two points which, although hinted at, are not often dwelt

on in the literature. Certainly I believe that they deserve more at tention than

they received. Briefly the points that I wish to make are as follows.

In order to ensure the existence of a probabi l i ty measure, which represents

the individual's degree of belief, all axiomatisations must demand that the

space of events and his perception of it are sufficiently rich. Namely there

must be a sub-field o f events whose probabilities the subject essentially accepts

as given and, furthermore, these probabilities must form a dense subset of the

interval [0, 1 ]. This subfield has been variously termed the auxiliary, canoni-

cal or reference experiment. I shall use the first tenn. Some authors (e.g.

Savage, 1972; Luce and Krantz, 1971) do not acknowledge auxiliary experi-

ment explicit ly in their axiom systems, whereas others do (e.g. DeGroot,

1970; Pratt et al., 1964). Mathematically the former is the more pleasing ap-

proach; it leads to beautifully concise sets of axioms. Nonetheless I strongly

prefer the latter. Consider an analogy with the measurement of length.

I accept that football pitches in London and Manchester are the same

Theory and Decision 14 (1982) 19-33. 0040-5833/82/0141-0019501.50. Copyright �9 1982 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

20 SIMON FRENCH

length. I do not believe this because I can compare the length of one pitch directly with that of another, but rather I take the comparison to be made though the indirect means of a tape measure. Nor do I imagine that there are thousands of tape measures already lying - strictly, ready to be constructed - along all the possible lengths that I might wish to compare. When I wish to compare two lengths, I must lay out my tape measure along each first. Now any axiomatisation of subjective probability, which does not explicitly acknowledge the auxiliary experiment, is essentially insisting that 'tape measures' for comparing degrees of belief are already laid out along all con- ceivable events of interest. Axiomatisations, which describe the auxiUary experiment separately from the events of real interest, simply put a 'tape measure' in the hands of a subject and tell him how to use it. Actual decision analyses introduce explicitly artificial randomising devices (e.g. spinners on a wheel of fortune) to enable the decision maker to scale the uncertainties in his problem (see, e.g. Hampton etal., 1973). Thus I prefer the auxiliary ex- periment to be axiomatised separately. As an aside, it is perhaps noteworthy that the axiomatisation of length in Krantz etal. (1971) does not treat the tape measure separately. Perhaps too I should admit that this metaphor rather unfairly paints a black and white picture of a grey landscape, but it does, I hope, express my point.

Now, given that the auxiliary experiment is to be considered separately from the events of real interest, there is another point to be made. There is never any need to consider the likelihood of events formed by unions and intersections of events drawn both from the field of real interest and from the field of the auxiliary experiment. There is no need to form the smallest field containing both fields and extend the individuals' relative likelihood ordering to this. Returning to the length analogy, there is no need to consider the length formed by concatentating a football pitch with a tape measure. In the next section I present an axiomatisation of subjective probability which does keep the auxiliary experiment very separate from the field of real interest. Indeed there is no common field of events.

In the third section of the paper I extend the axiomatisation developed next to justify the use of Bayes Theorem to update beliefs. This renown theorem underlies much of modem statistics and decision theory. This is essential that the conditions, under which it may be applied, should be fully explored. For instance, recently I came upon an example which highlighted

SUBJECTIVE PROBABILITY 21

some rather unusual aspects of these; how should one update ones beliefs in

the light of another's opinion (French, 1980). Here it transpires that the

likelihood function is, in a sense, conditioned upon the prior; a possibility de-

nied by the usual statements of Bayes' Theorem. Although I choose to make my points in the context of a particular axiom

system, most have more general applicability. Savage (1972) and Fine (1973)

give extensive bibliographies of alternative axiomatisations. Finally, before beginning I should like to thank Professors J. B. Copas,

J. M. Dickey, D.V. Lindley, A. F. M. Smith, and D. J. White together with Drs. R. Hartely, T. Leonard, and L. C. Thomas for many discussions, which

have led to the following. "

2. THE AUXILLARY EXPERIMENT

In this Section 1 develop a simple axiomatisation of subjective probability,

one which has much in common with that of DeGroot (1970, Chapter 6), but

one which, even more than DeGroot's, separates the auxiliary experiment

from the space of events of real interest. It will never be necessary to con-

struct the smallest field containing the real problem and the auxiliary experi-

ment. With this in mind I emphasise that all unions of fields are set-theoretic

and do not imply closure of the union under the field operations.

