on subjective probability and related problems

GUNTER MENGES

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

R E L A T E D P R O B L E M S

ABSTRACT. Of late, probability subjectivism was resuscitated by the development of statistical decision theory. In the decision model, which is briefly described in the paper, the knowledge of a probability distribution over the states of nature plays a decisive role. What sources of probability knowledge are legitimate, or at all possible, is the main point at issue. Different definitions, evaluations, and foundations of probability are narrated, discussed, and weighed against each other. The typical research strategy of the statistician is set against axiomatics of subjective or mathematical probability. Finally, the epistemological roots of the probability concept are located by the author in what he calls the "etiality principle".

I. THE BASIC MODEL OF DECISION-MAKING

I shall begin this paper with a brief description o f the background which

has led up to the renaissance o f probabil i ty subjectivism: statistical

decision theory. The central figure o f modern statistical decision theory as

developed by Wald (1950) is the decision maker. He has certain courses

o f actions at his disposal and he faces certain states o f nature. The states

o f nature are supposed to be probabil i ty measures o f a certain r andom

variable X w h i c h can take values x in a sample space 3L 3~ is made measur-

able by the indication o f a a-algebra 27 x. Any probabil i ty measure # on Zx

is called a state. By (3E, 271, p) a stochastic experiment concerning X is

uniquely determined. The space o f the measures p is called 12.

A single action o f the decision maker may be called a, the space o f

possible actions A. A decision function d is a map of 3; into A. The space

o f all possible decision functions is D = A x. Each deD unambiguously

attaches an act ion a = d ( x ) ~ A to every x ~ . On the cartesian produc t f2 • A a bounded real function L is defined.

L(IJ, d(x)) is the loss the decision maker has to suffer when d ( x ) ~ A is taken while kt is true. L is supposed to be known. By means o f L a risk

funct ion r is defined on 12 x D :

r(li, d) = f L(#, d(x)) d~.

Theory and Decision 1 (1970) 40-60. All Rights Reserved Copyright �9 1970 by D. Reidel Publishing Company, Dordrecht-Holland

S U B J E C T I V E P R O B A B I L I T Y A N D R E L A T E D P ROBLEMS 41

Let Z2 be a a-algebra on f2 transforming f2 into a measurable space. If the decision maker knows the 'true' probability measure 2 on 272 he can calculate the risk

R (2, d) = f r (p, d) d2

with respect to 2. 2 is called the a priori distribution on the states; it is the subject of the

personalistic controversy. For if 2 is known to the decision maker he can apply the Bayes type of solution to his decision problem. A decision function d* is called a Bayes solution of the decision problem if and only if it minimizes the value R(2, d) as d runs through D:

R (2, d*) = inf R (2, d). d e D

Decision problems with known 2 are called decision problems under risk. If the decision maker does not know the 'true' 2 he has to deal with a

class F of probability distributions in (f2, 2;2). Decision problems with unknown 2 are called decision problems under uncertainty. The decision maker in this case has to look for other methods than the Bayes method; Wald (1950) suggested the application of the minimax criterion. A decision function d is called a minimax solution of the decision problem if it minimizes sup R (2, d):

sup R (2, d) = inf sup R (2, d). ;~EF d ~ D 2 e F

II . K N O W L E D G E OF 2

Most people agree in that the application of the Bayes type of solution is a direct manifestation of rationality. Some go even so far as to think that the application of the Bayes criterion is dictated by logic. The important point, however, is whether the decision maker who is facing a certain problem knows his 2.

The decision maker comes to know 2 by three different classical ways: (a) objectively a priori (so-called 'logical' conception), (b) objectively a posteriori fso-called 'frequentist' conception), (c) subjectively (so-called 'subjectivistic' conception).

42 GLrNTER MENGES

A fourth way is the one suggested by Wald: namely, to solve the decision problem independently of any knowledge of 2.

Numerous researchers tend to consider the above order of succession also as a ranking order of declining quality. This does not apply to modern probability subjectivists:

Some, I among them, hold that the grounds for adopting an objectivistic view are not overwhelmingly strong; that there are serious logical objections to any such view; and, most important of all, that the difficulty a strictly objectivistic view meets in statistics reflects real inadequacy. (Savage, 1954, p. 4).

Every topic in statistics ought to be reviewed in the light of the concept of subjective probability. (Savage, 1962, p. 33.)

The 'logical' position comes in with the sentence 'S is P ' . I f S is general, this sentence is incomplete. It can be completed to a universal sentence: 'All S are P ' . This universal sentence can be true or false. Its truth is founded either on logical laws, i fP is contained in S (All natural numbers are real numbers), or on matters of fact, if there is no S which is not P (All men are mortal). If the sentence 'All S are P ' is true, it is given the numerical value 1, if it is universally false, its value is 0. The sentence 'S is P ' can also be completed to the particular sentence 'Some S are P ' . This involves the difficulty of assigning a number between 0 and 1 to this sentence. Moreover we meet with difficulties, when trying to give em-

pirical relevance to it. (Cf. Section VI). Although the subject notion can incorporate a large number of prop-

el'ties, logical probability in most cases refers to a single property with a definite, often finite, number of pairwise disjoint elementary modalities M1, M2,. . . , Ms. I f m~(i = 1, 2,.. . , N ) is the 'objectively justifiable' share falling to the modality M~ when decomposing M, then, in logical interpretation, the probability of Mi, given the property M, is defined as

P(M, /M) = mJ(ml + m2 + "'" + rn~).

