Degroot Chap 6.pdf

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SUbÍGCÍÍV9

6

5.1

probabilíty

INTRODUCT!DN

Much work in probabíiity and statistics deals with the áerivation of the

probabilities of certain complicated events from the speciñeá probabilities ef simpier events, with the study ef the manner in which certain specified pmbabilities change in the tht of new information, and with procedures for making effective decisions in certain situations that can be character— ized in terms of specified probability dietributions. In the remaining chaptere of this book, other than the present one, we shall consider the soiutions of problems in these areas.

It should be noted, however, that

each problem invºlves the manipulatien of the specified probabiiities of certain fundamenta] events. In this chapter, we shall consider the following question: How does the statistician initialiy assign to these events the probabilities on which ali of his subeequent ceiculations are based? In many problem areas, the aseignment of & probability distribution over the relevant a—ñeld of events has become reutine and standardized

among workers in these areas.

This assignment is usua]iy based on a

combination ºf tradition and experience. In some other problems, there are natural probability models that represent the situations. Some typical examples were mentioned brieñy in Chaps. 4 and 5. Many other

CHAP. 5

70

SUBJECTIVE PROBABILITY

examples can be found in the references given at the end of Chap. 1. Hence, suitable probabilitjes can often be assigned objectiver and quickly because of wide agreement on the appropriatenese of a specific distribu— tion for a certain type of problem. On the other hand, there are some situations for which it would be very difhcult to ñná even two people who would agree en the apprºpriate— nese of any specific prebability distribution. In such & situation, the statistician'e assignment of probabilities must be highiy subjective and must reflect his own information and belief5. We shall new discuss 'm detail the conditions under which the statistician can represent his infor— mation and beliefs in terms of probability distributions.

6.2

RELATIVE LIKELIHÚOD

Consider & sample space S together with a a—field & of events, and suppose that it is desired to assign & probability to each event in a. A funda— menta1 concept is that of one event being at least as Iikeiy to occur as another event. and this concept will be accepted as & primitive one in the

development to be presented here.

Thue it will be assumed that a

person, when considering any two events in &, can decide whether he regards one of them as being more likely to occur than the other or whether he regards the two events as being equally likely to occur. When any two events A and B are compared, we write A -< B to indicate that B is more likely to occur than A and we write A … B to indicate that A and B are equaliy likely to occur. Furthermore, hy analºgy with the inequaiíty relation between real numbers, we write A 5 B to indicate that B is at least as likely to occur es A or, equivalently, that A is not more likely to occur than B . Henee, if A 5 B , then either A < B ar A … B . Finally, we define A > B to mean the same es B < A, and we define A R B to mean the same 33 B 5 A. Since the probability of an event is supposed to be & numerical measure of the likelihood that the event will occur, any probability dis— tribution P which is assigned to the events in the o-ñeld & should have the property that P(A) 5 P(B) if, and only if, A 5 B. A prehahiiity dis— tribution P which has this property is said te agree with the reiation 5…

SEC. 6.2

Assumption SPL For any two events A and B, exactly me of the fe¡lowing three relations must hold: A < B, A >- B, A … B.

The next assumptien has a simple interpretation which makes it

intuitiver p]aueibíe.

Assumption SP,

H A;, A2, B;, and B:; are four events such that

U B;. A1-Az = 3132 = 9 and A;“ 5 B; for i = 1, 2, then A1 U A2 5 31

If, 7:71 addition, either A1 '< B) 03” A2 < B;, ¿hm A1 U A2 < B]. U Bº.

This assumptien can be interpreted ae follows: Suppose that each of

the events A and B can occur in either of tw0 mutually exclusive ways.

If each way that leads to A is not more likely to occur than the corresponding way that leads to B, then A 5 B . Furthermere, if at least one of the ways that leads to A is actually less likely than the corresponding way that leads to B, then A < B. Several results can be derived from these two elementary assump— tiens… One interesting and important result is the tmnsitivity of the relation 5, as proved in Theorem 1 below. Before preceeding to that theorem, we shall present a simple 1emma. Lemma 1

Suppo.ee that A, B, and D are events such that AD = BD = &.

ThemA 5Bzf,andonlyíf,AUD 5BUD.

Proof Suppese A 5 B. Then the desired resuit follows from Assump— tien SP,. Cenversely, suppese that A > B. Then, again by Assump— tion SP2, A UD > B UD.. Theorem 1

A 5 D.

If A, B, cmd D are events such ¿hat A 5 B and B 5 D, then

Proof Consider the seven disioint events shown in Fig. 6.1 whose union is A U B U D. Since A 5 B, it follows ¡rem Lemma ] that

A

relation 5 in order for there to exist & unique probahiiity distribution that agrees with it. The aesumptigns that will be made in regard to the relation 5 are suggested by our intuitive use of the concept of the relative of prohability distributions, as given in Chap. 2. The basic assumption is the following.

(1)

ABºD“ U ¿Bºb 5 AºB.0º U AºBD.

We shall now investigate the conditions which must be satisfied by the

likelihood of occurrenee of two events and by the mathematical properties

n

RELATIV£ LIKEL¡HOOD

ABCD“

Fig. 5.1 The partítioning of

A U B U D.