So to defme a few terms, let

S . . .

j - . . .

be the set of possible futures facing the subject. Any subset of S is a

a possible event of interest to him. be a field of events of interest (i.e. ~ ' - is a set of subsets of S which is

closed under finite additivity and complementation). be the subject's feeling of relative likelihood between events. ~ is thus a binary relation on a set, initially ~ a~-, later extended to include the

auxiliary experiment. Throughout I take the meaning of ~ to be intui- tive to the subject. I know to my own satisfaction that, being August in Manchester, I believe it more likely to rain tomorrow than that a fair die would fall 'six', which,in turn, I believe more likely than snow. I assume that the subject is happy to make the same kind of judge- ment. As a point of notation: A ~ B is read as "the subject holds A to be at least as likely as B "

22 SIMON FRENCH

A "-~B ~* (A ~ B and B ~ A ) by definition and is read as "the subject holds A to be equally likely as B" .

A >- B r (A ~ B and B ~ A ) by definition and is read as "the subject holds A to be strictly more likely than B" .

I have chosen to adopt the term 'event' to refer to the objects of interest. I would be equally happy to use almost any other of the terms current in the

literature. Chapter 2 of De Finetti (1974) is an appropriate reference and I accept fully his discussion of random events, entities, quantities, variables etc.

I assume that the subject can imagine an auxiliary experiment such that

the outcome must lie in the unit interval [0, 1 ] (c.f, DeGroot op. cir.). For example, an idealised wheel of fortune would be suitable. The axioms assume that ~ can be extended to comparisons with events generated by this experi- ment. Note however that it is never necessary to consider events formed by intersections of events in the auxiliary experiment with events in ~" .

3. AXIOM SYSTEM

Let ~ be the Borel field of all finite unions and intersections of open and closed intervals on [0, 1]. Then the subject ima~nes an auxiliary experiment with events in ~5~ such that ~ extends to ~ U 3r-and

SP1 is a weak order, viz. V A, B, C E ~ U ~r-

(a) either A ~ B or B ~ A or both.

(b) (A~B,B~C)~(A~C).

SP2 Let I, J be intervals, open or closed, in B with lengths I z and Ij respectively.

T h e n I ) z J r li >~ la.

SP3 restricted to ~J' is a qualitative probability, viz;in addition to being a weak

order

(a) [0, ~1 ~" 4.

SUBJECTIVE PROBABILITY 23

(b)

SP4 (a)

(b)

and

SP5

VA,B, CE ,~ such thatA r C = ~ = B (3 C: ~ (A ~ B ) r (A U

C~BUC)

S ~ [ O , 1].

V A , B E ~,, C, DE .~ such than A ~ B = ~ = CAD (A ~ C,B ~ n ) ~ (A U B ~ CUD)

(C~A,D>,.~B)~(CUD ~ A UB).

VA E ~r sets {p E [0, 111[0, p] ~ A } a n d {iv E [0, 1][A ~ [0,p] }are both closed.