E.g., let M be the property 'roulette number' with the modalities MI =0 , M2 = 1,..., M3v = 36. It seems plausible to assume that m~ . . . . . m37 = 1 so that P (O/M) = 3@. But even when there is no sufficient reason for this numerical specification of mi, Laplace and many other 'logicians' found their arguments on the principle of insufficient reason (or principle of indifference, as Keynes called it) and set mx =m2 = ... =mN.

In frequentist conception, M is considered as a property of the outcome

SUBJECTIVE P R O B A B I L I T Y AND RELATED PROBLEMS 43

of a stochastic process, n experiments are carried through (or observations made, respectively) and the outcomes recorded with respect to the modalities of M. If the modality M~ is observed z~ times in n experiments, the probability, in frequentist interpretation, is P,(M~/M)= z,/n. For instance, when in 1000 roulette turns at a particular roulette table zero comes out 25 times, the frequentists will consider the value 0.025 as the probability of zero. Von Mises (1931) and other 'strict' frequentists interpret probability as the limiting value which relative frequency zJn approaches as the number of experiments is indefinitely increased:

lira (zJn)=P(MJM). 11--*oo

The subjectivistic or personalistic conception derives probability from the opinion, feeling, intuition, or belief of a human individual. There are different lines of probability subjectivism to be distinguished; the principal ones, in my opinion, are the following:

(i) Introspective or intuitive subjectivism:

The intuitive thesis in probability holds that ... probability derives directly from the intuition, and is prior to objective experience; it holds that it is for experience to be interpreted in terms of probability and not for probability to be interpreted in terms of experience ... (Koopman, 1940, p. 269.)

(ii) Bayesian subjectivism: The so-called Bayesian criticizes intuitive subjectivism on the grounds that

... it is not easy to ascribe numbers to the intensity of feelings; but apart from this it seems observably false, for the beliefs which are hold most strongly are often ac- companied by practically no feeling at all; ... (Ramsey, 1926, p. 71.)

For the Bayesians, not feelings, but 'dispositional belief' is the source of probability statements. The degree of belief can be found out by hypo- thetical bets. Someone who is ready to bet on M 1 three times as much as on M2 and twice as much as on M 3 - these being the three possible modalit ies- 'believes' that P (MI/M) = ~-r, P (M2/M)= 1At, P (MAIM)= ! 1 1 "

III. H I S T O R I C A L NOTES

The discussion on the three 'positions' of probability is as old as probability theory itself. The founder of probability theory, Jacob Bernoulli, had

44 G U N T E R MENGES

a peculiar ambivalent attitude towards objectivity and subjectivity of probability 1, as is particularly obvious in his correspondence with Leibniz. 2 Bernoulli's objectivistic arguments were taken up especially by de Montmort (1708) who went so far as to postulate that probability calculus may be applied only to such phenomena which are beyond human will. J. Bernoulli's subjectivisfic arguments were first taken up by his nephews Nicolaus (1709) and Daniel (1730-31). Nicolaus Bernoulli applied subjectivistic probability concepts to juridical issues; Daniel was the first to combine probability with utility, in a way that anticipates Savage's approach and strikingly that of modern decision theory altogether. For good reasons, modern decision theory recognizes Daniel Bernoulli as one of its most distinguished and, in any case, earliest pre- cursors. He gave strong impulses to Laplace, too. The next station on the way of the subjectivism controversy is frequently seen in the works of Bayes (1763). In my view, however, it can be shown (Menges, 1967) that Bayes was indifferent towards this controversy; in any case, he was not a subjectivist. The fact that his name is frequently associated with probability subjectivism has two causes; one is that many people have become accustomed to set the name of Bayes as, in a sense, a synonym for 'a priori distribution', the other that the so-called Bayes theorem is used when subjective a priori distributions, in the light of informations, are trans- formed into (subjective or objective) a posteriori distributions. In particular the advocates of so-called Bayesian subjectivism (cf. Section II) see their connection with Bayes expressed in the proposition that:

... a person who succeeds in conforming to the principles of coherence will behave as though there were a probability measure associated with all events in terms of which his preference among statistical procedures is such as to maximize expected utility under this measure. In particular, he will behave in accordance with Bayes' theorem as applied to his personal probability measure. (Savage, 1964, p. 180.)

The real and unquestionable ancestor of probability subjectivism is Laplace (1812, 1814), a genuine apriorist among the subjectivists. Ellis (1842) and Cournot (1843) were the first to oppose Laplace's 'alchemy of logic', Ellis by setting a frequency interpretation against the apriorism of Laplace, Cournot with a critique of Laplace's so-called principle of indifference (see below) and the postulate that not insufficient, but suJficient, reasons should be the basis of probability statements. The

SUBJECTIVE P R O B A B I L I T Y AND RELATED PROBLEMS 45

frequency theory of Ellis was worked out in a masterly manner by Venn (1866) who is known to have exercised a considerable influence upon the English school. The great controversy on statistical methods took place during the second half of the last century. Not Laplace, however, but Quetelet (1869) was the target for criticism. The justification of subjectivism was but a marginal topic in that controversy, featuring C. Stumpf (1893) as apologist of subjectivism in the sense of Laplace, yet going far beyond him in mathematical meticulosity, and yon Kries (1886, 1888, 1916), the founder of a new objective apriorism, as its main opponent. Different schools emerged from this methodological controversy, among them: the English, the Russian and the German school. Well-known is the English school with R. A. Fisher as its central figure. The Russian school is characterized by the names of D. L. Tchebycheff, A. A. Markoff, and A. M. Ljapunoff, the German school by W. Lexis, A. Tschuprow, and L. yon Bortkiewicz. In contrast to the English and Russian school, the German school has ceased to exist. Common to all three was the rejection of probability subjectivism.