X

A ºB cD

/

”

CHAP. 6 SU BJECTW£ PROBAB1LITY

12

Similarly, since B 5 D, it follows from Lemma 1 that (2)

ABDº U AºBDº :; ABºD U AºBºD.

Since the left sides of the relations (1) and (2) are disjoint and the right sides are also diejoint, it follows from Assumption SP¡ that

AB=D= U AB=D U ABD= v AºBDº 5 AºBDº U AºBD U ABºD U AºBºD. (3)

If the common event ABºD U AºBDe is ehminated from both sides of this relation, it follows from Lemma 1 that (4) ABºDº U ABDº 5 AºBD U A=B=D.

It can now be seen from Fig. 6.1 and Lemma 1 that A 5 D.! It follows from Assumption SP; and the transitivity property exhib— ited in Theorem 1 that the relation 5 yields & complete erclering of the events in Ct. Of course, in this ordering it is possible that two distinct events A and B will be equivalent, that is, A … B. The next result is an extension of Assumptien SP; to unions of any ñnite number of disjeint events.

Theorem 2 If A;, . . . , A,. are n d£sjeint events and B¡, . . . , B.. are ateo n disjo£?tt events suchtha.t A.— 5 B¡f0fi = 1, . . . ,n, then U?_,A,— 5 U?_¡B,. If, in addition, A.— < B¡ fer at Least ¡me value oft' (¿ = 1, . . . , n). then U3_1A¿ < U$,,B,.

This theorem is easily verified by an induction argument, and its proof is omitted. The next theorem reñeets &. basic property of relative likehheod which has previously been noted in Chap. 2 for prebahi1ities. Theorem 3

For any events A and B, A 5 B ¿f, and onty t'f, Aº 2; Bº.

The proof of this theorem serves as Exercise lb at the end of this chapter. We new make the natural aesumption that no event is less likely than the empty set 9, and we avoid & compietely trivial situation by assuming that the entire sample space S is actually more likely than 9. Assumpiion SP;

¡¡ A te any event, ¿hen B 5 A.

Furthermore, 9 < S.

This aseumption leads to the following fundamental property of relative hkelihoeds, which has 3130 been previoust noted fer probabilities (see Exercise lc).

SEC. 6.2

73

RELATWE LiKEL1HDOD

Theorem 4 If A and B are events such that A C B, then A 5 B. particular, tf A is any event, ¿hm 9 5 A 5 8.

In

We now introduce en assumption whose distinguishing feature is that it involves an inñnite sequence of events. Assumpt¡on SP.; If A1 3 Ag :) - - - is ¿: decreasíng sequence of events and B t'.s seme fmed event such that A; > B for i = 1, 2, . . . , then ('X,…':_,A, > B. To help ciarin the meaning of Assumpt'10n SP., consider an example in which the events under eoneideratíon are subsets of the real line. Suppose that eachinñniteintervalof the form (n, ac),forn = 1,2, . . . , ie regarded 33 more liker than some ñxed but small subeet B of the line. Since the interseetáon of all the infinite intervals is the empty set 9, it

then follows from Assumption SP.; that the small subset B must itself be

equivalent te 9. In other words, if B is any set such that B > 9, then regardless of how small the set B is, it is impossible for every infinite interval (n, ºº) to be at least as likely es B. A property of this type distinguiehee &. probabihty distribution that is ceuntably additive (i.e., one that satisñes property 2 in the definition of &. probabiiity distribution given in Sec. 2.3) from one that is only ñnitely additive. The following theorem is the dual of Aesumption SP¿. The statement of this theorem could equally well have been taken as Assumptien SP. and the statement in that aesumption then derived as ¡¡ theorem. Theorem 5 If A¡ C A; C - - - is an increasing sequence ef events and B t's some jízed eventeuchthat A¡ 5 Bfore' = 1,2, . . . ,then U?_,A, 5 B. Proof It follows from the hypotheees of the theorem and from Theorem 3 that ¿if :) Agº :) - ' - is a decreasing sequence of events such that A,—º ¿ Bº for ¡“= 1, 2, . . . . Henee, by Assumption SP., (U3_¡A¿)º : ñ;,A; > Bº. The desired result now follows from another application of Theorem 3.I

The next result extends Theorem 2 to unions of inñnite sequences of diejoint events. Theatem 5 If A1, Ag, . . . is an t'n£níte sequence of dt'sjot'nt events and B,, B;, . . . is another tn¡€m'te sequence ef díejo»£nt events such ¿hat A; 5 B.—

for ¿ == 1, 2, . . . , then U:_,A,- 5 U,-"_,, ¿.

If, in addition, A.— < B¡for

at least one value ofí (i = 1, 2, . . . ), Hum U,Z_¡A¡ < U;"_1B¿. Proof

It follows from Theerem 2 that U?_,A¿ 5 U;_,B, for any value

74

CHAP. 5 suwscrws PRGBAB!LITY

of 91 (n = I, 2, . . . ).