SP6

I f A 1 2 A 2 ~ _ A 3 2 . . . . AnE if,, n = 1 , 2 , 3 . . . . and if An ~ [ 0 , p] for some fixed p E [0, 1 ] and for all n, then

o o

fl A,, ~ [o,p] r t = l

Axiom SP1 makes the usual assumption of a weak order. Axiom SP2 describes

the uniformity required of the reference experiment. (c.f. DeGroot, op. cir.) Axiom SP3 demands that ~ must be a qualitative probability over the reference experiment. For arguments supporting this requirement see, inter alia, Fine, (1973). Axiom SP4 ensures that all certainties are equally certain and also makes an assumption linking the likelihood of unions in ~ - with that of unions in ~ . It may be motivated by arguments similar to those used to justify SP3. Indeed it is only necessary to make this as a distinct assump- tion because i f " and ~ are being held separate. Usually ~ applies to the whole of the smallest field containing ~ - and ~ and ~ is assumed to be a qualitative probability over all this field. Axiom SP5 is a continuity axiom used to obtain the subjective probability representation. It might be more honestly written:

A C ~ , 3 p A @[0,1] such that A ~ [ 0 , pA].

24 SIMON F R E N C H

However, there is enough of a mathematician in me to feel that this alterna- tive form is only an expression of the connectivity of the unit interval and hence the statement of SP5, which follows Herstein and Milnor (1953).

Axiom SP6 is necessary if the probability representation is to be countably

additive, in which case ~'-and ~ must be taken as a-fields. It is a version of Villegas' (1964) axiom appropraite to the present system.

THEOREM. Given axioms SP1-5, 3 P : J - ~ [0, 1] an unique, finitely additive probability measure agreeing with ~ on J ' . Moreover, i f J - is a a-

fieM and ~ is taken as the Borel o-fieM on [0, 1], then under SP6 P is

countably additive. Proof. Let A E ~.. By SP3 and SP4 we have

H, say,

and

0 E { p E [ 0 , 1 ] I A ~ [ 0 , p ] } = K, say.

By SP1 H U K = [0, 1 ] and so by connectivity of the unit interval and the closure of H and K (SP5),

H ~ K ~

P.4 such that A ~ [0, PA]. Further SP1 and SP2 s p A is unique. Define P (A)=PA VA E ~'~. Clearly P(A) ~ O.

By SP4, P(S)= 1. LetA,B E ~ , A n B = ~

A ~ [0,P(A)] ~ [0,P(A)) by SP2

8 ~ [0,e(B)] ~ [0 ,P(B) )~ [P(A) ,P(A)+P(8) ) by SP2.

So by SP4(b),

A UB ~ [0,P(A))O [P(A) ,P(A)+P(B))

"~ [0, P(A) + P(B)] by SP2

Hence P(A U B) = P(A) + P(B) when A r B = ~.

SUBJECTIVE PROBABILITY 25

Thus P(.) is an unique, finitely additive probability measure on f . Clearly P(.) agrees with ~ on fl'- by SP1 and SP2. We note that, since ~ may be represented by a probability measure, ~ is necessarily a qualitative proba- bility on f (Fine (1973)).

Suppose now that ~f-and ~ are o-fields and that SP6 also holds. Now, from, e.g., Ash (1972) �82 P(.) is countably additive if and only if

'r ~-A2 ~-A3 ~- . . . E ~ , f q ~ = ~ A . = O ,

lim P(An) = O. n - . ~ oo

So letA1 ~-A2 ~-A3 ~ - . . . E ~-,, and A~=IA n =4). Since ~ is a qualitative probability and P represents it

P(A1) >~P(A2) >~P(A3)>~. �9 �9 >10 ~ lim P(A , ) exists. t,l---> rm

Let this limit be b. ThenA, ~ [0, b] Vn = 1,2, 3 . . . =~ by SP6

I~ A , ~ [ O , b ] . 1'1=1

So if b > 0 , 4)=(-1~=1A n [0, b] ~-4),acontradiction. T h u s b = 0 a n d P i s countably additive. Q.E.D.

The above axiom system may appear more complex than previously suggested systems, but, in fact, it requires rather less of the subject than they do. It does not require him to invent the smallest (a-)field containing 3~ and ~'-and consider relative likelihood comparisons of all pairs of events in this field.

For completeness, I have extended these ideas elsewhere to an axiomatis- ation of expected utility with act-conditional futures (French, 1979). How- ever, it should be clear that this approach may be applied to any axiomatis- ation of subjective probability (with or without a simultaneous treatment of utility).