The discussion was revived with the appearance of Keynes' (1921) book. In Keynes' view, probability rests upon two pillars: apriority and intuition. Keynes understood probability theory as a theory of "logical reasons which make us believe one thing more strongly than another" (Bortkiewicz, 1923, in his review of Keynes' book). Keynes took over the principle of indifference from Laplace, presenting it in a stronger version, and adding, in particular, the so-called rule of symmetry.

Ramsey (1926), inspired by Keynes, argued a decided probability subjectivism, basing it less strictly on logical apriorism than Keynes, and substituting the concept of belief for that of intuition. Ramsey and some subjectivistic interpretations in Venn were adduced by Savage and his school. Von Mises (1931) came forward with the limit definition of probability, a pure frequentist concept. De Finetti's (1937), and later on Koopman's (1940), Halphen's (1955), and Morlat's (1960) works, in their turn, were intuitionistic, moderately frequentist the much talked of theory by Reichenbach (1935), and logical-aprioristic the broadly conceived probability theory by Carnap (1950).

With the book (1954) and numerous additional papers by Savage, probability subjectivism gained ground, in particular among the American decision-theorists.


IV. PROS AND CONS

The preceding historical notes are far from being complete. To give a complete survey, one would have to study an immense philosophical, mathematical, and statistical literature. I merely wanted to block out the principal lines of development, and to give an impression of its changes and fluctuations. In the following I shall try to outline the arguments for and against the three conceptions of probability as put forward in the course of its long and varied history.

Proaprioristic position

(1) Interpersonal communicability. (2) (Physical) verifiability.

Contraaprior&tic position

(1) Difficulty of associating 'objective' numerical values to the modalities Mx, Mz, ... of the property considered. Savage, e.g., maintains that there exists no satisfactory method of association.

(2) The information upon which association is based is criticized to be ascertainable only with a regressus ad infinitum; a regress which, moreover, is circular.

(3) The field of practical relevance is small, limited to a few phenomena in physics.

Profrequentist position

(1) Interpersonal communicability. (2) (Experimental) verifiability. (3) Direct and easy application in insurance business and other fields

where many individual cases occur.

Contrafrequentist position

(1) Difficulty o f - real or imagined - endless repetition of experiments. (2) Problem of stability: Long series of experiments involve the risk

of changing conditions. This objection holds in particular with respect to social and economic phenomena.

(3) Problem of random series: The required randomness does not exist in reality, not in most cases, anyway.

S U B J E C T I V E P R O B A B I L I T Y A N D R E L A T E D P R O B L E M S 47

(4) Circularity: Relative frequency at best converges to a fixed limit. This passing to the limit can only be interpreted in terms of the probability concept.

Prosubjectivistic position

(1) Practicability: Subjective probabilities can be gained in almost all practical cases.

(2) Applicability to memory~ and perception. 4

Contrasubjectivistic position

(1) Limited interpersonal communicability. (2) "Strength of belief and degree of probability diverge." (This

statement by Meinong (1915) has been confirmed in the light of modern experiments on probability.) 5

(3) Emotional and intellectual circumstances exercise an uncontrol- lable and troublesome influence.

(4) Insufficient differentiation: An individual is (in most cases) not able by feeling or belief to distinguish two probabilities of say ~ and ~ .

(5) Psychological measurement of degrees of belief is very complicated. Hence an untrained person cannot be expected to be capable of measuring his belief without technical help.

It is in fact a matter of one's personal taste, particular standpoint and scientific origin which arguments or counterarguments are considered to outweigh the others. I have the impression, after all, that in spite of all divergence in the opinions most statisticians would accept the following research strategy.

If the research worker possesses reliable 6 objective a priori knowledge or can obtain such knowledge, he will make use of it. A die, for instance, can be analyzed with respect to its physical properties, whether it is 'practically ideal', i.e., whether its six sides are at right angles to each other, whether its mass is evenly distributed, whether it is not loaded etc. If the test is positive, we can without any throw (objectively a priori) indicate the probability as ~.

If the research worker does not possess reliable objective a priori knowledge, nor can obtain such knowledge, he will make experiments in order to infer the unknown probabilities (objectively a posteriori). This is


where statistical inference comes into its rights. Casting a die with unknown probability distribution of the number of points very often one may infer the probability distribution.

If the a posteriori way is also blocked up, because experiments cannot be made, for technical or pecuniary reasons, e.g., or because the stability problem is too serious, some people will resign and consider the task as unsolvable, others will turn to their belief or intuition and try to determine the unknown probabilities subjectively, in particular when there is a strong necessity, e.g., in the framework of a decision problem, to come to know the probabilities.

