Origins of subjective probability

Acta Psychologica 34 Subjective probabiIity (G. de Zeeuw et al., eds.) 1970, 287-299 0 North-Holland Publishing Company, Amsterdam

ORIGINS OF SUBJECTIVE PROBABILLTY JAMES A. WISE Ohio State University, Colu?nbus, Ohio, U.S.A.

ABSTRACT

A geometricmodelof cognitivestructure was proposed and its implications for non-frequentistic subjective probabilities were investigated in an empirical, subjective estimation task. The basic idea was that distance relations in the cognitive structure can generaatesubjective probabilities that meet the formal requirement of a probability measure. These probabilities have certain consistency properties that are analogous to those possess4 by frequentistic probability. The implications of this geometrical interpretation were discussed. The conclusion is that the results suggest interrelationships among our present classifications of subjective probability that may allow our theories the comprehensiveness they lack at present. 1. INTRODUCTION

At the core of decision theory lies the concept of probability, In terms of relative frequency, i.e. ‘objectively’, probability has been rigorously defined by several mathematicians, notably MOLMOGOROV (1950) and VON MISES(1964). Sulz,$ctively, the situation is less clear. Everyone seems to appreciate probabilities; as evidenced by the lengthy and colorful history of games of chance and mathods of divination. This cognitive USA of probability has been called ‘subjective’ by the decision theorist, in order to distinguish it from its supposedly realistic, ‘objective’ counterpart. However, experimental investigations of subjective probability have commonly used a relative frequency measure. The idea of non-frequentistic probability has remained relatively uninvestigated due to lack of appropriate models in the applied situation. The purpose of this study is to partially rectify this imbalance. It seeks to demonstrate that personalistic probabilities can be derived in an experimental situation within a general theory of cognition. The approach is in two stages. First, a general cognitive model is proposed to be descriptive of the subject’s cognitive categorization for this particular experimental situation, Second, a measure operation on the structure of this categorization is defined to meet the axiomatic requirements of 287

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a probability measure. This model is then empirically tested with respect

to some accuracy and consistency predictions. 2.

MODEL

categorization model characterizes a set of stimuhrs events as points mapped into an n-dimensional cognitive space. For experimental purpu%es,this is treated as a linear vector space. One of these stimulus events, designated the test point, is given the class name (1). The subject is told that one and only one of the remaining stimulus events is also identifiable under the same class name, and that the present stimulus events exhaust the alternatives. The question is: What is the individual’s subjective probability that each of the unidentified events also belongs under the class label (1). These are unitary events, and no distribution measures for the class 1 are given. With only this information, the situation is unanalysable in terms of frequentistic probability. People are willing to give subjective probability estimates under these conditions however, so the question is, ‘What are the characteristics of their cognitive structure that contribute to the derivation of their subjective estimate? The possibility pursued here is that their forthcoming estimate is a function of the difference between the test point stimulus and the alternative stimuli. This is a consequence of assuming that if the test point and any of the alternative points were exactly the same, they would be coincident and the subjective probability should be 1.00. The difference of each alternative point from the test point (I) can be measured in terms of its distance from t. The method of determining the appropriate distance metrile in these representations has been the subject of extensive controversy (see, for example HAKE, 1966), but these arguments will not be considered for this experiment’s purposes. The only criterion to be met here is that the particular distance function will accurately predict the subject’s estimates when substituted into the model. For this reason, the appropriate distance function will be rreated as a power transform of the standard Euclidian distance function defined in the Cartesian coordinate system. What this means is that the subject’s cognitive categorization of the stimuli will be considered to be a Cartesian organization, except for a power transform of the distance metric. This approach is similar to that taken by STEVENS(1957) except that here the power transform is introduced in the cognitive organization, not in The

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the sensory transmission system. As will be seen, the nature of the experiment makes this approach ;5 reasonable one. 2.1. A measureof subjectiveprobdility As stated earlier, the operation used to define a probability measure uses the distance of each alternative stimulus from the test point (t). In the problem considered, the simplest situation requires two alternative stimuli. In this case, the hypothesis is that for any two stimuli E’and j, the probability that either one belongs under the same class as the test point is inversely proportional to its distance from the test point. Because the stimuli are exhaustive and mutually exclusive, the probability for j must equal (1 - probability for i). Therefore, the ratio of the probabilities assigned to the alternatives must be equal to the inverse ratios of their respective distances from the test point. Symbolically, p4

dkjt

iLp,=&

and simplifying

where CPUis the Euclidian distance to some power k, and d > 0. Taking the log of this expression provides a simple means of determining the value of k. All that is necessary is to plot the ratio of the distances against the ratio of the probabilities on log coordinates. This should result in a straight line whose slope is k. Solving for one probability in the simple case gives a clue to generalizing the model beyond the two alternative situations. Here, 1 Pi = 4 (4dkdkrt. The function of the first quantity on the right hand side of the expression is to act as a normalizer. Call it s, so that Pi=s&.

