Subjective Probability Subjective Probability

File ID: risk0539Subjective ProbabilityShawn P. CurleyCarlson School of ManagementUniversity of Minnesota Shawn P. CurleyDepartment of Information & Decision SciencesCarlson School of ManagementUniversity of Minnesota321 19th Avenue SouthMinneapolis, MN 55455USAChapter in:Encyclopedia of Quantitative Risk AssessmentEd Melnick & Brian Everitt (eds.)DraftNovember 27, 2006Subjective ProbabilityI. IntroductionThe fundamental reality that motivates interest in any form of probability is variability. There is variability over instances, over space, and over time. As a consequence, we experience uncertainty: a lack of surety in the face of variability. Although uncertain, we intuitively recognize that some things are more uncertain than others; variability itself is variable. That uncertainty has degrees also supports our attempts at measurement. Thus, the overarching definition of subjective probability is that it is such a measure, one that captures a degree of belief, a degree of judged certainty. Surrounding this general notion are several major variant approaches to subjective probability.A first approach is to treat subjective probabilities as an interpretation of mathematical probability theory. Thus, Section II of this chapter will begin with probability theory as a mathematical system and derive subjective probability as an interpretation of a correspondence between this theory and reality. Subjective probabilities, being personal, need to come from the person. There are two general variants for assessing these degrees of belief: the direct expression of likelihoods and the inference of likelihoods from expressed preferences. Section III derives these variants from an axiomatic system for a qualitative probability structure designed to connect probability more directly to the underlying behavior it is intended to represent. In Section IV, a broader argument is presented as a second general approach to subjective probability. The claim is that subjective probability is not just one of many interpretations; but, rather, subjectivity is an inherent aspect of all uses of probability, i.e. that all probability statements are subjective. Bayesian inference as an outgrowth of this approach is described. Section V describes a third general approach to subjective probability, one which disconnects from traditional mathematical probability theory. Treated as a measure of a degree of belief or of subjective uncertainty, it is noted that uncertainty has not always been seen as a unitary construct. Specifically, from the early days of probability theory, a distinction has been claimed between uncertainty arising from the weight of evidence and uncertainty arising from the balance of evidence. Dempster-Shafer degrees of belief are outlined to exemplify meaningful measures of uncertainty that are distinct from probability theory.A good resource for many of the topics and issues raised in this chapter is the book edited by Wright & Ayton [1]. The book provides a good next step for those interested in pursuing subjective probabilities further.II. Subjective Probability As an Interpretation of Mathematical Probability TheoryA. Mathematical Probability TheoryProbability theory is mathematical. Like all branches of mathematics, e.g., algebra or geometry, the theory is an abstract, self-contained representation, with no necessary connection to thereality in which we live. However, although this theory can be viewed separately from reality, like other mathematical theories, probability theory derives its strength and interest from the correspondence to reality that can be formed. In that we can draw correspondences between elements of geometry (points, lines, planes) and elements of the real world (objects, edges, surfaces), geometry is critical to good architecture and construction. Before addressing the correspondences for probability theory, we begin by outlining the theory itself.As a mathematical theory, probability theory starts with basic principles and definitions and, from these, other laws and theorems are derived. Kolmogorov [2] is generally credited with the mathematical formulation of modern probability theory. He defined a field of probability as a system of sets and a an assignment of numbers that satisfy the following axioms:i. E is a collection of all elementary events [the complete set of possibilities].ii. All subsets of E, and their sums [A + B, i.e., A or B], products [AB, i.e., A and B] anddifferences [A - B, i.e., A but not B] are defined.ii. Each subset A can be assigned a probability P(A) $ 0. [Sets the lower bound at 0,which gets assigned to the impossible event.]iv. P(E) = 1. [Sets the upper bound at 1; that one of the elementary events will occur ispresumed certain.]v. If A and B are mutually exclusive (have no elementary events in common), then P(A +B) = P(A) + P(B).These axioms, along with the definitions