V. AXIOMS OF U T I L I T Y A N D OF S U B J E C T I V E P R O B A B I L I T Y

As has been mentioned above, Savage and his adherents do not accept the research strategy outlined at the end of Section IV. Instead, they re- commend to determine probability subjectively in all cases. One is greatly tempted to give and to adopt such a recommendation because subjective determination of probability is cheap, and because this method never fails for its believer. For the subjectivists the Bayes criterion is in all cases applicable. If the a priori distribution 2 (cf. Section I) can be - subjectively - ascertained in all cases, the expected risks R (2, d) which are the inevitable basis for the Bayes solution can always be determined.

Due to this close connection between subjective probability and decision theory, as understood by most decision theorists, the transformation of the axioms of decision theory, i.e. the axioms of utility by von Neu- mann and Morgenstern, in a way to fit also subjective probability ap- peared quite obvious. This, in fact, has been done by Savage (1954).

We first consider the axioms of utility in the decision-theoretical con- text of von Neumann's and Morgenstern's (1947) work.

To any possible coincidence of an action a=d(x)sA with a state #~f2 we associate an outcome e; A x g2~{e}. Let the set of outcomes e and probability mixtures of outcomes be denoted by E. The problem of Neumann-Morgenstern utility consists in deriving, from a preference preordering ~ on E, a cardinal utility function. The respective axioms are listed below (together with Pfanzagl's (1959) supplements):

(1) A preordering ~ is given on E.

SUBJECTIVE P R O B A B I L I T Y AND RELA TED PROBLEMS 49

(2) For any two e 1, e2~E and any probability p(0~<p~<l) an element e l p e 2 7 e E is defined such that et-<e2 implies el-<el pe2 for every pc(0, 1).

(3) It follows from et-<e2-<e3 that there exist p ,p ' e (O , 1) with

el pes-<(e2-< et pe ' s . (4) From e2,,~2 follows et pe2 " e t P # 2 .

(5) et pe2 "~ e2 (1 - p ) e 1. ~ 8

(6) (et pe2)p' e2 ~ et (PP') e2 J (7) e 1 pe t ,,~ e 1.

The Neumann-Morgenstern theorem on expected utility says: If the conditions 1 to 7 are fulfilled in E, then there exists a cardinal utility function u on E with the properties of expected utility. With appropriate new interpretation (e.g. : L = - u ) the utility function becomes the loss function of Section I.

The first postulate (axiom) of subjective probability according to Savage runs thus:

A complete preordering ~ is given on the set A of actions.

One may ask why, when this postulate is fulfilled, the decision maker has not yet reached his goal. When he is able to establish a complete ordering on all his actions, why is he interested in probabilities, utilities, losses, and states of nature? Well, in fact, he is no longer interested in these matters. The decision maker adopts the action which he prefers most. Savage, with his first axiom, has suspended the idea of probability. We need no longer interest ourselves in the following six axioms by Savage, as the first one has killed all problems.

Or is it not so? Maybe Savage wished to give the decision maker some help for consistent behavior. For this end, utility theory with its axioms is altogether sufficient.

Or is it just to show that the subjective probabilities he has produced, together with his preferences with respect to E, are consistent with his preordering on A? Well, such consistency can hardly be of any interest when he already knows which action to adopt. And even if he is interested in consistency he can produce it in his own heart and does not need any help in this intimate business: A little more utility here, a little less probability there. If, however, he could not bring about consistency in his own heart, the theory will hardly bring him help: Who, besides him, could

50 GUNTER MENGES

find out whether the error was on the utility side or on the side of probability?

In fact, the first axiom by Savage is strong enough to suspend not only the idea of probability, but of decision theory altogether.

The axiomatization of utility according to yon Neumann and Morgen- stern starts - nota bene - from a (complete) preordering on the set E of outcomes and mixtures of outcomes with exactly known numerical probabilities. 9 It does not postulate that all situations of uncertainty be such that the decision maker exactly knows the resulting mixture of outcomes, in the above sense, for his possible actions. The Neumann- Morgenstern axioms allow for other forms of uncertainty, they do not force the subject to act as if he knew everything; they help him, however, to discover inconsistencies in his behavior. The further de,,elopment of the Neumann-Morgenstern system of axioms, in my view, should not aim at a still greater rigidity and determination of the principles of behavior, but, on the contrary, at more openness with respect to an eventual revision of the evaluation of probabilities. With increasing weight, new empirical and a priori information can bring about such a revision, can increase or modulate uncertainty. Utility theorists do not give us any details as to how this should be carried out. We can learn much more, in this instance, from Fisher's research strategy and philosophy: More precise formulation of a vague law through estimation, testing the precise law in the light of new observations, etc.

Against Fisher's type of the learning individual one might set the Bayesian learner 1~ and compare their behavior. It appears utopian, on the other hand, to take the adoption of this or that learning behavior in itself as a decision problem which could be solved all at once; the correct (or better: successful, in a particular situation) learning strategy is acquired and refined by experience. I shall not, in this paper, enter into details of the problems here involved.

One could certainly think of setting up a system of axioms of subjective probability and utility which, contrary to Savage's system, were not subject to the obvious objection discussed above. But what would it profit us? Utility and probability, although the pillars of decision theory, are entirely different concepts: Probability, a property of random events, is not tied to an individual, verifiable by experiments, and not a personal experience. Utility, on the other hand, is tied to an individual, and does not

SUBJECTIVE PROBABILITY AND RELATED PROBLEMS 51

make sense except with respect to the individual. I t is ascertainable, but

not verifiable, by experiments, and is a personal experience (or rather the

comparison as for preference is such an experience). Probability, when rid of the useless and detrimental linkage with

utility, can be axiomatized at choice. An almost everywhere accepted

system of axioms is that by Kolmogoroff (1933) which, for completeness,

we shall consider now. In modern terms (Menges, 1969, pp. 250f.):

Let H be a a-ring on the nonempty set F whose elements (called elementary events) are the possible outcomes of a definite random observation. Every element of the a-ring H in E is called a random event. Probability is a map P: H ~ R , with the following properties (axioms):

(1) From A, BEH, with A A B = 0, follows p(A fl B) =P (A) + P (B). (2) For any A ~H: P (A) ~< 1. (3) P is normalized, P (E) = 1.

Kolmogoroff (1933, pp. lf.) insists that the semantic relevance of the

probability concept is not increased through the axiomatization. Never-

theless he devotes some reflections to the "relationship with the world of

experience", not without dismissing the mathematical reader for that

part. Any axiomatization of probabili ty is in fact completely neutral

f rom the ontological viewpoint. Epistemologically, the three positions of

probability discussed above are also indifferent, i.e., they do not answer the question if objectively valid probability statements are feasible, nor

which are the conditions for such feasibility.

vI. LOGICAL AND ONTOLOGICAL NECESSITY AND POSSIBILITY

Before going in search of a principle which ensures the real validity of

probabili ty statements, we shall try to find an answer to the question whether such a principle is necessary at all. Is it not sufficient to have a

system of axioms of probability together with the three positions dis-

cussed above? Let me give an example to illustrate the necessity of such a

principle: We have just seen the failure of the Apollo 13 mission; an

explosion in the supply module of the spaceship forced the astronauts to give up their landing on the moon. What is the probable cause for this explosion? For many people, the cause was that the spaceship had number 13. How do we know that the number 13 is not to blame for the failure? The various positions of probability help as little in answering this question as do the axioms of probability.

52 GONTER MENGES

We need insight into the objective facts and circumstances. We want to understand why - with probability (with certainty we shall never know it) - it came to the explosion. Understandable is what can be explained from sufficient reasons. Is it, therefore, the causality principle which we must draw upon?

The principle of causality explains real events which are subject to necessity. In our case, as in statistics, the relationship is not one of necessity but of possibility; not truth, but probability is involved.

To elude confusion which may easily arise, let us say that we are dealing here with ontological relations of necessity and possibility, respectively. Mere deduction of conclusions from given premises which exist in thinking, for instance, belongs into the domain of logical relations. Ob- viously there exist four kinds of connections:

(1) Logical necessity. For instance: All human beings are mortal. P is a human being. Therefore, P is mortal (necessary reason-conchtsion relationship).

(2) Logical possibility. For instance: All human beings are either males or females. P is a new-born human being. Therefore, it is (logically) possible that P is a male (possible reason-conclusion relationship).

If there is no reason to suppose that the new-born human being is rather female than male, then the 'probability' that the new-born child is male is just 0.5. This statement has obviously nothing to do with reality. Its verification through experience is neither necessary nor possible. Meinong (1915) called this kind of 'only logical' probability 'Vermutungs- wahrscheinlichkeit' (presumptive probability). In fact, it is very similar to the subjective probability of the 'Bayesians'.

(3) Ontological necessity. Here we are not concerned with reason- conclusion but with cause-effect, or causal, relationships. If a definite cause is given in reality, a real effect is the necessary consequence (necessary cause-effect relationship). If we cut off a crow's head (cause), the crow dies (effect). We can at any moment prove the truth of this statement by cutting off the head of another, a third, fourth, etc., crow. The crow's death is every time the necessary consequence.

(4) Ontological possibility. As in (3) we are concerned with a cause- effect relationship which, however, lacks necessity. It is 'only possible' (possible cause-effect relationship). We shoot with a gun at a crow at a distance of 120 meters. The shot is the cause, but there are different


possible effects. I f the crow is missed, it does not die at all (effect Wo); if

the bullet hits its lungs, it m a y die (effect wx), etc. As soon as the effect has come abou t the sequence can be causally

explained. The crow is dead because it was shot. But in advance we cannot tell which effect shoot ing at the crow will have. There are different

possible effects Wo, wl, w2 .. . . , p robab le to a definite degree. This p roba- bility (Meinong: 'vermutungsfre ie Wahrschein l ichkei r - p resumpt ion-

free probabi l i ty) is an ontological category. The principle of causali ty

is obviously not dismissed, but the relat ionship 'cause - possible effects'

is no t accessible to an entirely causal explanation. The cause does not

lead to a necessary effect but to a 'd is t r ibut ion law', i.e., a bundle o f

possible effects with definite probabil i t ies:

WO~ WX~ W2~ . . .

Po, P l , P2 . . . . Z Pi = 1 i

We should also ra ther speak of a complex of causes, or o f general causes,

here, than of a single cause. I t does mat te r at which distance the crow is, whether it moves, whether we have a good shot, etc.