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Now, just as with two alternatives, in a multiple-alternative situation with r stimuli, i, j, . . ., r, the sum of the probabilities over the istimuli is Pt + Pj +

=s =

l

l

l

-t- p

(l+l &+“‘J.Kt &

1 >

1.00.

If this is solved for s, we have

Then, for any stimulus i, the probability that it belongs to the same class as t when there are I alternatives is

This expression says simply that the probability that any alternative stimulus belongs in the same class as the test stimulus equals the reciprocal of the psychological distance between the test and alternative stimulus normalized with respect to the other exhaustive alternatives. To the careful reader, this model should seem familiar. Formally, it is equivalent to LUCE’S(1959) generalization of the strict binary utility model for preference probabilities, and to RESTLE’S (1955) model for the probability of a correct response in a discrimination learning task. The important difference, of course, is in what is predicted. The latter two use the formal properties to predict frequentistic choice probabilirties, while 1 have developed them to predict subjective probabilities. Among these different representations, the formal cartespondence is due to the general way of proportioning the quantities involved so that the measure fulfills axioms of probability theory. It is interesting that a model of subjective probability can be equivalent to frequentist moldels except for a theoretical labelling of elemetts. I shall have more to say about such interrelationships later. It has been shown (WISE,1970)that this measure does fulfill the axiomatic requirements of a probability measure. Here, a brief intuitive examination will show that it behaves in the manner that one expects of probability measures.

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First, because it is a simple, normalized ratio of the distances involved, the measure varies between 0 and 1.00. Second, the measure is invariant over common multiple extensions or contractions of the vectors in the generating structure. This is analogous to saying that 8 = g = t, etc. Third, only the relative locations in the space are important. A configuration of test stimulus and alternatives may bc transformed by any of the congruences and the measure would be unchanged. This intuitive examination reveals certain consistency properties that can be experimentally tested. Together with the predictive accuracy of this model, this was done in an empirical subjective estimation task. 3. EXPERIMENTAL STRATEGY Evidence from correlational studies (PETERSON and BEACH, 1967) indicates that subjects frequently weigh stimulus dimensions unequally, and that they also bring unpredictable, personal biases (DE KLERK, 1968) into consideration when making complex judgements. To minimize these influences, the stimuli and judgement task were chosen to be as novel to the subjects as possible. In this way it was hoped that all subjects would similarly categorize the stimuli given the same information. 3.1. Stimuli The stimuli were 24 schematic faces of extraterrestrials that differed on only three independent dimensions. These were (1) length of antennae, (2) placement of ears on side of head, (3) distance between nostrils. Every face was identical except for these three dimensions. The 24 faces were divided into 2 groups of 12 with the values on the three dimensions in one group being a common multiple (1.5) of those in the other. For example, a face with dimension values of (10, 20, 18) in one group would have values of (15, 30, 27) in the other. Within each group of 12, the faces were experimentally organized into 4 triads so that the relationships within B :ri;ld of group ! corresponded to those within a triad of group 2, and vice versa. Each of these triads had an isosceles structure, when the faces are experimlentally categorized into the Cartesian system defined by the three dimensions. This structure allows some interesting manipulations, if, regarding the task described earlier, successive faces are allowed to become the ‘test point’ on different presentdons. Fig. 1 is a two dimensional analogue of this organization.

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Here, if face (1) is the test point, then the probability that either one of the other faces belongs in the same class is 0.50. If (2) is the test point, then the probability ratio of course favors 3, and the actual probability depends on the distances involved. If (3) is the ,test point, the probability for 2 and 1 should be the same as for 3 and 1 previously, because the distances are equivalent on either side of the bilaterally symr&ric triadic structure.

Fig. 1. The isosceles conetrwion

of an experimental triad.

