The Improvement of Probability Judgements
D. V. Lindley · Journal of the Royal Statistical Society Series A (General) · 1982
SUMMARY A subject S assesses the probability for an event A as q. In this paper, it is shown how the assessment can be improved. Improvement requires two functions fi(q) (i = 0, 1); the probability for q when A is true (i = 1) and when A is false (i = 0). It is shown that data on S's probability assessments can be used to update these functions. Specific, normal forms for them are examined. The connection with calibration is explored. 1. THE TRANSFORMATION OF PROBABILITY JUDGEMENTS CONSIDER a subject, S, who has provided his personal probability for an event A. In this paper we address the problem of whether his probability assessment can be improved, not by providing information additional to that which S already has, or even by S remembering something he had temporarily forgotten, a process Brown and Lindley (1980) have termed digging in S's psychological field; but by a better understanding of the assessment mechanism. It is shown that improvement is possible and the method of making it is described. Let the event A under consideration initially have probability y. Because Bayes' theorem in log-odds form is linear, it is convenient to suppose S's statement of probability assessment is that the log-odds for A is q. Our question is: can we transform q to a new value, q* say, which, in some sense, is a better assessment of the log-odds? Two examples may be illuminating. Example 1 S is given an almanac question, with two possible answers, say an upper and a lower one. Data set (a) below provides an example. Before seeing the question he is told that one, and only one, of the answers is correct and that they are equally likely to be correct. On seeing the question he is asked for the probability that the upper answert is correct. Here y =-. The