Har twig (1956) designated the epistemological principle for the relation-

ship ' complex of causes - dis tr ibution law' as ' ,~t ialprinzip ' (etiality principle):

... indem wit das Kausalprinzip dahingehend formulieren, dab gleiche Ursachen die gleiche Wirkung tiberall zeitigen, mtissen wir ... yon dem ganz analogen Grundsatze als Verkntipfungsregel uns leiten lassen, dab gleiche allgemeine Ursachen das gleiche Verteilungsgesetz zur notwendigen Folge haben. Dieses letztere 'Verursachungsprinzip', tier Form nach dem ersteren tihnlich ... m6ge in Anlehnung an das ... griechische Wurzelwort ... A, tialprinzip heiBen, ohne ein solches Prinzip gtibe es nirgends ein Wahrscheinlichkeitsurteil yon objektiver Gtiltigkeit ... (1956, pp. 257t").

The following table summarizes the four relat ionships:

Necessi ty Possibility

Logical Principle of sufficient reason Principle of

insufficient reason Ontological Causal i ty principle Etiality principle

VII. THE P R I N C I P L E OF ETIALITY AND SUBJECTIVE PROBABILITY

One m a y follow Har twig ' s reasoning or not ; to me it appears inevitable

54 GONTER MENGES

that only by recurring to the linkage between general causes and distribution law we can arrive at objectively valid probability statements.

The analogy between etiality and causality is illustrated by the fact that in the case of etiality a change of general causes involves a change of possible real effects and their probabilities, just as in the case of causality a different cause produces a different unique effect. The probability of the crow's death, for instance, diminishes if instead of a good marksman a bad one shoots at it.

The distribution law belongs inalienably to the complex of causes which actually existed when the observations were made. As long as this complex remains exactly the same, we can, just as with causality, verify the etiality conclusion by another experiment. If, however, there is a change in the complex of causes, we fall back into ignorance which can be overcome again by searching out the distribution law.

Etiality, just as causality, is not provable. It is a relationship which

als transzendentale Voraussetzung unserer Wirklichkeitsvorstellung zugrunde gelegt werden muB, werm anders eine wohlgeordnete Erfahrung auch auf jenen Gebieten menschlichen Wissens zustandekommen soil, wo die Kausalkonzeption wegen der Kompliziertheit der betreffenden Ursachenzusammenh~inge unserem Erkermtnisstreben uniJberwindliche Hindernisse entgegensetzen wiirde. (Hartwig, 1939, pp. 79f.)

What about the relationship between etiality principle and subjective probability? In a word: Badly! Subjective probability, according to the etiality principle, is just as legitimate as subjective causality. Does there exist anything like subjective causality? Well, I don't think so. Certainly there are subjective judgments on probabilities, subjective guesses on logical relationships of possibility. But in order to make valid probability

statements we need more. In the Apollo 13 case we need, to begin with, hypotheses on the com-

plexes of causes. These hypotheses must be considered as premises relat- ing to reality, as, e.g., impact of a meteorite, chemical initial process, breakdown of the electric system, sabotage, interference from outer space, or even the number 13, if one likes. Consider the real premise 'impact of a meteorite'. The probability that a meteorite strikes a flying object of the size of Apollo 13 on its flight to the moon does in no way depend on our knowledge of the phenomena, but only upon the conditions and circumstances existing in outer space with respect to meteorites. We must study these conditions in order to come to a probability statement. It is just

S U B J E C T I V E P R O B A B I L I T Y A N D R E L A T E D P R O B L E M S 55

the same with the other real premises. From the real premises we may draw conclusions about the event. Finally we select that premise which, given our state of knowledge (!), of all competing premises possesses the highest (ontological) probability that the complex of causes of which it is characteristic produces an effect like the observed one. This selection, by the way, is a decision problem.

In conjunction with the principle of insufficient reason we meet with subjective probability as an expression of the relationship of logical possibility (Meinong: Vermutungswahrscheinlichkeit). Above we have attributed yet another function to subjective probability: that of a sur- rogate for objective a priori or a posteriori probability, respectively. Although none of the three 'positions' can put probability statements in the rank of possessing real validity, i.e., bridge the gap between thought and reality, they are characteristic of the methods of numerical specification of probability (in the sense of ontological or presumption-free probability). Every mortal day use is made of all three possibilities of numerical specification of probabilities. There is on principle no objection to this. Yet, in order to be able to conclude from the probability, however specified, to the "Leichtigkeit der Verwirklichung eines realen Ereignisses" (Hartwig) (easiness of a real event's realization) we are in need of a principle with respect to the relationship of ontological possibility. In my view, the etiality principle does meet this purpose.

VIII . H O W TO A S S O R T T H E W H O L E C O M P L E X OF P R O B L E M S

I would like to summarize the results of the above consideration in the form of eight theses, together with a few supplementary remarks.

(1) Axiomatization of subjective probability is nonsensical and useless. (2) Axiomatization of probability, as put forward by Kolmogoroff,

e.g., is sufficient for probability calculus and for the stochastic parts of decision theory.

(3) Beside the axiomatization of probability, that of utility (let us say, in the sense of yon Neumann-Morgenstern) is legitimate. It plays a part in judging of decision-making behavior.

(4) Subjective probability, together with the principle of insufficient reason, expresses the relationship of logical possibility (Meinong: Ver- mutungswahrscheinlichkeit).

56 GUNTER MENGES

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(5) The three classical 'positions' (objective a priori, objective a posteriori, subjective) characterize different methods of numerical specification of probabilities.