UA.—5UB,-

¿»1

¡»1

Hence, by Theorem 4,

n=1,2,....

(5)

Since the left side of the relation (5) yields an increasing sequence of events for n = 1, 2, . . . , it foliowe from Theorem 5 that U?_,A, 5 U_"_,B 11", in addition, A,- < B, for at least one value of:! (.? = 1, 2, …), then it follows from Theerem 3 that for 11. _2 j, ”

ll

in 1

5—1

UA.— < UB¡.

(6)

Furthermere, it followe from the first part of Theorem 6, which has just been proved, that

U A; 5

í=n+l

U B.—.

(7)

€-n+1

Hence, from the relations (6) and (7) and Aesumption SP;, we can obtain the following result: '

UA = (UA.-)U( U A.—) < (UBJU( U B¿)= UB¿.I

€—l

¡=n+l

¿II

(8)

It should be dear that the relation ;; must eatisfy Asenmptions SP; to SP,, if there is te exist any probehility distribution which agrees with it.

However, these four assumptions are not suíñcient to guarantee the

existence of such & probability distribution. In the next section we shall introduce a ñfth aseumption on the basis of which it is possible te construct & unique probebility distribution that agrees with the relation 5. Further Remarks and References

Sometimes the statistieian,s decisiºn es to which of two events A and B is more Iikely te occur can be facilitated hy auxihary censideretions. Fer instance, he may ask himself whether he would prefer to participate in &… contest in which he would receive a. valuable prize if the event A occurred and receive nothing if A did not occur or & Contest in which he would receive the same valuabie priee if the event B occurred and receive nothing if B did not occur. Although the statistician would typically prefer to participate in the cºntest which he considered had the greater ¡ikeiihood of yielding the priee, there are limitations and dangers in this reason'mg. An ordinary person would consider it more likely that he will be exterminated in a. nuclear war within the next ten years than that he

SEC. 6.3 THE AUXILIARY EXPERIMENT

75

will become President of the United States within that period. But if he were given & choice, he would certainly prefer the promise ef & prize at his ineuguration rather than the promise of the same prize at his extermination. This example is clearly extreme, but it illustretee the díiheulties which can arise when one must compare events that are not, according to Ramsey (1926), ethieaily neutral. These diñicultiee reveal the advantage of making, whenever poesih1e, & direct comparison of the relative 1ikeiihoods of two events without considering consequences which depend on their occurrence or nenoecurrenee. The question of whether there always exists at 1east one probebility distribution that agrees with the relation 5, when this relation eetieñee Aesumptiens SP; to SP4, wee raised hy de Finetti and Savage ¡see Savage (1954), p. 40]. It was answered in the negative by Kraft, Pratt, and Seideuberg (1959), who constructed an example using a sample space S with just & ñníte number of points. 6.3

THE AUXIL¡ARY EXPER!MENT

In this section we shall introduce one ñnai essumptíon which will make it poseible to essign ». probability to each event in an unambigueue manner. Any assignment of & probebility distribution indicates not only which of two events is more likely to oecur but also how much more likely its occurrence is. It is not always possible to give & meaningful numerical evaluation of these reiative ¡ikelihoods 301er on the basis of the assumptiene which have been made in Sec. 6.2. For example, consider en experiment or, equivalently, & samp]e epe,ee which has just two possible outcomee, A and Aº. The statistieian may fee1 that A > Aº, but it will e1early be impossible for him to eesign meauiugful numerica1 prohehilities to these two events without additional consideratiens. In particular, he must be able to compare the relative likeiiheods of A and Aº, not only with each other but else with many other events whose various probabilities have already been eetabhshed. The effect of these remarks, 'm informal terme, is that it must be assumed that there existe a clase 05 of events having the following two properties: (l) Each event in the clase (3 has & known probability, and (2) for any number ¡0 (0 < ¡0_ < 1), there exists an event B e (B whose prohability18 p. Henee, when aseigning e prohebility to some event A in which he13 interested, the statistieian simply finds an event B e (B such that A … B and assigns to A the same pmbability ae that of B. The aesumption which is needed, however, cannot properly be formulated in this way since this description pmvidee no indicetion of the manner in which the probabiiíties of the events in the clase (B were chosen by the statistician or became known to him.

The aesumptien can be

CHAP. £ SUBJ£CTWE PROBABILITY

?5

stated more precieer in terms of the existence of a particular type of random variable.

_

It will be reeelled from Sec. 3.1 that a random variable X is an (t—meaeurable function whose value is specified at each point s e S. Hence, for any random variable X and any intervaie ¡¡ and I: of the real line, the events ¡X ¿Id and ¡X ¡I;! belong te the a-ñe¡d G., and by Assumption SP,, either ¡Xehl 5 ¿X ¿I,30r1X eh! ¿ ¡XeIgl. For any interval 1 with ñníte end points ¿¡ and b (a 5 b), let MI) = b — & denote the length ef ! . Note that )t(í) has the same value regardless of whether or not either of the end points ¿¡ or b is included in I . The interval with end points ¿¡ and 6 will be denoted by (a, b) if neither (¡ nor !) is included in the interval, by ía, b) if a but not b is included in the interval, by (a, ¡7! if !) but not 0 is included in the interval, and by

[a, 5] if both ¿: and !) ere included in the interval.

