Credibility Procedures Laplace's Generalization of Bayes' Rule and the Combination of Collateral Knowledge with Observed Data

The casualty insurance business has used credibility formulas or procedures for many years in making rates or in experience rating plans. These formulas have been used to detelTnine the weight to be given to the indications of actual observations in a combination of such indications with a priori expectations which were based either on other actual data, on prior knowledge or on reasonable assumptions made before actual observations were available. Such formulas have invariably provided that the weight to be given to actual observations increase as the volume of such observations increases. Last December the discussion of a paper by Mr. T. O. Carlson, entitled "Statistical and Actuarial Procedures in Liabilit~ Insurance"*[1],pointed out that casualty insurance underwriters and actuaries believe that they are not devoid of knowledge before they have acquired any statistics from observed data, and that this belief results in the use of credibility formulas to produce weighted averages of that prior knowledge and the information provided by the observed data. The remarks made at that time were general and unsupported by any demonstration. In fairness to other statisticians and to students of casualty insurance, it appears desirable to present a complete development from basic principles to show exactly the basis upon which credibility formulas rest and to make evident the point at which the classical statistical theory, particularly that of statistical estimation, departs from that used by casualty actuaries. The basis for these credibility formulas has been a profound mystery to most people who have come in contact with them. The actuary finds them difficult to explain and, in some cases, even difficult to understand. Paradoxical as it may be, the more contact a person has had with statistical practices in other fields or the more training a person has had in the theory of mathematical statistics, the more difficult it has been to understand these credibility procedures or the validity of their application. The credibility formulas for casualty insurance have been accepted in the past, and continue to be accepted at the present time, because it appears to most people to be logical and reasonable to give the indications of a large volume of data more consideration or weight than the indications of a small volume of data. How much weight the indications of specific volumes of data