4. CONDITIONAL SUBJECTIVE PROBABILITY AND BAYES' THEOREM

Dempster (1968) describes Bayesian inference as "starting with a global probability distribution for all the relevant variables, observing the values of

26 SIMON FRENCH

some of these variables, and quoting the conditional distribution of the re- maining variables given the observation."

Undoubtedly this is the bare mathematical bones of the Bayesian process; Dempster claimed no more. However, this sketch ignores a very important point. The initial global distribution is specified in a particular manner. First, beliefs about the variables of interest, i.e. the variables that will not be ob- served, are encoded as probabilities. Then, perhaps at some point distant in time the subject specifies the experiment which provides the observations. This separation in time is central to Bayesian inference. It is, after all, impossible to design an informative experiment until one has thought seriously about ones under-lying physical model. Thus the introspection necessary for the prior must precede that for the conditional distribution of the observations.

Now all subjectivists agree that beliefs depend both on current knowledge

and current mood (psychological state). As the subject moves through time these may change. Indeed it is arguable that they must change for him to recognise the passage of time. Thus, if the two distributions involved in Bayesian inference are specified at different times, then they may be based upon different sets of knowledge and different moods. How then can Bayes theorem be applied? The answer is, of course, that the application of the theorem is justified provided that there is no change in information or mood that would cause the subject to change either distribution.

My purpose in this section is simply to expand upon and discuss these remarks. In doing so I draw heavily upon DeGroot's CP axiom (op. cir.). His statement, however, corresponds to the bald mathematical statement of Bayesian inference given by Dempster. Here I explicitly introducethe time at which beliefs are encoded and also the notion of relevant data sets, which rather nebulously express the knowledge and mood which led to the subject to hold a particular belief. I should remark that my axiomatic model is very naive. It is intended as a vehicle for general comments rather than a specific suggestion.

The first task is to take the axiom system of the last section and explicitly acknowledge the time at which the subject specifies his beliefs. Also, since the subject may well not have conceived of an informative experiment at the time that he constructs his prior. I begin to introduce a mechanism whereby the field of events expands and contracts as he plans an experiment and observes its result.

S U B J E C T I V E P R O B A B I L I T Y 27

Let ~ (t) be the set of conjectures, i.e. verifiable, but as yet undecided,

events, in the subject's mind at time t.

There are two important qualifications to note. The phrase 'at time t ' is not to be taken too specifically. Rather than a particular time point, it refers to a period of time in which he does not introduce nor delete any conjecture

from his consideration. Similarly 'in the subject's mind' does not mean

'consciously aware of ' , but rather that each event in ~ ( t ) is easily recallable. It may also be appropriate to reiterate that by "event" I mean any random

entity discussed by De Finetti (1974, Chapter 2).

(t) then is the set of events available to the subject for consideration at time t without need for further imagination. It is tempting to make f ( ( t ) a

field or a o-field so that the subject's beliefs may be expressed as probabilities immediately but that seems too strong a requirement. In real life people do not express beliefs about all possible futures, but only those that seem pertinent to a particular decision. Instead I suggest the subject has a mental

operation called, say, concentration. In concentrating he focuses attention on a subset o f ~ (t) and it is only over such subsets that he expresses beliefs.

Let St and Ot be Jthe certain and null events respectively at time t. The

field of events generated by any f'mite set of events { A 1 , A 2 , . . . , A n } will be denoted by f ( A 1, A2 . . . . . An ) . From now on I shall work simply with fields and Finite sets of events. The extension of o-field and infinite sets of events is straightforward.

Assumpt ion 1. If the subject concentrates on the events A1, A 2, �9 � 9 An E

~ ' (t), then ~ " ( A 1 , A 2 , . . . , A n ) c_ ~" (t).

N.B. c_ is simply set inclusion, c~(t) is not a field.