We can new define what is meant by a uniformly distributed random variable in the present context. Let X be a random variable such that 0 _<__ X (e) 5 1 for every? s e S. Then the random variable X is said to have & uníferm distribution en the interval [O, 1] if the fol!owing property helds: For any two subintervals I; and I: of the interval (O, 1], [X e ¡1! 5 [X ¿ Ig! if, end enlyif,h(1,) 5 ¿(L.) Nate that this definition of a. uniform distribution does not mention pmhahilitie& The ñnal eeeumptien can now be fermulated very simpiy as follows:

Assumptien S P. There existe a random variable which, has a uníferm dels— tr£butíon on the interval [O, 1].

Since Assumption SP;, or one similar to it, underlies at] Work on the construction of explicit eubjeetive probabilitia, some further remarks in regard to this assumption may be helpful. As mentioned above, many interesting experimente have only a. ñnite number of outcome5, and all ef these are not necessarin equaliy likely to occur. Cleerly, it is not possible to deñne on the sample space of such an experiment & random variable which has ». Uniform distribution en the interval it), 1]. The etatistieian must enlarge the sample space by considering, along with the original experiment, an euxiliary experiment in which the value of a random verieb1e having the appropriate uniform distribution ie observed. Henee, each point in the enlarged samp1e space of this composite experi— ment eompriees an outcome of the original experiment together with an outcome of the auxiliary experiment. It is assumed that Aesumptiens

SP, te SP, concerning the relation 5 still held in the composite

experiment. It is not necessary that an auxiliary experiment actually he per—

550. 5.¿ CONSTRUCTION OF THE PROBABILITY DIS1'RIBUTION

??

formed, or even that it be poesih1e for such an experiment te he performed. The statisticien need only be able to imagine an idea] auxiliary experiment in which & uniforme distributed random variable X can be generated, and be able to compare the relative likelihood of any event A which was

originally of interest to him with that of any event of the form IX ell. Further Remarks and References

The statistician may think of the value of the random variable X es being determined by seme sort of rendemization device, such as & spinning pointer which eeleets “at random” & point en & circle having unit eircumferenee. Alternatively, he may think of a. “fair” coin toseed repeatedly. Suppoee that for any ñnite number n oí toeees, any possible sequence ef n heads end taile has the same Iikeliheod of being obtained es any other sequence. A random variable X having & uniform distribution on the interval [U, 1] een then be constructed es follows: For ?¿ = 1, 2, . . . , let T,, be the random variable such that T,. = 1 if the result of the nth tose ie heads and T,. = 0 if the result of that tose ie ta.ils. Let X be the random variable defined by the equatíon X = 2:_,2“'“T,,. Then it can be shown [see, e.g., Feller (1966), p. 341 that X has & uniform distribution on the ínterva1 [O, 1]. 6.4

GONSTRUGTION OF THE PRDBABILITY DISTRIBUTION

Suppese now that the sample space 8, the a—ñeld & ef events, and the relation 5 satisfy Aseumptione SP, to SP¡. It wiii be shown in this section and the next one that under these aseumptione there exists & unique pmhability distribution P that agrees with the relation 5. By Assumption SP,, there existe a. random variable X which has & uniform distribution on the intervai [O, 1]. For any eubinterval (a, 5) of the interval [O, 1]. let G(a, b) denete the event that the random variable X lies in the interval (a, b). Then for any two intervels (a,, bl) and (0-3, bg), with 0 5 e¡ 5 b.- 5 1 for e' = I, 2, it follows that G(a¡, b,) 5

(Km, bz) if, and only if, bl _ a¡ S bz _ ag.

MOI'GQVGI', G(G]_, bl) """'

G(ºh b23 "” G[ºl,— 51) '““ G[ºl; b¡¡.

Theorem ]. If A is ¿my event, there existe a unique number a"' (0 5 a* 5 1) such ¿hat A r—-' G'[0, a*]. Proof

For any event A, let U(A) be the eubeet of the interval [O, 1]

defined by the following equatien:

…A) = 102 G[º: 6] a Al-

…

Since GEO, I] = S 2; A, the number 1 be10nge to the set U(A), end hence,

SEC. 6.5

CHAP. 5 SUBJECTIVE PROBAB!LITY

78

79

VERIFICAT¡ON OF THE PROPERTIES OF A PROBABILITY DISTRIBUTIÚN

If e¡ 2 a; 2 — - ºie U(A) is not empty. Let a* = inf ta: a ¿ U(A)l. ges to the vah1e eonver that U(A) from values of ce amy deereesing sequen ption SP. Assum from e*, thenG[0, a*] = ñZ_,GIO, a,]. Henee it feliows that

Furthermore, since S = GEO, 1]. it follows that P(S) = 1. Te complete the x_reriñcation, it is only necessary to show that fer any sequence of

this inequaiity If a* = 0, then since GH], 0] M B 5 A, it followsfmm A. … and the inequelity (2) that GH), 0] the definition of a* Suppese then that a* > 0. It follows from

