During the years since 1982, research in a small number of leading statistics departments in the United States has undergone a dramatic change as these departments have acquired significant computational resources of their own. Primary support for the acquisition of this equipment has come from the instrumentation programs of the National Science Foundation and the Department of Defense. The purpose of this study is to provide a timely assessment of the current state of computing resources in statistics departments and a projection of future needs. During 1985 the Workshop conducted a mail survey of leading research-doctorate statistics departments and held extensive discussions during two two-day meetings. We have arrived at a series of recommendations directed, variously, at statistics departments, university administrations, professional organizations, and research sponsors.

Three decades ago L. J. Savage published The Foundations of Statistics, in which he argued that it is normative to make choices that maximize subjective expected utility. Savage based his argument on a set of postulates for rational behavior. Empirical research during the past three decades has shown that people often violate these postulates, but it is widely believed that this is irrelevant to Savage's argument. This article re-examines Savage's argument and concludes that his postulates cannot be so thoroughly insulated from the empirical facts. The argument actually relies heavily on assumptions that have been empirically refuted. Savage's normative interpretation of subjective expected utility must therefore be revised. The revision suggested here emphasizes the constructive nature of probability and preference. It also emphasizes the constructive nature of small worlds, the frameworks within which probability and utility judgments are made. According to the constructive understanding, an analysis of a decision problem by subjective expected utility is merely an argument, an argument that compares that decision problem to the decision problem of a gambler in a pure game of chance. This argument by analogy may or may not be cogent. In some cases other arguments are more cogent.

Ill-posed inverse problems arise in many branches of science and engineering. In the typical situation one is interested in recovering a whole function given a finite number of noisy measurements on functionals. Performance characteristics of an inversion algorithm are studied via the mean square error which is decomposed into bias and variability. Variability calculations are often straightforward, but useful bias measures are more difficult to obtain. An appropriate definition of what geophysicists call the Backus-Gilbert averaging kernel leads to a natural way of measuring bias characteristics. Moreover, the ideas give rise to some important experimental design criteria. It can be shown that the optimal inversion algorithms are methods of regularization procedures, but to completely specify these algorithms the signal to noise ratio must be supplied. Statistical approaches to the empirical determination of the signal to noise ratio are discussed; cross-validation and unbiased risk methods are reviewed; and some extensions, which seem particularly appropriate in the inverse problem context, are indicated. Linear and nonlinear examples from medicine, meteorology, and geophysics are used for illustration.