So, in each triad there is the possibility of two independent probability measures of equivalent distance relations on repeated exposures, plus a relationship that should be judged -W/50.The equivalent distance relations measure the subject’s consistency regarding a particular relative distance, even though different distances are involved. The 4 types of triads were constructed so that the Euclidian distances generated a wide range of probabilities. Judgements involving a t.riad of group 1 with its corresponding triad in group 2 measured the invariance that should be found with respect to extensions of the stimulus vectors. Repetition of the SO/SOjudgement across triads reveals if equal probability holds across different distances and structural configurations. 3.2. Procedure For the experiment, each triad was mounted on heavy white cardboard with one of the faces above and directly between the other two. The

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top face was always the test face and the bottom two were designated alternatives. Each triad*was presented 3 times - glevertwice consecutively - with a different member of the triad appearing as the test face on each presentation. The subjects were told to make their probability judgements with respect to the following imaginary situation: They were told to imagine themselves in the future when the earth had made contact with an alien race humanoid in character, etc. This race relied on the concept of ‘brotherhood’ to determine an individual’s social status, and friendly contacts should respect this hierarchy. ‘Brothers’ in their race tend to look alike, as do brothers on Earth. So the subjects were to consider themselves to be a group of judges who must make some subjective estimates as to the likelihood of the ‘brotherhood’ relation based on the schematic pictures they were to see. They were told what the relevant facial characteristics were, and were asked to consider these equdy in making their judgements. They were further told that the 2 bottom individuals were not brothers, and that these two exhausted the alternatives of brotherhood to the top individual. Although they may set some faces more than once, they should concentrate on making each judgemend solely in terms of the configuration before them. Each group of 3-6 subjects then saw the triads in different, randomly ordered presentation sequences, and made their estimates by drawing a line across scales that were marked in, units of 0.01 from 0.00 to I .OO. These scaIes were printed on separate pages in personal answer booklets. For this first study, Ss were 29 male college undergraduates. 4.

RESULTS

Medians of the 29 subjects* estimates were computed for the 24 dispiays. The mean of the medians for each of the 4 basic triads’ probability split other than 50/50 was taken as the representative probability split for that triad. These representative mean ratios and the mean ratio of the SO/SOeslimates were then plotted on log coord’inates against the ratio of the objective distances. 4. I. Estimationof k Fig. 2 shows the results. The dark line is for slope k = 2, and the open circles are the obtained estimates. The closed circles represent the probability split that should have occurred if the square of the Euclidian distance was the appropriate transform for these estimates. The proper interpretation is how near each open circle is to its closed

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circle, for even if the subjects were perfectly consistent with dz, their

points would not have fallen precisely on the line due to rounding errors in the calculations of the triads. This value of k generates probability splits of 97103, 90/10, 77/1!3, and 69/21 other than 50/50, for each of the four types of triads. Subsequently, predictions were made with this value of k = 2 substituted into the model.

THEORETICAL

BEST FOR SLOPE K:t

EXPERIMENTALLY OBTAINEO FROM MEDtAN OF ESTIMATES

8

3

4 56;B$io CALCULATED

20

30

’

djldi

Fig. 2. m-log plot of calculateddj/drfor each of the four types of triads vs. mean af mAian estimatedPtPj for each of the four types of triads.

4.2. Predictive accuracy Fig. 3 is another way of examining the data. It shows the relationship between the predicted probabilities for the four triadic structures and the obtained ss’ medians. Points on the 45” line represent i,dentity between the two sets of values, The thin line connects the mean of the medians at each probability level. It seems that the Ss overestimate SO/50and underestimate in the middle of the range with respect to the model. This may be in part a statistical artifact, because errors in the

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predicted estimates are all summed in one direction. Still, the Pearson product moment correlation is r = 0.96 and b = 0.84,

x Mean of median estimates 0.90-

$

33

0.60-

040

o.tio

0:70

Predicted

o’s0

O.&O

values

Fig. 3. Predictedvs. obtained estimated for k=2.

4.3.

Consistency

The consistency of this model is describable in terms of transformations of the model that leave certain predictions invariant. The transformations examined here involved symmetric equivialent distances within triads, and a uniform expansion of the stimulus vectors composing a triad. Fig. 4 shows the consistency of the median subjective estimates with respect to the two equivalent distance relations within each triad. This is generally quite good, t = 0.92 and 6 = 0.89, and would have been better were it not for two deviant points. It was later found, that as an accident of randomization, the displays for these two points followed each other for 11 out of 29 subjects. Some faces in the two displays resembled each other - so it appears the subjects were trying to be consistent in their judgements, and estimated the same value for each

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display. Subjects’ post-experimental written coi~.~en!~ support this interpretation. The other consistency prediction was invariance between corresponding triads in different groups. Fig. 5 shows the consistency between triads in groups 1 and 2 for each median of the subjective estimates. As predicted, the consistency is quite high, t = 0.96, b = 0.93.

ok0

0:70 GO 0190 Obtained median ‘3’ estimate

Fig. 4. Ccmsistency within triadx

median estimates of @valent relationships on repWe prcsentationk

distance

4.4. Related results In this experiment, the variability of the subjects’ estimates, as computed by the semi-interquartile range (Q), was similar to the variability of estimates collected in more traditional experiments. For half of the distributions of estimates Q was 0.05 or less, while fsr the remaining, Q was less than 0.10, with one (0.12) exception. The results of this first experiment were replicated in a second experiment (WISE, 1970) using 3 alternative fws and two dimensions. Again high predictive accuracy and high consistency was apparent, and

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equi-probable predictions were overestimated by the same relative amount. In both of these experiments, subjects’ post-experimental written comments indicated that no one had reliably recognized repeated faces or used a strategy other than his intuition - which he had been instructed

to rely on.