. The next theorem shows that Eq. (1) is eerreet et least for the limon ef two diejeint events.

sequence of values that if tt1 < a; <— - - is any strictly increasing 0, a;]. Henee, hy U?_,G[ = eonverges to the veiue e'“, then G[0, a*) a*) ;3 A. This GIO, … a*) GIO, that e The0rem 5 of See. 6.2, it feI¡ow the result that yie1de again (2) lity inequa the inequnlity together with G[O, a*1 … A. ¿tg are any other The value of a* must be unique since if a; and al] < Giº, a*] < 610, that s follow numbers such that a, < ¿=* < ag, it lent to the equiva be can events three these Gí0, a,]. Hence, only one of event A.!

P(A U B) = PM) + P(B).

(2)

GIO, a*] h A.

0 $ a < a*. that GEO, a] < A for any number & such that

Furthermere,

now he defined directly The desired probability distribution P can is defined to be the P(A) then , event frºm Theerem [. If A is any , P(A) een be determined by number a* specified in Theorem 1. Hence the relation

,(3) A … GIO, P(A)]. deñnitien, P agrees with In the next theorem, it is shown that with this Let A and B be any two events.

By Eq. (3), A 5 B if, and only if,

Giº, PM)] 5 G[01 P(B)1.

Theorem 1

(1)

U A and B are any two events such that AB = 9, then

Proof By Eq. (3) ofSec. 6.4, A … G[0, P(A)] and A U B … G[0, P(A U B)]. Furthermere, since A 5 A U B, then P(A) _<_ P(A U B). It will now be shown that (2)

B … G(P(A), P(A U B)].

Suppose first that B < G(P(A), P(AUB)]. Aesumption SP, that

Then it foIlowe from

A U E < G[O, P(Á)] U G(P(A), P(A U B)] = G[O, P(A U B)], which is &. eentradietien.

(3)

A eimiiar eontradietion is obtained when it is

supposeed that B > G(P(A), P(A U B)]. mee

Hence, Eq. (2) must be correct.

G(P(A), P(A U B)] … GH), ¡“(/1 U B) - P(A)]

(4)

Then A 5 B z'f, and ¿mty if,

(4)

By an elementary induction argument, which will not be given here, the result of Theorem I can now he extended to any finite number of die¡oint events.

P(A) 5 P(B). Proof

P(€K:1;Aí) '"- ¡;1 P(Á<)-

and since by deñnition B … [O, P(B)], it now fel¡ows from Eqs. (2) and (4) that (5) P(A U B) -» P(A) = P(B).I

the relation ;5. Thootem 2

dlsjmnt events A1, A;, . .

of the uniform distribution Furthermere, it follows from the dehm'tion if, P(A) $ P(B).I that the relation (4) helds if, and only ES OF A PRDBABILITY 5.5 VER!FICATÍDN OF THE PROPERTI DISTRIBUTION ion P, es deñned above, actually 1t must new be veriñed that the funct ty distribution ae given in See. bili pmba meets all the requirements of a that P(A) 2 0 for any event A. 2.3. It is clear from the deñnitien of P

Corollary 1

U" A¡, . . . , A,, are any d£sjoinl events, then

P(__%Á.-) = E P(Á¡)l'==l

Theorem 2 Let A, :) Az :) - — — be ¿my decreasz'ng sequenc:e of events such that ñ?_,A, = B. Then lim P(A…) == 0.

(6)

CHAP. & SUBJECTNE PROBABILITY

lll!

Proof Since A1 D A, D- - - , then P(A¡) 2 P(A2) 2- ' - . Henee, the numbers P(A,.) must converge to some nonnegative limit !) es n —+ oo . Since P(A,-) 2 ¡: for :: = 1, 2, . . . , it fehows that A.— z 610, b] for ¿ = 1, 2, . . . . By Assumption SP¿, this implies that

(7)

¡a = 9, A…- ¿ G[o,b1. If ¡; were any positive number, then it would be true that

Since this relation eontradicts relatiºn (7), it must. be true that ¡: = D..

The combination of Corellary 1 end Theerem 2 can now be used te prove that Eq. (1) is correct. Theorem 3

The function P is a prebehíl£ty distribution.

Proof Suppese that A,, Ag, . . . is any sequence of disjoint events. follows from Corollary 1 that

UA,) A,)+P( i-n+1 )=EP( P(ÓA, ='-¡ ,—_1

n=l,2,....

It

(9)

events Since the events A; (t" = 1, 2, . . .) are diejeint, the sequence ef that such sequence ng deereaei & is .) . . 2, 1, = (n B,, = U?_,,+,A, lun P(B,,) == 0. (“X“ ,B = 9. Renee, it follows from Theerem 2 that f|-—DG “ "we Therefore, by taking the limit of the right side ef Eq. (9) es n —* ºº,

ºbtain Eq. (l).l

By showing that P is the only probability dietribt_1tion which , satieñes Theorem 2 of Sec. 6.4, we can now establish the followmg theorem section. this of results the which summarizee that the Theorem 4 If the retatíon 5 sattafíes Assumptiens SP1 ¿a 313.5, desm— hty probaba unique the ¿3 6.4 Sec. of function P es defined by Eq. (3) bu£íon which agrees with the relation 5.

ees ef Proof All parts of this theorem except that relating to the uniquen _ hed. establis P have aireedy been Censíder any probebility distribution P" which agrees With the

re13tien 5.