Assumption 1 says that if the subject concentrates on a set o f events, he is

also aware of all the events in a field containing them. Perhaps I should allow him time to construct these extra events. However, for notational convenience

as much as anything, I choose not to. N.B. It is not necessary that he is only

aware of the events in the smallest field containing A 1, A 2 . . . . . A n ;he might consider a larger field. Hndley, Tversky and Brown (1979) suggest reasons why he might do this.

The next assumption simply draws in the axiom system developed in the

last section.

28 SIMON FRENCH

Assumption 2. If the subject concentrates on some field of events ~r-at time t, then he has an intuitive feeling of relative likelihood between events which may be extended to an imaginary auxiliary experiment witlvevents in ~r the Borel field on [0, 1 ], such that SP1 to SP5 are obeyed.

Thus the subject will be able to represent his beliefs about the events, upon which he is concentrating, through probabilities.

Because I have not demanded that he expresses beliefs over all W (t) and because 'at time t ' refers to a period of time, it is possible that he will be con- centrating on two fields at the same time. If these fields have a non-trivial intersection, i.e. larger than {r St}, then he might hold different beliefs over the same event depending on where he considered it. To eliminate such a clearly irrational possibility, I assume

Assumption 3. If Y and ~ are two fields upon which the subject concen- trates at time t and if ~ : - a n d ~ are his respective relative likelihood order- ings, then ~ : is identical to ~ on J - n c~.

So far I have neither let the individual note the passage of time nor paid any attention to his knowledge and his moods upon which his beliefs depend. So let Y (t) represent the subject's body of knowledge and psychological state at time t. In considering his beliefs about a field of events, Y , the subject will discard much of Y ( t ) as irrelevant. Let ~ ( Y ; t) be the subset of ~r which he considers relevant to his beliefs over f l" . There are many qualifi- cations that I should make here. First, it is by no means obvious that know- ledge and moods can satisfactorily be collected together into one global set Y ( t ) and then partitioned into subsets -~( fl'~; t) and ~re( y ; t). However, if such naivety is taken with a 'pinch of salt', it will hopefully help express my general point. Second, I do not expect the subject to be able to recognise . ~ ( ~ - ; t). Instead I expect him to be aware of changes as time passes. Thus I do not expect him to list exhaustively the grounds for all his beliefs at time t. However, I do expect him to notice if a fact, before considered irrele- vant, suddenly acquires relevance as he sees it in a new light, or equally to notice a similar disappearance of relevance. Also I expect him to notice any logical arguments (i.e. tautologies) which become clear to him. He may also

SUBJECTIVE PROBABILITY 29

note changes in his moods; "Today, I feel lucky". So, whilst he cannot identify all of -~( ~ , t) at any one time, he can say if ~ ( ~ ' - , t) has changed over a period of time.

It is of interest to contrast _~(.Y; t) with Jeffrey's (1961) H (i.e. Y (t) here) and De Finetti's (1974, p. 134) implicit inclusion of H in the proba- bility/prevision symbol. These authors make no distinction between those parts of a subject's make up which he considers relevant to his beliefs and those parts which he does not. In the vast majority of cases this distinction is a fussy irrelevance. However, it is not always so; see French (1979). I shall illustrate this briefly in the closing discussion of this section.

The judgement of relevance and the judgement of belief are clearly very much related. The assumptions below, essentially tie to the coherence de- manded of the subject's beliefs to the consistency demanded of his feelings of relevance.

Firstly, it seem reasonable to demand that if the data considered relevant to a set of beliefs do not change then neither do the beliefs.

Assumption 4. If t < t ' , Y _ ~ ' ( t ) f3 ~ '( t ' ) , ~ - a field upon which the subject concentrates at t and at t ' , then

( -~( f f - ; t ) = _~ ( J ' ; t ' ) ) =* ( V A , B E J - , A ~ t B =~ A ~ t , B ).