F .-:

2 0.60 0

1

0.60 Obtained

Fig. 5. Consistency across triads:

0.70 median median

I

1

0.60 estimate, estimates

a90 group of

I

ti~G\~it-ni

riistance

relationships for similar triads of groups 1 and 2. 5.

DISCUSSION

The results of this experiment provide substantial

support for the and the technique

hypothesis about the derived probability measure chosen to investigate it. Given different types of stimuli, it is entirely conceivable that a different distance function would be appropriate .it even a different model of cognitive structuring. Here the value of k = 2 thaj was found should not be surprising, since all three facial dimensioni\ involved real distances. What has been shown is that if WC: have good1 reason to believe that subjects structure stimulus information in a particular way, we can use mathematical hypotheses about their structuring to derive a model for the estimates they give

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in a subsequent complex judgement task. In this experiment the ‘good reason’ was supplied by the experimental conditions, designed specifically to encourage a certain type of subjective categorization. The high predictability and consistency obtained indicates this technique was successful. Another interesting possibility would be to use a multidimensional scaling analysis to make a categorization in terms of a generalized model, such as the Minkowsky metric. These results could then be used in a measure like the one developed here to genIerate experimental prediction. Perhaps m.ost intriguing is what results like this mean for the theory of subjective probability. Estimates of this nature can be questioned on the grounds that they are merely a form of relative similarity scaling. But such criticism misses the point. In this case, the interpretation would be that subjective probabilities could be generated by measuring the relative similarities of stimuli. This is not really different from the notions that subjective probabilities are generated by measuring relative frequencies of stimuli, or that choice and response probabilities can be inferred by measuring relative outcome utilities and relative numbers of conditioned cues, respectively. In the lack of situations where probabilities can be inferred from observing repetitive occurrences in the environment, they must arise from an individual’s processing of other environmental information. If this information is expressible in terms of similarities, all the better, for it allows us to bring the techniques developed in this area to bear on the problem. What remains to be demonstrated is whether similar findings will be obtained for these types of estimates when other traditional paradigms (e.g. betting) are used. Even at the present, these results suggest a different interpretation of subjective probability. Different ‘types’ of subjective probabilities have been regarded to result from different generating factors. And so we have probabilities resulting from relative freuqency, symmetry, or logical considerations. But perhaps these so called ‘varieties’ of subjective probability are not as r;eparate as psychologists have semantically isolated them. A subjective probability could easily be but one type of output from a generaliable cognitive structure that is expressible in many ways. By an appropriate measure function, a subjective probability could be taken from the same cognitive representation that upon different occasions c&d yield judgements of proportionality, similarity, complexity, etc., each with a distinct output through a corresponding me.asure. This characterization, contrasted with the usual theoretical approach of labelling a process in terms of the response used to study it, em-

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phasizes the coherence we should expect to find in an integrative cognitive system that behaves like an ‘intuitive statistician’. Designing experimental investigations to utilize these interrelationships should bring our theories to the general comprehensiveness t’hey lack now. REFERENCES DE KLERK, L. F. W., 1968. Probabilistic concept learning. Voorschoten, The Netherlands: Vam. HAKE, H. W., 1966. The study of perception in the light of multivariate methods. In: Cmttell,R. B.ted.), Handbook of multivariable experimental psychology. Chicago: Rand McNally, 502-534. KOLMOGOROV, A. N., 1950. Foundations of the theory of probability (Trans. by Morrison, N. D.). New York: Chelsean Publishing Company. LUCE, R. D., 1959. Individual choice behavior: a theoretical analysis. New York: Wiley. PEIERSON, C. R. and L. R. BEACH, 1967, Man as an intuitive statistician. Psychol. Bull. 68, 29-46. RESTLE, F., 1955. A theory of discrimination learning. Psychol. Rev. 62, I l-19. STEVENS,S. S., 1957. On the psychological law. Psychol. Rev. 64, 153-181. VON MISES, R., 1964. Mathematical probability and statistics. New York-London: Academic Press. WISE,J. A., 1970. Origins of subjective probability. Unpublished doctoral dissertation. University of Washington, Seattle, Washington.