It can be shown from the properties of the umform d13tm-

that butien that for any event of the form G[0, a], it must be true

P'ÍGÍO, Gl] = G-

81

If A is any event, it was shown in Theorem 1 of See. 6.4 that A … G[O, a*] for some number a*. Since P' agrees with the relation 5, then P'(A) = P'[G[O, a*]) = ¡t* = P(A).

(10)

Hence, P'(A) = P(A) for any event A, and the uniquenese of P is eetablisheá.l Further Remarks and References

(8)

GIO, Z)] > G [G, %] z 9.

SEC. 5.5 COND¡TÍONAL UKEUHOO DS

The presentation given here is related to that of Villegas (1964), and some of the exercises at the end of this chapter are based on his work. An excellent bibliography on subjeetive probability is given hy Kyhurg and Smokler (1964), who also reprint important papers hy Ramsey (1926), de Finetti (1937), Keepman (1940), and others. Savage (1954) gives & thereugh development of eubjeetive pmbahility and & highly informetive bibliography. Other interesting cºntributions are by Anseembe and Aumann (1963), Fiehhum (196%), and Scott (1964). Fishburn (1964) also diseueeee eubjeetive probabiíity. Several of the famous books on this topic were mentioned in Chep. 1. Other references are given at the end of See. 6.6. 6.6

CONDITIONAL LIKELIHDDDS

Suppoee now that the relation 5 satisñes Assumptione SPI to SP5 given in preeeding sections and there existe a unique probahihty distribution P which has the properties described in Theorem 4 ºf See. 6.5. In this section we shall extend the concept of the relation 5 so that we can consider not only the relation A 5 B but also the relation (All)) 5 (BED) for any three events A, B, and B. This letter re!ation, which compares

eenditional likelíhoods, has the following meaning: The event B is at

least ae likely to occur as the event A when it is known that the event D has occurred. Henee, we must new aseertain what further conditions should be impoeed on the relation 5, in this extended sense, in order that all the eenditional probebiíity distributione which are constructed from P will also agree with this relation. In other words, we now seek conditions under which, for any three events A, B, and D, the relation (All)) 5 (BÍD) will hold if, and only if, P(A]D) 5 P(B]D), Clearly, this equivalenee can be required only for events D such that P(D) > 0, since conditionaf prehabi]ity is defined only for such events.

For any event D such that P(D) > 0, the inequaííty P(AID) _<_

P(BID) is equivaient to the ineque]ity P(AD) 5 P(BD).

Furthermore,

CHAP. 6

82

SUBJ£CTIVE PROB.ABILITY

this inequality is equivalent to the relation AD 5 BB. naturally lead to the next assumption.

These remarks

Assumption CP For any three events A, B, and D, (AID) 5 (BID) t'f, and ¿mty if, AD 5 BD.

As we remarked above, when Aseumptiene SP. te SP; are made and there existe & pmhebility distribution P which can be uniquer epeciñed, it is sufhcient for most purposes to apply Aseumption CP only to events D such that P(D) > 0. However, in order to keep the aseump— tiens independent of each other, we have impoeed the slightly stronger requirement that Assumption CP applies to all events D. Furthermere, it now follows from Assumptiens SP, and CP that for any three events

A, B, and D, either (AID) 5 (BID) er (BID) $ (AUD). It is possible, of course, that both relations wil] be correct. —1u fact, both relations will always be correct when D is an event such that P(D) = 0 (see Exercise Ga).

EXERCISES

83

(a) H ..4 and B are events such that A C B, then A 5 B. . (d) If A1, . . . , A,. and B;, . . . , B.. are events such that B,B,— = 9 for : ;£j and A¡ 5 B¡ for i = 1, . . . , n, then U','_,A, 5 U“;_,B,. If, in addition, .4¿ < B.- for at least one value ef :" (t" - l, . . . ,a), then U','_,.4, < U'I_,B,. (This is an extension ef Theorem 2 of See. 6.2 since it is not assumed here that A;, . . . , A… are disjoint,) Furthermore, the statements remain true if the condition that B.-B,— == 9

for t" ;¿ 3" is replaced by the weaker condition that B¡B, … 9 lor t' ;5 j. 2. Suppoee that Assumptíene 3Pl to SP; are made. Let A ¡, . ..,A_hedís-

joint events, and let B,, . . . , B., also be disjnint events such that U?_,A, = U:…'_,B, =S,Al 5 --- 5 A…andB. 5 --- $B.,andm gn, Provethat3. 5 A…. Prove also that if m < 11, then 31 < A,... 3. Prove that the following resulta follow from Assumptione SP¡ to SP.: (a)

II A¡ 3 A2 3 - - -is & deereaeing sequence of events and ¡( B is an event

stlch that f)?_,.4¿ < B, then A¡ ¿ B far ahi); a ñnite number of values of 1" (t = 1, 2, . . …). State and prove the analegeue result for an increasing sequence of event3A1C.-th — , » .

