Historically, the study of artificial intelligence has emphasized symbolic rather than numerical computation. In recent years, however, the practical needs of expert systems have led to an interest in the use of numbers to encode partial confidence. There has been some effort to square the use of these numbers with Bayesian probability ideas, but in most applications not all the inputs required by Bayesian probability analyses are available. This difficulty has led to widespread interest in belief functions, which use probability in a looser way. It must be recognized, however, that even belief functions require more structure than is provided by pure production systems. The need for such structure is inherent in the nature of probability argument and cannot be evaded. Probability argument requires design as well as numerical inputs. The real challenge probability poses to artificial intelligence is to build systems that can design probability arguments. The real challenge artificial intelligence poses to statistics is to explain how statisticians design probability arguments.

Arguments are adduced to support the claim that the only satisfactory description of uncertainty is probability. Probability is described both mathematically and interpretatively as a degree of belief. The axiomatic basis and the use of scoring rules in developing coherence are discussed. A challenge is made that anything that can be done by alternative methods for handling uncertainty can be done better by probability. This is demonstrated by some examples using fuzzy logic and belief functions. The paper concludes with a forensic example illustrating the power of probability ideas.

The development of expert systems in medicine has generally been accompanied by a rejection of formal probabilistic methods for handling uncertainty. We argue that a coherent probabilistic approach can, if carefully applied, meet many of the practical demands being made, and briefly illustrate our claim with three current projects.

The environment in which W. S. Gosset (Student) worked as a brewer at Guinness' Brewery at the turn of the century is described fully enough to show how it forced him to confront problems of small sample statistics, using the techniques he picked up from Karl Pearson. R. A. Fisher's interest in human genetics prompted biometrical applications of his mathematical training even as an undergraduate. As soon as he considered Student's work, he perceived its importance and began to extend its applications. Consequently, when he started work at Rothamsted Experimental Station in 1919, he was ready to respond to the experimental problems by developing statistical theory along with appropriate methods of experimental analysis and design.

In this paper we introduce certain numbers, called collinearity indices, which are useful in detecting near collinearities in regression problems. The coefficients enter adversely into formulas concerning significance testing and the effects of errors in the regression variables. Thus they provide simple regression diagnostics, suitable for incorporation in regression packages.