(6) Subjective probability as 'degree of dispositional belief' is a sur- rogate for numerical a priori or a posteriori specification of probability.

(7) Probability as component of a statement on matters of fact is the expression of the relationship of ontological possibility.

(8) Relationships of ontological possibility are founded upon the etiality principle which connects complexes of causes with 'laws of distribution', or, mathematically speaking, distribution functions.

For decision theory, from which our consideration has started, the following consequences ensue:

The process of decision-making, as a rule, relies upon a measure of utility, on the one hand, and a statement on probability, on the other. For the acquisition and utilization of probability statements we need the calculus of probabilities which, for its part, rests on (e.g. Kolmo- goroff's) axioms. Moreover, probability must be numerically specified - a problem of statistics. Such measurement can be a priori, a posteriori, or (at last need) subjective. The (objective) real validity of the probability statement is warranted by the etiality principle. For the acquisition and utilization of the utility measure we need utility theory which, for its part, rests on (von Neumann-Morgenstern's) axioms. The utility measure must be numerically specified (i.e., measured in reality). This measurement can also be a priori, a posteriori, or subjective; it is again a statistical problem. (I don't know if there exists anything analogous to the etiality principle with respect to utility, warranting the (subjective) real validity of the utility measure. If so, it must be a problem of psychology.) In the following survey table (see Figure) the place is indicated where such a 'utility principle' would have to stand.

Saarbriicken and Burnaby

REFERENCES

Bayes, T., 'An Essay Towards Solving a Problem in the Doctrine of Chances', Philo- sophical Transactions 53 (1763) 376-398; 401-403.

Becker, G. M. and McClintock, C. G., 'Value: Behavioral Decision Theory', Annual Review of Psychology 18 (1967) 239-286.

Bernoulli, D., 'Specimen Theoriae Novae de Mensura Sortis', Commentarff Acad. Petrop. 5 (1730-31) 175-192.

58 GUNTER MENGES

Bernoulli, N., Dissertatio inauguralis mathematico-juridica de usu artis conjectandi in jure, quam ... ad diem 14 junii a.d. 1709 ... publice defendet M. Nicolaus Bernoulli, Basel.

Bernoulli, J., Ars conjectandi, Basel 1713. Deutsche (2bersetzung: Wahrscheinlichkeits- rechnung (iibers. v. R. Haussner), Leipzig 1899.

von Bortkiewicz, L., 'Wahrscheinlichkeit und statistische Forschung nach Keynes', Nordisk Statistisk Tidskrift 2 (1923) 1-23.

Carnap, R., Logical Foundations of Probability, Chicago 1950. Cournot, A., Exposition de la thdorie des chances et des probabilitds, Paris 1843. Ellis, R.L., On the Foundations of the Theory of Probability, Cambridge Phil. Soc. 1842. de Finetti, B., 'Foresight: Its Logical Laws, Its Subjective Sources', Annales de l'Institut

Henri Poincard 7 (1937). Reprinted in Studies in Suhiective Probability (ed. by H. E. Kyburg, Jr. and H. E. Smokier), New York-London-Sydney 1964, 93-158.

Friedman, M. and Savage, L. J., 'The Utility Analysis of Choices Involving Risk', Journal o f Political Economy 56 (1948) 279-304.

Halphen, E., La notion de vraisemblanee. Essai sur les fondements du calcul des probabilitds et de la statistique mathdmatique, Publ. Inst. Statistique, Paris 1955.

Hartwig, H., Die stochastischen Grandbegriffe der Statistik, Dissertation Frankfurt am Main 1939.

t-Iartwig, H., 'Naturwissenschaftliche und sozialwissenschaftliche Statistik', Zeitsehrift fiir die gesamte Staatswissenschaft 112 (1956) 252-266.

Keynes, J. M., A Treatise on Probability, London 1921. Kolmogoroff, A. N., Grundbegriffe der Wahrscheinliehkeitsrechnung, Berlin 1933. Koopman, B.O., 'The Axioms and Algebra of Intuitive Probability', Annals o f

Mathematics 41 (1940) 269-292. von Kries, J., Die Principien der Wahrscheinlichkeitsrechnung, Freiburg i.B. 1886. von Kries, J., '~ber den Begriff der objektiven M6glichkeit trod einige Anwendungen

desselben', Vierteljahresschrift fiir wissenschaftliche Philosophic 12 (1888). ,con Kries, J., Logik, Tfibingen 1916. de Laplace, P. S., Thdorie analytique des probabilitds, Paris 1812. de Laplace, P. S., Essai philosophique sur les probabilitds, Paris 1814. Leibniz, G.W., Gesammelte Werke (edited by Pertz and Gerhardt), Vol. 3, Halle

1855. Luce, R. D. and Suppes, P., 'Preference, Utility and Subjective Probability', in Hand-

book ofmathematicalpsychology (ed. by R. D. Luce, R. R. Bush, and E. Galanter), Vol. III, New York 1965.

yon Meinong, A., f3ber M6glichkeit und Wahrscheinliehkeit, Leipzig 1915. Menges, G., 'f]ber Thomas Bayes (1702-1761) und das Theorem - Versuch einer

Wiirdigung', in Geschichte und Zukunft. Anton Hain zum 75. Geburtstag (ed. by A. Diemer), Meisenheim/Glan 1967, 485-498.