. (5) If 111 C A: C - - , is an increasing sequence of events and B, :) B, 3 . — — 18 a decreasng sequence of events such that A; 5 B¡for £ = 1, 2, . , . , then

U:.,A, 5 nf.,B,. (c)

If A1 D A: 3 - - - is a decreasing sequence of events and B¡ :) B—¿ 3

º - is also ¡¡ decreasing sequence Df events such that A; 5 B.— for 1' = I, 2, , . , ,

By eombining Aesumptien CP with Theerem 4 of See. 6.5, we obtain the following general theorem.

then ()?_,A,— 5 ñf_,B¿.

Theorem1 Ifthe relation 5 eat£s£esássumpttms SP1¿oSP5end Assump— tt'on CP, then the Junction P dejned by Eq. (3) 0)“ See. 6.4 ¡'s the unique probab¿l£ty dtstr£but£on which has the fe£¿mmlng property: For any three events A, B, and D such that P(D) > 0, (AID) 5 (BID) tf, ami ¿mty ij,

. (d) If.4.,A=, . . .a.nd B;,B1, . . .aresequencesofeventssuch thatB;B; …9 for 1 ¡ºf and A.- 5 B.— for ¡' == 1, 2, . . . , then U?_,A, 5 U:_,B,. If, in addition, A; < B.— for at ¡east one value of :" (=" = 1, 2, . . .), then U?_,A,- < U?_,B,-. (Note that this is an extension of Theorem 6 of Sec. 6.2, sinw it is not assumed here that the events A1, A;, . . . ere disjeint and the condition that B,, B,, . . . are dísjeint is replaced by the weaker eonditien that B.—B,— … 9 for ¿ ¡¿ j.)

Further Remarks and References

B > 9, then A.- 2 B for only ¡¡ fmite number of values of ¿ (¿ = 1, 2, _ . .). 4. Let I” be a given probability distribution on the set S of positive integers ll, 2, . . ,) such that each integer in 3 is assigned &. positive probability. Let

P(A|D) 5 P(BID).

Luce and Suppcs (1965) give a good review of subjective probability, ine!uding the relativer small amount of work that has been done on the experimental measurement of subjective probebilities. An interesting and controversial group of papers deehng with the existence and measure— ment of subjeetive probebilitiee are those hy Ellsherg (1961), Fe11ner (1961, 1963), Re.ina (1961), Brewer (1963), and Roberts (1963). The book by Fellner (1965) is also of interest here. Exercises ”¡ to 9 at the end of this chapter require the actual evalua— tion of the reader'e subjective probabilitiee of some speeiñc events. EXERCISES 1.

(a.) A < D.

(t)

Prcwe that the following results fellow from Assumptions SPI to 8sz

If A, B, and D are any three events such that A 5 B and B < D, then

I! A and B are any two events, then A 5 B if, and only if, Aº ¿ Bº.

ñ?_,B¿.

Furthermere,if.d¿ … B¡íor ¡” - 1, 2, . . . , then f)?_,.4¿ …

State and prove the analogoue results for increasing Bequenees of events

A1CÁ:C ' ' '3nd31C32C ' ' '-

(3)

[¡ A., A,, . . . is a sequence of disjoint events and B is an event such that

31 - II, 3, 5, . . ,], and let S; .. [2, 4, 6, . , .). Hence. every subset 1—1 of S can be expressed in the form A = (AS,) U (AS;). Suppoee that & relation 5 is deñned

between subsets nf S es follows: It A and B are any two eubsets of 8, then A 5 B if either P*(AS¡) (. P'(BS¡) er P'(ÁS¡) = P'(BS¡) and P*(AS:) S P'(BS;). Show that the relation :S satieñes Aseumptíons SP. tu SP; but not Assumption SP..

5. Coneíder the problem described in Exercise 4, but suppoee now that the relation 5 between suheete of 3 is deñned as follows: If A and B are any two sub-

sets of 8, then A :S B if either P”(A) (. P'(B) or P'(Á) = P'íB) and P*(ÁS¡) :S P*(BS¡). Show that the relation 5 satísñes Assumptions SP¡ to Blºg but not

Assumption SP.. 6. Prove that the following results can be derived frºm Aesumptiens SP1 to SP4 and Assumptien CP: (11)

If D is any event, then D … 9 if, and only if, (AID) N (BID) ¡or all events

(c)

Let .4, B, D, and E be events such that ADE N BDE N 9.

¿ and B. (b) 15 A and D are any events, then (BED) 5 (AID) 5 (D1D).

(BIE) it, and only ¡J, ((A u D)1E] 5 [(B U D)1E].

Then (AIE) S

“

CHAP. 6 SUBJECTIVE PROBABIL1TY

(r!) If A, B, D,andEare events such that (AIE) 5 (BIE) and (BEE) 5 (DIE), then (ALE) 5 (DIE). (e) event D.