Note that - ~ ( J ' ; t ) = . ~ ( ~ ' - ; t ' ) implies among other things that the sub- ject's knowledge of the passage of time is irrelevant of his beliefs. Also note that I am misusing my notation slightly. J ' i s a field relative to St when ~'-<__ ~ (t) and relative to S t, when ~ f ' E ~'( t ' ) . Thus ~Z-in the above is technically ill-defined. Nonetheless, my meaning here should be so clear that it is not worth setting up the notation to correct this.

Assumption 3 guaranteed that the subject's beliefs remain constant over a common sub-field when he changed his attention from one field to another. Here we need a similar assumption for his judgement of relevance.

Assumption 5; If 5 r" ~ ~' _c ~ (t) are two fields upon which the subject concentrates at time t, then ~ ( ~ - ; t) c_ N ( ~ ; t).

The assumption says that he does not gain or lose any information relevant to ~ - in enlarging his discussion to ~ .

30 SIMON FRENCH

The last pair of assumptions say, in effect, that the subject is irrational if he changes his beliefs without reason. The next assumption prescribes how he should change his beliefs in the light of information of a particular sort.

Suppose at some time, t, he concentrates on a field ~r-. At some later time t', he learns that some event E E ~ - h a s happened. Clearly any events A ~'-such that A r E = Ot are now known to be impossible. In fact the re- maining possibilities form a field in cg (t '): viz.

(~'-[E) = {(AIE)IA E J ' }

where (A [E) is the event in ~ (t ') formed by deleting from A N E that part which has happened. Thus (A IE) is the part ofA E ~'-that is still conjecture at t ' . That ( J " [E) is a field is clear from its natural isomorphism with ~ n E = {A f3 E IA E ~'-}. I use the notation (A IE), rather than A[E, to denote that at time t' (A IE) is an unconditional event.

Suppose that the subject feels that the occurence of E is the only ad- ditional relevant information that he acquires in the period [t, t']. Essentially, all the extra information has done is to restrict his attention to ~ " A E, now (~q-[E). Thus there seems to be no argument for changing his relative beliefs within , Y A E. Assumption 6 says precisely tiffs.

Assumption 6. If at time t the subject concentrates on a field ~ c__ ~ ( t ) and at time t ' he concentrates on (~- IE) , where E E ~ - i s , the totality of

events in J - t h a t have happened by t ' , then, if

t ' ) = t) u

V(A IE), (B IE) (5 (,N-IE),

(A]E)~ t'(BIE)~'A NE ~ t B OE,

for any A, B ~ ~ such thatA AE andB r become identified with (ALE) and (B [E) respectively at time t ' .

It might be argued that the condition

-~ ( ( Y l E ) ; t ' ) = - ~ ( Y ; t ) U {E}

is too strong. After all the occurence of E may have eliminated the majority of the possibilities in ~r-. Thus much of - ~ ( J ' ; t) may be no longer relevant

SUBJECTIVE PROBABILITY 31

to the judgement of likelihood within (~"IE) . However, I am leaving .~(~a,-; t) as a very nebulous object. The subject is only aware of changes in it and

cannot list exhaustively its members. Thus he cannot search all of _~( a~; t) throwing away those items that are no longer relevant now that his attention has been restricted to (~r-lE). In any case this discussion is somewhat irrele-

vant to the main point. Namely the only change in his knowledge and mood, which is relevant to his beliefs over ~- , is the occurence of E. Consideration of items in .~ ( Ar~ t) which are now irrelevant to his beliefs over ( ~ ' I E ) are surely of a secondary nature.