of a conditional probability [probability of A given B, i.e., conditional on the assumption that B occurs for certain]:(1)P A B P B (|)()=P AB ()and of independent events:A andB are independent if P(AB) = P(A) * P(B)are the key specifications of mathematical probability theory.B. Views of Theory CorrespondenceHow then to apply this theory in the real world? I first present two non-subjective interpretations of probability to provide context for a third, subjective view that is our focus.1. Classical ViewWhat is the probability of a coin turning up heads in a single toss?In the history of probability theory (e.g., as described by Hacking [3]), games of chance provided an early motivation for the theory's development. The view begins with the specification of the elementary events of Axiom (i). Further, the situation provides no differentiating knowledgeabout the elementary events, and a perceived symmetry is observed among them. For example, the usual six-sided die provides no information as to any side being favored, and each side is roughly equivalent. This state of affairs leads to an application of the Principle of Insufficient Reason, whereby all elementary events are accorded the same likelihood (for the die, each face has probability 1/6 of appearing).This view has been most usefully applied in using probability as the metric for games of chance, e.g., in casinos and lotteries. Under this interpretation, the emphasis is upon counting: translating the numbers of elementary events to numbers of more complex events using the relationships derived within mathematical probability theory. For example if drawing each individual card in a standard deck of 52 cards is an elementary event having a 1/52 chance, what is the probability of getting exactly three matching card values out of five cards (a three-of-a-kind)? The mathematics of probability theory combines with the mathematics of combinatorics to derive complex probability statements like these.2. Frequentist ViewWhat proportion of days in the past ten years has there been a traffic accident involving at least one fatality, within the city limits of Minneapolis, Minnesota (USA)?What is the probability of a traffic accident involving at least one fatality tomorrow, within the city limits of Minneapolis, Minnesota (USA)?Probability is also recognized as describing proportions, as in the first traffic example above. Proportions clearly correspond with the tenets of probability theory; indeed, probability theory is well-characterized as the mathematics of proportions. So, the proportion of days with exactly one fatality can be added to the proportion of days with 2 or more fatalities and this sum will exactly equal the proportion of days with 1 or more fatalities (an application of Axiom (v)). The frequentist interpretation takes this correspondence and extends it into the realm of uncertainty. As a measure of likelihood, the frequentist interpretation views probabilities as idealized proportions (as the number of observations approaches infinity). Thus, the proportion of the first traffic example could be idealized to a representation of a process producing potentially indefinite traffic fatality values over time according to some idealized proportion. To the extent that tomorrow's value is drawn from this process, the idealized proportion provides a likelihood estimate for tomorrow. Reversing the logic, as the number of tomorrows increases (approaching the unreachable, idealized value of infinity), the observed proportion of days with fatalities converges to a value that is then accepted as the probability.In general, this view is treated as useful in situations where there is: (a) repeatability (identical events are recurring so that some large number of occasions can be observed) and (b) stability (the same process is operating over time to generate these occasions, so that the events are identically distributed). Unlike the classical view, the elementary events are not assumed to be equally likely. However, they could be, making this interpretation a more general case of the classical view, with an empirical grounding.3. Subjective ViewWhat is the probability of the euro being adopted by the United Kingdom within the next ten years?It is hard to imagine any set of similar, equivalent events for which a frequentist interpretation of this probability would make sense. And yet, making such a judgment does make sense to most people. The subjective view partly arose from such considerations.In the subjective view, probabilities are characterized as degrees of belief, more specifically as judged likelihoods. Probabilities under this interpretation are presumed to derive from the evidence that underlies the qualified belief. This evidence could include the presumption of equal likelihoods or evidence of proportions, making this interpretation a more general case of the classical and frequentist views, while