11" 1—1 and B are events such that 1—1 C B, then (AID) 5 (BID) for any

(I) If A, B, and D are any events, then (AID) 5 (BID) if, end on!y if, (A=ED) ¿ (B=|D).

(9) If .4¡ :) A1 3 , - - is & decreasing sequence oí events and if B and D are any given events such that (¡%,—10) ¿ (B[D) for ¿ = ], 2, , . . , then (ñ?_,1—1,|D) Z

(BID).

(h)

If ¡h. Ag, . . . and B., Bº, . . . and D are events such that B¡B,-D …9

for 1" ;£jand (.'1.-1D) 5 (B,]D) for i = 1, 2, . . . , then (U,'_“,A,ID) 5 (U,—º_º,B,]D). If, in addition, (¡tch) < (B;!D) ¡or at least one value of ¿ (:" = 1, 2, . . .), then (U¡ÍIAíID) '( (U¡:º]B¡ID)-

.

(1) Let D¡, D;, . . . he events such that D.-D,— … 0 forí 96 j and U,:,D, … 8. If A and B are events such that (A1D.—) 5 (BID.) for i = 1, 2, . . . , then A 5 B. If, in addition, (MD.) < (BID.) for at !eaet one value oí ¿ (i =- 1, 2, . . .), then A -<'_ B. Spech Note: The purpose of Exercises ? to 9 is to provide epeciñc examples involving the evaluation of eubjeetive prohabílities and showing how these proba—

hilitics change when additional information is gained. In order that Exercises 7 and 8 may be effective, each part of an exercise must be worked out without aeeuming any knowledge obtained from &. later part.

Accordingly, you should place a. sheet

of paper se es to expuse each exercise to your view one line at a time. You eheuid be able to see only the line you are eurrent!y reading and earlier lines, while the remainder of the exercise ie etili covered.

7. (a) Consider four events A1, A,, A., and A., which are described as fo!lows: A. is the event that the year of birth of John Tyler, former President of the

United States, was no ¡aber than 1?50; Av; is the event that John Tyier was born during one of the years 1751 to 1775, inclusive; A3 is the event that he was born during one of the years 1776 to 1800, inclusive; A. is the event that he was born during

the year 1801 or later.

Ailow yourself sufñcíent time to evaluate the relative !ikeli—

heads of these four events, but do not use any outside sources of information. of these four evants do you coneíder the most likely?

least likely? (b)

Which

Which do you eonsider the

What probahi1ity do you aseign to the most like13,r event?

You are now given the information that John Tyler was the tenth Presi-

dent of the United States. Use this information to reevaluate the relative likelihoods of the events A., Az, A;, and A.. Before going on to part c, answer again the questions that were asked in part 0.

(c) You are now given the information that George Washington, the first President of the United States, was born in 1732. Before going en te part ¿, answer again the questions that were asked in part a. (d)

You are now given the information that John Tyler was inaugurated as

President ¡n 1841. Answer again the questions that were asked in part a.. 8. (a) Cnnsider four events A., A., A;, and A., which are described ae fol—

lows: 1—1, is the event that the area of the state of Pennsylvania ie less than 5,000 eq miles; A3 is the event that the area of Pennsylvania is between 5,000 and 50,000 eq

milm; Aa is the event that the area ¡3 between 50,000 and 100,000 eq miles; A… ¡3 the event that the area ¡a greater than 100,000 eq miles. Which of these four events do you consider the most likely? Which do you consider the least likely? What probability do you eeeign to the most likely event?

EXERCISES

35

(5) You are now given the following information: The area of Alaska, the largest of the 50 states, ie 586,400 sq miles, and the area of Rhode Island, the smallest of the 50 states, ie 1,214 eq miles… Before going on to part 6, answer again the questions that were asked in part ct. (c)

You are now given the information that when area 13 considered, Pennsyl—

(d)

You are now given the information that the area of New York, the

vania is the thirty—thírd largest of the 50 states. again the questions that were asked in part ct.

Be¡ore going en te part ¿, anime:

thirtieth largest of the 50 states, is 49,576 eq miles. Answer again the questions that were asked in part a.. 9. Think of & fixed site outside the building in which you are at this moment.

Let X be the temperature at that site at neon tomorrow. that:

(º)

Choose a number :, such

P(X <m.) - P(X >:n) *” ¡-

Next, choose a. number z: auch that: (to)

P(X < z:) - P(:t: € X < rn) - —i-

Finally, choose numbers ::; and :… (a:; < 21 < z.) auch that: (I:) P(X <x;) +P(X > n) - P(a:; (X ¿, x¡) = P(r.¡ ¿. X <:c;) = %.

Using the values of 31 and :tg that you have chosen and tahiee of the standard normal distribution, find the unique normal distribution fer X that satíeñes the relations ¡11 parte a and b. Assumíng that X has this normal distribution. find from

the tables the values which a:. and :. must have in order to satisfy the relation in

part c and compare them with the values that you have chosen. Decide whether or not your distribution for X can be represented approximateiy by &. normal distribution.