If I subscript the subjects agreeing probability distribution with time, then it may be shown quite trivially the Assumptions 1, 2, and 6 imply

Pt(A n E) Pt' ((A[E)) -

Pt(E)

Notice the double brackets in Pt'((A IE)). By this I mean to emphasise the un- conditional nature of (,4 IE) at time t'. Defining conditional probability at time t in the unusual (mathematical) fashion

Pt,((AIE)) = Pt(AIE)

and by Bayes' identity

Pt(EIA)Pt(A) Pt' ((=4 IE)) =

Pt(E)

Note that the above gives the conditional probability Pt (A IE) its semantic

meaning. It is what the subject would belie~ve if he knew E had happened. Now we are in a position to consider the process of inference. Typically it

proceeds as follows. At some time tl the subject is interested in a set of events {A1,A2 . . . . . A,}. He concentrates on them so, by assumption 1, J-1 = ~'-(A1,A2,. �9 �9 , A , ) _ ~ ( t l ) . By assumption 2 his beliefs over ~'~1 may be represented by an agreeing probability distribution and these beliefs will be based upon - ~ ( ~ ; t~ ). If he decides that he needs more information, he will perform an experiment. Put formally, at a later time t2 he enlarges the discussion to field ~a'-2 ~_ ~-1. I f .~ ( ~ ; t2 ) = 9 ( ~ ; t a ) then, by assump- tions 2, 3, and 4 his marginal beliefs over ~a~ 1 at t2 will be identical to those originally over ~-1 at tl �9 By marginal beliefs, I mean his beliefs over ~ 2 marginalised to be over the sub-field~'-i c_ ~'-'z-

32 SIMON FRENCH

Consider the extension from ~'-1 to ~"2 more closely. Let the possible results of the experiment be described by the field W. Then ~'-2 is the field generated by 3r'1 and W, i.e.

= { A n D I A e J r l , D e

Suppose at time ta after t2 he observes E E ~ and this is his only gain in in- formation. Then it is not hard to see that ( ~ IE) is naturally isomorphic to Y l and that by assumption 6 for A ~ ~ r 1 .

Pt2 (EIA) Pt2 (A ) Pta (A ) =

eta(e)

= Pt~(EIA)Ptl (A) (e)

Thus we see a set of conditions under which Bayes Theorem is justified as a prescription for updating beliefs and one which allows that the prior distri- bution may be specified at a completely different time to the conditional distribution of the observation.

In closing I should like to make a number of points. First, these conditions justify the use of Bayes theorem only when the change in information takes the form of the occurence of an event E in the field upon which the subject is concentrating. Changes in belief due to, say, a change in mood or the realis- ation of a tautology are left outside the axiom system. In some small way this may answer the criticism in Fine (1973, p. 228) that Bayes theorem, if applied too universally, far from helping subjectivists, puts their theory in a

straight-jacket. Second, I should perhaps remark that, although I have suggested that the

subject designs his informative experiment, the above all carries through if he just 'happens' upon an event E in the field upon which he is concentrating.

Third, it may be of some benefit to describe briefly the example that first draw the above to my attention. Suppose the subject is interested in a field of events J ' . Having expressed his opinions as probabilities he decides to ask another's opinion. First he must decide what he believes the other will say in each set of circumstances where particular events in ~ turn out to be true. In other words, he must specify the conditional distributions of the observation- here the other's op in ion- given the events of interest. Now,

S U B J E C T I V E P R O B A B I L I T Y 33

since the subject has just been through all the arguments concerning the

events of interest in his own mind, these arguments together with his know-

ledge of how the other generally thinks will be very relevant information for

specifying the conditional distributions of the observations. Thus here we

have an example where the information, upon which the l ikelihood function

in Bayes' theorem depends, cannot possibly be available until after the prior

distribution has been specified. Further details are given in French (1980).

Finally, may I repeat that the above should not be taken too literally. The

above set of assumptions are too naive to stand in their own right, but

perhaps they do convey one or two general points.

University o f Manchester

R E F E R E N C E S

De Finetti, B.: 1974, 'Theory of Probability', Vol. 1 (John Wiley, New York). DeGroot, M. H.: 1970, 'Optimal Statistical Decisions', (McGraw Hill, New York). Dempster, A. P.' 1968, 'A Generalisation of Bayesian Inference', J. Royal Statistical

Society B 30, 205-247. Fine, T.: 1973, ~l'heodes of Probability', (Academic Press, New York). French, S.: 1979, 'Some Comments on Conditional Expected Utility plus Yet Another

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