stepping back from the more direct empiricism of the frequentist view.Thus, one approach to subjective probability is as an interpretation of the mathematical theory of probability. In this approach a subjective probability is a correspondence between the theory and a reality defined as a subjective degree of belief based on evidence. Before turning in Section III to the development of two variants of this correspondence, the next section discusses the implications of this view for the important issue in risk assessment of the quality of subjective probability measures.C. Quality: Internal CoherenceUnder this perspective, what does it mean for a subjective probability to be good, or at least better than another? What standard of quality can be meaningfully applied? One difficulty with the subjective view as a generalization of the frequentist view, as described above, is the disconnect that is created from empirical evidence. Whereas frequentist probabilities can be meaningfully compared to observed proportions (e.g., see Section IV. B.), subjective probabilities do not necessarily afford such a comparison. If I judge that the p(Rain tomorrow) = 0.7 based on the evidence I have and you judge p(Rain tomorrow) = 0.6 based on your evidence and it rains tomorrow, was my judgment better? Since the judgment is personal and based on different evidence, we might be hesitant to draw this conclusion, particularly for non-repeatable events.The primary means of assessing quality that has developed from the view of subjective probability as an interpretation of mathematical probability is the criterion of internal coherence. This criterion is not applicable to a single judgment like the above; no quality assessment of a single subjective probability has been convincingly argued. Internal coherence does provide a quality standard for certain groups of judgments, however. The mathematical theory, e.g., as illustrated by Axiom (v) and by the definition of conditional probability (Eq. 1), makes statements about how probabilities are related to each other. So, if independent assessments of P(A), P(B), and P(A + B) are made for events A and B that are disjoint, then they should relate to each other according to the specifications of Axiom (v): P(A) + P(B) = P(A + B). For subjectiveprobabilities to satisfy this condition represents internal coherence. The three probabilities cohere as a group to the required relationship set forth by the mathematical theory.Numerous studies have tested different relationships required by the theory. Generally, the results are not positive; subjective probabilities typically do not cohere. For the relationship in Axiom (v), individuals tend to be superadditive, underjudging the likelihood of a disjunction, i.e., stating judgments such that P(A) + P(B) > P(A + B) for mutually exclusive events A and B. For example Barclay and Beach [4] asked students to imagine someone they know getting a car for graduation. They assessed the probabilities of the car being a Ford, being a Chevrolet, and being a Ford or Chevrolet. The average responses showed: P(Ford) + P(Chevrolet) > P(Ford or Chevrolet). A special case of the effect is for partitions of the collection of all possibilities: {A i } = E. Wright and Whalley [5] showed partitions having 5-6 constituent elements to have a probability sum of about 2 (vs. p(E) = 1); and, for partitions of 16 elements, the sum rose to 3.One of the earliest lines of research (e.g., [6]) on the coherence of subjective probability judgments demonstrated conservativism in the updating of probabilities with evidence, as compared to Bayes's Theorem. The theorem, which can be derived from the definition of a conditional probability given in Equation 1, can be stated as:P H D (2)P H p D H p D (|)()(|)()=where H is interpreted as some hypothesis and D as data relevant to the hypothesis. In thisequation, p(H) is the prior probability of the hypothesis, before the data are received; and p(H|D) is the posterior probability, after the data. P(D|H) expresses the likelihood of the data for this hypothesis and p(D) normalizes the calculation so that the posterior probability remains within the interval [0, 1]. The usual finding was that judgments of p(H|D) were less than thosecalculated using Equation 2 from the judgments of the components on the right-hand side of the equation, i.e., the updated probability is too conservative.An even more dramatic example of a failure of internal incoherence is the conjunction fallacy identified by Tversky and Kahneman [7]. The mathematical theorem tested in this case is that:p(AB) # p(A) (3)i.e., the joint occurrence of two events cannot be more likely than either event alone. For example, a head on the first toss of a coin cannot be less likely than a head on both of the first two tosses. As intuitive as this principle appears, systematic violations of the relationship have been demonstrated in a number of studies. The Linda problem, one of those used by Tversky and Kahenman, provides an oft-cited example. Consider the following description:Linda is 31 years old, single, outspoken, and very bright. She majored inphilosophy. As a student, she was deeply concerned with issues of discriminationand social justice, and also participated in antinuclear demonstrations.Then, presented with several statements including the following:[A] Linda is a bank teller.