Menges, G., Grundmodelle wirtschaftlicher Entseheidungen, K61n-Opladen 1969. yon Mises, R., Wahrscheinlichkeitsrechmmg und ihre Anwendung in der Statistik und

theoretisehen Physik, Leipzig-Wien 1931. de Montmort, P. R., Essai d'analyse sur les /eux de hazard, Paris 1708. Morlat, G., 'L'incertitude et les probabilit6s'. F~conomie appliqude 13 (1960). Mosteller, F. C. and Nogee, P., 'An Experimental Measurement of Utility', Journal

of'Political Economy 59 (1951) 371-404. yon Neumann, J. and Morgenstern, O., Theory of Games and Economic Behavior,

2nd ed., Princeton 1947.


Pfanzagl, J., Die axiomatischen Grundlagen einer allgemeinen Theorie des Messens, Wiirzburg 1959.

Preston, M.C . and Baratta, P., 'An Experimental Study of Auction Value of an Uncertain Outcome', American Journal oJPsychology 61 (1948) 183-193.

Quetelet, L. A. J., Physique sociale, Paris 1869. Ramsey, F. P., 'Truth and Probability' (1926), in The Fotmdations ofl~athematics and

Other Logical Essays, London-New York 1931. Reichenbach, H., Wahrscheinlichkeitslehre, Leiden 1935. Robbins, H., 'The Empirical Bayes Approach to Statistical Decision Problems', in

The Annals of Mathematical Statistics 35 (1964) 1-20. Savage, L. J., The Foundations of Statistics, New York-London 1954. Savage, L.J . , 'Subjective Probability and Statistical Practice', in The Fotmdations

of Statistical Inference (ed. by M. S. Bartlett), London-New York 1962, 9-35. Savage, L. J., 'The Foundations of Statistics Reconsidered', in Studies in Subjective

Probability (ed. by H . E . Kyburg, Jr. and H .E . Smokier), New York-London- Sydney 1964, 175-188.

Stumpf, C., '~ber den Begriff der mathematischen Wahrscheinlichkeit', in Sitzungs- berichte der philosophisch-philologischen trod historischen Klasse der krniglichen bayrischen Akademie der Wissenschaften zu Miinchen, Jahrgang 1892, Mi.inchen 1893.

Venn, J., The Logic of Chance, London 1866. Wald, A., Statistical Decision Functions, New York-London 1950.

N O T E S

1 The strength of his objectivistic engagement may be seen from the following quota- tion: ". . . es ist zur richtigen Bildung yon Vermuthungen tiber irgend eine Sache nichts anderes zu thun erforderlich, als dass wir zuerst die Zahl dieser F/ille genau ermitteln und dann bestimmen, um wieviel die einen F/ille leichter als die anderen eintreten k/Snnen. Und bier scheint uns gerade die Schwierigkeit zu liegen, da nur ftir die wenig- sten Erscheinungen und fast nirgends anders als in Glticksspielen dies mrglich ist; die Glticksspiele wurden aber yon den ursprtinglichen Erfmdern, damit die Spiel- theilnehmer gleiche Gewinnaussichten haben sollten, so eingerichtet, dass die Zahlen der F~ille, in welchen sich Gewinn oder Verlust ergeben mug, im voraus bestimmt und bekannt sind, und dass alle F~ille mit gleicher Leichtigkeit eintreten k/Snnen. Bei den weitaus meisten andern Erscheinungen aber, welche yon dem Walten der Natur oder yon der Willktir der Menschen abh~ingen, ist dies keineswegs der Fall. So sind z.B. bei Wtirfeln die Zahlen der F~ille bekannt, denn es gibt fiir jeden einzelnen Wtirfel ebenso- viele F/ille als er Fl/ichen hat; alle diese FNle sind auch gleich leicht mrglich, da wegen der gleichen Gestalt aller Flfichen und wegen des gleichm/issig vertheilten Gewichtes des Wtirfels kein Grund daftir vorhanden ist, dass eine Wiirfelfl~iche leichter als eine andere fallen sollte, was der Fall sein wtirde, wenn die Wtirfelfl~ichen verschiedene Gestalt bes~issen und ein Theil des Wtirfels aus sehwererem Materiale angefertigt w/ire als der andere Theil." (Bernoulli, 1713, German translation, 1899, pp. 88f.)

Leibniz (1855), pp. 71-97. The letters were written between 1703 and 1705. 3 I remember with high probability, says the witness, to have seen the suspect at the locality of the crime. 4 I hold, says the soldier on the telescope, something has moved over there.

For instance, by Friedman and Savage (1948), Preston and Baratta (1948), Mosteller and Nogee (1951), Luce and Suppes (1965), and Becker and McClintock (1967).

60 G f.JNTER MENGES

6 The subjectivists often criticize that a judgment of the reliability of knowledge is always subjective. In principle, this objection is sound. Yet a refined technical apparatus can set rather narrow limits to the field of subjective judgment. 7 el p e~ denotes the situation of uncertainty of obtaining el with probability p or e2 with probability 1 --p. s el p e2 denotes the situation of uncertainty of obtaining el with probability p or e2 with probability 1 --p.

It is altogether possible, by the way, that there exists a complete preordering on the set of pure outcomes, and an incomplete one on the set of mixed outcomes. a0 Cf. Robbins ' 'empirical Bayes approach' (1964).

on subjective probability and related problems

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