The two phrases presented by Lindley describe two completely different probabilities which are connected to the same pair of events (P[Death/Men] and P[Men/Death]). In the first phrase, the probability is about the rate of mortality, given the gender; in the second one, the probability is about the rate of gender, given the mortality. Whereas the second phrase implies that P(M/D) is equal to 2/3, the first does not.Footnote 1 The confusion between these two notions is a very common phenomenon, and has great implications for reasoning and decision making. According to Dawes (Reference Dawes1988, p.80), “words are poor vehicles for discussing inverse probabilities.” A main question is about the nature of this confusion and, consequently, the understanding and the use of the conditional probability.
We propose a pragmatic explanation of the phenomenon of the confusion between inverse probabilities and of the base-rate fallacy. In particular, concerning the kinds of problems considered in the literature on the base-rate fallacy, a pragmatic analysis of the texts/problems allowed us to identify, as responsible for the fallacy, the ambiguous formulation of a likelihood, instead of an intrinsic difficulty to reason in Bayesian terms (Macchi Reference Macchi1995; Reference Macchi2000; Macchi & Mosconi Reference Macchi and Mosconi1998).
Let us consider, for example, three kinds of problems formulated as follows:
“If a woman has breast cancer, the probability is 80% that she will get a positive mammography.” (Medical Diagnosis problem)
“The percentage of deaths by suicide is three times higher among single individuals than among married individuals.” (Suicide problem)
“The witness made correct identifications in 80% of the cases and erred in 20% of the cases.” (Cab problem)
This sort of formulation does not seem to express the intrinsic nature of a conditional probability, which conditions an event (A) to the occurrence of another event (B). Nor does this sort of formulation represent the cases in which, given the occurrence of B, A also occurs. In other words, it transmits just a generic association of events: A & B. For statistically naive subjects, this kind of formulation is not informative even if all the data were literally spelled out. The distinction between sentence and utterance is at the core of Grice's communication theory, according to which phrases imply and mean more than what they literally say (Grice Reference Grice, Cole and Morgan1975). What is implied is the outcome of an inferential process, in which what is said is interpreted in the light of the intentions attributed to the speaker and of the context (unavoidably elicited and determined by any communications).
Common language is ambiguous in itself. The understanding of what a speaker means requires a disambiguation process, centered on the attribution of intention. Differently, specialized languages (e.g., logical and statistical ones) presuppose an univocal, unambiguous interpretation (the utterance corresponds to the sentence).
The formulation of these kinds of problems uses common language. Data are all available, but subjects are able to understand them in the specific meanings proper to a specialized language, only if they are adequately transmitted. Then, the particular interpretation of the data, required for a correct solution, needs a “particularized,” marked formulation (see Grice Reference Grice, Cole and Morgan1975; Levinson Reference Levinson and Goody1995; Reference Stanovich and West2000).
In the sentences from Lindley (Reference Lindley1985) quoted earlier, the effective transmission of the independence of the hit rate from the base rate does not seem guaranteed. This is a crucial assumption for proper Bayesian analysis (Birnbaum Reference Birnbaum1983), because the a posteriori probability P(H/D) is calculated on the basis of the base rate and is therefore dependent upon it. If the hit rate depended on the base rate, it would already include it and, if this were the case, we would already have the a posteriori probability and it would be unnecessary to consider the base rate itself. This is what often underlies the base-rate fallacy, which consists of a failure to consider the base rate on account of the privileging of hit-rate information. In our view, the confusion sometimes generated between the hit rate and a posteriori probability is due to an unmarked formulation of conditional probability, or, in other words, to the absence of a partitive formulation.
If this is true, the partitive formulation should not be considered a facilitation, as Barbey & Sloman (B&S) argue, but a way of transmitting a particular information, able to translate a specialized language into a common, natural language, differently from the “step by step” question form used to compute the Bayesian ratio (adopted by Girotto & Gonzalez Reference Girotto and Gonzalez2001). An example of partitive formulation of likelihood in the diagnoses problem is: “80 per cent of women who have breast cancer will get a positive mammography” (Macchi Reference Macchi, Hardman and Macchi2003).
The low performance with the Medical Diagnosis (MD) problem has usually been considered as evidence of the activation of System 1, which operates associatively (fast, automatic, effortless). Vice versa, the high performance with this kind of problem is ascribed to System 2, which is able to process rule-based inferences. However, in a recent study (Macchi et al. Reference Macchi, Bagassi and Ciociola2007), statistically sophisticated subjects who solved the MD problem (66% of 35 subjects) were not able to solve the computationally less complex Linda problem (only 14% of the subjects did not commit the conjunction fallacy). So, the conceptual distinction between two reasoning systems, which explains the biases recurring in System 1 and the normative performance as the result of the activation of System 2, gives rise to some doubts.
According to us, the ability of statistically sophisticated subjects to grasp the informativeness of the data and the aim of the task in Bayesian tasks is a pragmatic ability. Also, when subjects don't give the logical-normative solution to the Linda problem, they are again considering the informativeness of the data, which, in this instance, hinders the intent of the experimenter (concerning the inclusion-class rule), because of a misleading contextualization of the task.
We could further speculate that, instead of having an ability for decontextualizing the task (Stanovich & West 2000), those gifted subjects who give the normative solution to the Linda problem (14%) would have a high ability to understand which context the experimenter intended, thereby revealing an interactional intelligence.