[B] Linda is active in the feminist movement.[AB] Linda is a bank teller and is active in the feminist movement.The vast majority of respondents judged the likelihood of A as less than the likelihood of AB. (For additional information, including reasons for such behavior, Yates [8] has an accessible discussion of findings with respect to the internal coherence standard.)III. Subjective Probabilities As Assessed Degrees of BeliefThe mathematical approach of Section II is outcome-oriented. It focuses upon the resulting probabilities attached to events without considering how the degrees of belief arise. Addressing the latter leads to a more behaviorally oriented approach. There are two general means by which degrees of belief are obtained: by direct assessment of likelihoods and by inference from assessed preferences. And, an important theoretical foundation in either case is the use of an axiomatic theory that analyzes the conditions required of these assessments for probabilities to be derived from them. This section describes an axiomatic system for a qualitative probability structure designed to connect subjective probability more directly to the underlying behavior it is intended to represent.A. Qualitative Probability StructuresIn terms of an axiomatic subjective theory, de Finetti [9] provided the starting point. The idea is to provide minimal constraints on the judgments that would be used in assessing the probabilities and from which a coherent set of quantitative probabilities will be derived. Put another way, if I want to have a set of numerical probabilities that behave in a coherent mathematical way, what constraints must be placed on the judgments? In the axiom system, the judgment is described as a qualitative comparative relation, A š B, between sets (events) A and B. Typically, this relation translates to either: "is at least as likely than" or "is at least as preferred as." These two interpretations form the bases of Sections III.B. and C. The axiomatic structure below follows that described by Luce and Suppes [10].i. All subsets of E, and their sums [A + B, i.e., A or B], products [AB, i.e., A and B] anddifferences [A - B, i.e., A but not B] are defined [equivalent to Axiom ii above formathematical probability theory].ii. A š B and B š C implies A š C. [transitivity]iii. Either A š B or B š A [or both; ii and iii signify that there is a weak ordering of allsubsets, "weak" in that ties are allowed].iv. If A and C are mutually exclusive (have nothing in common) and B and C aremutually exclusive, then A š B if and only if A + C š B + C [common elements, e.g., C, can be ignored or included without effect, a principle of the irrelevance of commonalternatives].v. A ši [i.e., nothing is less likely than the impossible event; sets a lower bound].vi. Not (iš E ) [The entire set of possibilities is more likely than the impossible event;eliminates the trivial case where everything is impossible].To these axioms we can add the following definition:™ B if and only if B š A,Adefining the strict ordering relation (without ties). A conditional probability can also be defined, however, this is more complicated and will be left aside here. Suppes & Zanotti [11] provide an example. For our purposes, it is sufficient to note that constraints on relations involvingconditionality are similar to those above, and support the existence of a quantitative definitionlike that of Equation 1.The qualitative axioms present a series of constraints on subjective probability judgments. For example, Axiom (ii) claims transitivity: If rain tomorrow is judged to be at least as likely assnow and snow is at least as likely as fair weather, then the axioms require that rain be judged tobe at least as likely as fair weather. In terms of these axioms, probably the most controversial isAxiom (iv).As an example, Ellsberg [12], in introducing the concept of ambiguity and positing consistentambiguity aversion, focused on this axiom. The challenge can be demonstrated by the following example. An urn contains 150 balls, 50 of these are known to be blue, the remaining 100 are redand/or white in some unknown proportion. All 100 could be red, all could be white, or anythingin between. Several options, A-D, are defined with respect to this urn with the following payoffs associated with color draws in each game:50 balls100 ballsOption Blue Red WhiteA$100 $0 $0B$0 $100 $0C$100 $0 $100D$0 $100 $100Two choices are offered: a choice between Option