Conditioning (Support Theory)

Conditioning

In support theory, P(A, B) is interpreted as the conditional probability of A, given A or B. To obtain a general treatment of conditioning, we enrich the hypothesis set H by assuming that if A and B are distinct elements of H, then their conjunction, denoted AB, is also in H. Naturally, we assume that conjunction is associative and commutative and that (AB)′ = A′ ∩ B′. We also assume distributivity; that is, A(B ∨ C) = AB ∨ AC. Let P(A, B|D) be the judged probability that A rather than B holds, given some data D. In general, new evidence (i.e., a different state of information) gives rise to a new support function s indexed by D that describes the revision of s in light of D. In the special case in which the data can be described as an element of H, which merely restricts the hypotheses under consideration, we can represent conditional probability by

$$P(A, B | D) = \frac{s(AD)}{s(AD) + s(BD)}$$

provided that A and B are mutually exclusive, the evidence D is compatible with at least one of them, and the denominator is positive.

Several comments on this form are in order. First, note that if s is additive, then the above reduces to the standard definition of conditional probability. If s is subadditive, as this model assumes, then judged probability depends not only on the description of the focal and the alternative hypotheses but also on the description of the evidence D. Suppose D′ = (D1 ∨ D2)′, D1 and D2 are exclusive, and D is implicit. Then

$$P(A, B | D) = \frac{s(A(D_1 \lor D_2))}{s(A(D_1 \lor D_2)) + s(B(D_1 \lor D_2))}$$

But because s(AD) ≤ s(AD1 ∨ AD2) and s(BD) ≤ s(BD1 ∨ BD2) by subadditivity, the unpacking of D may favor one hypothesis over another. For example, the judged probability that a woman earns a very high salary given that she is a university professor is likely to increase when “university” is unpacked into “law school, business school, medical school, or any other school” because of the explicit mention of high-paying positions. Thus, the conditional probability formula extends the application of subadditivity to the representation of evidence. It also allows us to compare the impact of different bodies of evidence, provided they can be described as elements of H.

Consider a collection of n ≥ 3 mutually exclusive and exhaustive (nonnull) hypotheses, A1 . . . A**n, and let Ā**i denote the negation of A**i that corresponds to an implicit disjunction of the remaining hypotheses. Consider two items of evidence, B, C ∈ H, and suppose that each A**i is more compatible with B than with C in the sense that s(BA**i) ≥ s(CA**i), 1 ≤ i ≤ n. We propose that B induces more subadditivity than C so that s(BĀ**i) is discounted more heavily than s(CĀ**i) (i.e., s(BĀ**i) ≤ s(CĀ**i); see Eq. 7). This assumption, called enhancement, suggests that the assessments of P(A**i, Ā**i |B) will be generally higher than those of P(A**i, Ā**i |C). More specifically, we propose that the sum of the probabilities of A1 . . . A**n, each evaluated by different judges, is no smaller under B than under C. That is,

$$\sum_{i=1}^n P(A_i, \bar{A}i | B) \geq \sum{i=1}^n P(A_i, \bar{A}_i | C)$$

Subadditivity implies that both sums are greater than or equal to one. The preceding inequality states that the sum is increased by evidence that is more compatible with the hypotheses under study. It is noteworthy that enhancement suggests that people are inappropriately responsive to the prior probability of the data, whereas base-rate neglect indicates that people are not sufficiently responsive to the prior probability of the hypotheses. The following schematic example illustrates an implication of enhancement and compares it with other models.

Suppose that a murder was committed by one (and only one) of several suspects. In the absence of any specific evidence, assume that all suspects are considered about equally likely to have committed the crime. Suppose further that a preliminary investigation has uncovered a body of evidence (e.g., motives and opportunities) that implicates each of the suspects to roughly the same degree. According to the Bayesian model, the probabilities of all of the suspects remain unchanged because the new evidence is nondiagnostic. In Shafer’s theory of belief functions, the judged probability that the murder was committed by one suspect rather than by another generally increases with the amount of evidence; thus, it should be higher after the investigation than before. Enhancement yields a different pattern: The binary probabilities (i.e., of one suspect against another) are expected to be approximately one half, both before and after the investigation, as in the Bayesian model. However, the probability that the murder was committed by a particular suspect (rather than by any of the others) is expected to increase with the amount of evidence.