Subjective Statistical Inference Based on Pure Associations

The “Fourth axiom of probability theory” quickly yields Bayes Theorem – and hence how to deal with inverse probabilities (e.g., P(h|e) versus P(e|h)) – and shows in odds form the importance of likelihood ratios as opposed to simple likelihoods. This paper discusses a problem more basic than ignoring Bayes Theorem. The problem is making no comparisons at all! Just associating. Thus, if P(A|B) or P(B|A) is high (low) people tend to think that A and B “go together” (or don’t). A distressing number of examples ranging from Nazi ideology to unjustified inferences in the mental health field will be presented, as well as some experimental research. In fact, A and B are often thought to “go together” – simply on the basis that P(A) and P(B) are both high (or low). Criminal behaviors are unusual, minority group membership is unusual, so would you believe? This “Von Rostoff effect” was originally found in paired associates learning of words or nonsense syllables. If a few stimuli were long rather than short and so were a few to be learned responses, people tended to believe that they were paired, even though statistically independent. Following the conception of Fischhoff and Beyth-Marom (1983) as elaborated by Dawes (1998) the standard cognitive biases and heuristics in the irrational assessment of probability can be understood by considering simple forms of Bayes Theorem. Consider, for example, the relationship between a symptom S and a disease D, and suppose a diagnostician observes this symptom S. If the probability of the disease is assessed on the basis of a pure matching or association between the symptom and the disease, independent of considerations of conditional probability, there is no normative structure to which the judgment corresponds. More often, however, the judgment will be made on the basis of the conditional probabilities—a normatively correct judgment if the conditional is the probability of the disease given the symptom, P(D|S), but a representative judgment if it is the probability of the symptom given the disease, P(S|D). Unfortunately, there is a lot of evidence that the judgment is made on the basis of the latter relationship when conditional probabilities are considered at all. The relationship between these two probabilities is given by rewriting Bayes theorems as: P(D | S) = P(S | D)P(D) P(S) (1) Bayes theorem can also be written in terms of the “ration rule: