Forecasts, decisions and uncertain probabilities

Abstract

In the traditional decision theories the role of forecasts is to a large extent swept under the carpet. I believe that a recognition of the connections between forecasts and decisions will be of benefit both for decision theory and for the art of forecasting. In this paper I have tried to analyse which factors, apart from the utilities of the outcomes of the decision alternatives, determine the value of a decision. I have outlined two answers to the question why a decision which is made on the basis of a forecast is better than a decision which is based on a guess. Neither of these answers is universally valid. An assumption which is necessary for the first answer, i.e. Good's result, is that Bayes' rule is accepted as a correct and generally applicable decision principle. The second answer, which was given with the aid of probability intervals, departed from a more general decision principle, the maximin criterion for expected utilities, which was formulated in order to evade some of the criticism against Bayes' rule. However, the argument leading to the ansser is based on the assumption that the probability intervals associated with the states of nature represent certain knowledge. For this reason this answer is only approximatively valid. As a number of quotations in the section on “the weight of evidence” show, it is not sufficient to describe the knowledge about the states of nature by a single number, representing the (subjective) probability of the state, but something else has to be invoked which measures the amount of information on which a decision is based. Several authors have tried to characterize this mysterious quantity, which here was called the weight of evidence. However, there seems to be little agreement as to how this quantity should be measured.