A and Option B, and a choice between OptionC and Option D. In the first choice, the modal selection is Option A, avoiding the additional uncertainty, termed ambiguity, of Option B (How many winning balls are there?). In the second choice, the modal selection is Option D, similarly avoiding the ambiguity of Option C. Howeverthis pattern of choices is inconsistent with a qualitative probability structure, particularly withAxiom (iv), the irrelevance of common alternatives.To show this, we can set the utility or value u($100) = 1 and u($0) = 0, since value is generally recognized as being intervally scaled, i.e., any linear transformation of value retains its meaning. Then, the preference for Option A implies:p(Blue) > p(Red)Whereas the preference for Option C implies:p(Red) + p(White) > p(Blue) + p(White)or, cancelling the common alternatives:p(Red) > p(Blue)leading to incompatible inequalities.The problem highlighted by this particular example leads to an expansion of the concept of subjective probability that is taken up in Section V. In addition, the example shows a potential discrepancy between two forms of assessing subjective probabilities. These are identified next.B. Assessing Subjective ProbabilitiesBy far, the most common way of assessing subjective probabilities in practice is to do so directly. The relation A š B is generally interpreted as: "A is at least as likely as B." By this method degrees of belief as sensitive to likelihood are taken as the primitives that assessors can judge directly. Thus, a weather forecaster or physician, providing a forecast or diagnosis, directly reports the numerical measures. As demonstrated by the example in the previous section, an alternative is to derive subjective probabilities from preferences. In this case, the relation A š B is generally interpreted as: "A is at least as preferred as B."Decision scientists have tended to favor this latter approach, grounded in preference behavior. Savage's [13] landmark treatise particularly connected probability and utility theory, providing a theoretical and axiomatic basis for what has come to be known as subjective expected utility (SEU) theory. This theory incorporates a subjective (personal) probability measure into preference as modeled by utility theory. Another manifestation of the decision-analytic ties between probability and utility is the probability elicitation method for assessing utilities. In this method, utilities are derived from judgments like the following: For what probability p are you indifferent between receiving $50 for certain; or, receiving $100 with probability p, otherwise receiving nothing? This method essentially measures utilities as an assessment of preference along a probability scale.Under a strict decision analytic view, judgments are only of significance if they are connected to consequences. Direct assessments which are not grounded in outcomes do not qualify. Only preferentially-based methods, tied to real outcomes, lead to useable subjective probability assessments. Despite this argument, in practice, direct assessments predominate.C. Quality: ConvergenceConsidering subjective probabilities as arising from an assessor based on the evidence available, another view of decision quality can be proposed. However, for whatever reasons this approach is rarely used. From this view, quality can be defined as correspondence with a converging value with shared evidence. This idea has at least three components. First is that multiple judges of an event will tend to converge on a common probability judgment as they increasingly share their knowledge on which the judgment is based. Second is that, if two judgments are based on sets of information where one set of information is a subset of the other, then the judgment based on the more inclusive set of evidence is not worse than the other. Third is that quality can be measured as the degree of correspondence with the convergent value under the condition that the shared information is complete, i.e., as the information is fully shared. This notion of quality is more speculative than the others identified in this chapter, as this view has not been applied experimentally to the evaluation of subjective probabilities. However, authors have identified convergence as a potential expectation and, as such, it provides a potential standard of quality. IV. Probabilities as Inherently SubjectiveThe above characterizations all define subjective probability as against other means of interpreting probability, e.g., against classical or frequentist interpretations. An alternative approach is to accept that probabilities in practice are all subjective. The characteristics that lead to the interpretations are just different forms of evidence.Consider rolling a six-sided die. The probability of a single roll leading to an ace is generally judged to be 1/6. Such an assessment, from the classical view, arises from there being six elemental events, each of which is treated as equally likely. But, how does one determine that they are equally likely? Not all elemental events are so. If I am assessing the P(Rain in Minneapolis MN, USA, tomorrow), I can define Rain and Not Rain as elemental events, but they are not equally likely. To define equally likely events in this situation is probably not possible (a limitation of the classical view that led to its falling from favor). So, again, how does one determine that the six outcomes of a die roll are equally likely? One cannot prove this logically. One cannot prove this empirically either. Even under the frequentist view of a probability as a long-run frequency, the long run is not an empirical possibility but a theoretical concept: the frequency as the number of rolls approaches an unattainable infinite number of rolls.In the end, then, the answer is that one cannot prove the events are equally likely. Ultimately it is a subjective assessment. Consequently, the Principle of Insufficient Reason is understood, not as a statement of how to treat symmetrical events but, as a subjective recommendation of how to assign probabilities to elemental events when no other evidence is available.Similarly, probability is not equivalent to frequency, even in an idealized sense. To the extent that frequencies underlie a probability, wholly or in part, this also is ultimately a subjective judgement arising from the assessor's knowledge. If I am just analyzing how proportions arerelated to each other, no subjective assessment is necessary. But, the use of proportions as a basis for a probability is necessarily going beyond the available evidence to treat the proportion as an idealized long-run frequency. What justifies such a statement? The frequentist interpretation of probability is argued here as grounded in subjective assessments of repeatability and stability. Consider assessing the likelihood of a traffic fatality in Minneapolis tomorrow by using past frequency data. We are assuming that the events underlying the proportion are comparable to each other (e.g., each day is comparable to an identifiable set of other days) and that the likelihood is unchanging (e.g., the chances of an accident tomorrow are the same as on each of those other comparable days). How can we know either of these conditions? We cannot. It is a subjective assessment, based on available evidence, that leads us to operate on these assumptions.The argument, therefore, is that subjective probability is not one of several possible interpretations of probability, but rather that all real-world applications of the mathematical probability are necessarily subjective. Elaborating, the uncertainty being captured in a probability judgment is recognized as depending not only on the event being assessed, but also necessarily upon the knowledge that the assessor brings to the judgment.A. Bayesian InferenceThe recognition of a subjective aspect in using any probability extends to the use of probabilities in statistical inference, most notably in the Bayesian approach to inference. Traditional statistical inference, as manifest in null hypothesis testing and the use of p-values is frequentist.A p-value is interpreted as the probability of a sample result, in an idealized, long-run number of experiments, if the null hypothesis is true. In this, the p-value is most comparable to the likelihood in Equation 2, p(D|H), with a frequentist interpretation of this probability.The Bayesian inference approach considers inference from a more subjective viewpoint. The approach is to consider probability statements as arising from the consideration of evidence, from a process of updating assessments based on accumulating evidence. Bayes's Theorem, which describes the updating of an hypothesis in light of data, for example as expressed in Equation 2, is the conceptual centerpiece to the subjective approach to statistical inference. First, the focus is upon p(H|D) rather than p(D|H). Congruently, the interpretation of the posteriorp(H|D) is as a subjective degree of belief in light of the evidence of D.As to the first point, the Bayesian approach is accommodating a more natural way of viewing the situation. Anyone who has taught an introductory statistics course can testify to the non-intuitive nature of the p-value as p(D|H). Despite all warnings to the contrary, students tend to treat the p-value as p(H|D). This confusion of the two conditionals has also been observed notably by Eddy [14] in his analysis of reports in the medical literature. Eddy observed the same tendency among researchers interpreting medical data as observed in the introductory class, interpreting p(D|H) as p(H|D). Bayes's Theorem explicitly acknowledges that it is ultimately p(H|D) that is of interest and models this explicitly.The second complementary aspect is the interpretation of the resulting probability, the posterior。