L. J. Savage

The Utility Analysis of Choices Involving Risk

T vHE purpose of this paper is to suggest that an important class of reactions of individuals to risk can be rationalized by a rather simple extension of orthodox utility analysis. Individuals frequently must, or can, choose among alternatives that differ, among other things, in the degree of risk to which the individual will be subject. The clearest examples are provided by insurance and gambling. An individual who buys fire insurance on a house he owns is accepting the certain loss of a small sum (the insurance premium) in preference to the combination of a small chance of a much larger loss (the value of the house) and a large chance of no loss. That is, he is choosing certainty in preference to uncertainty. An individual who buys a lottery ticket is subjecting himself to a large chance of losing a small amount (the price of the lottery ticket) plus a small chance of winning a large amount (a prize) in preference to avoiding both risks. He is choosing uncertainty in preference to certainty.

Elicitation of Personal Probabilities and Expectations

Abstract Proper scoring rules, i.e., devices of a certain class for eliciting a persons probabilities and other expectations, are studied, mainly theoretically but with some speculations about application. The relation of proper scoring rules to other economic devices and to the foundations of the personalistic theory of probability is brought out. The implications of various restrictions, especially symmetry restrictions, on scoring rules is explored, usually with a minimum of regularity hypothesis.

Personal probabilities of probabilities

By definition, the subjective probability distribution of a random event is revealed by the (‘rational’) subjects choice between bets — a view expressed by F. Ramsey, B. De Finetti, L. J. Savage and traceable to E. Borel and, it can be argued, to T. Bayes. Since hypotheses are not observable events, no bet can be made, and paid off, on a hypothesis. The subjective probability distribution of hypotheses (or of a parameter, as in the current ‘Bayesian’ statistical literature) is therefore a figure of speech, an ‘as if’, justifiable in the limit. Given a long sequence of previous observations, the subjective posterior probabilities of events still to be observed are derived by using a mathematical expression that would approximate the subjective probability distribution of hypotheses, if these could be bet on. This position was taken by most, but not all, respondents to a ‘Round Robin’ initiated by J. Marschak after M. H. De-Groots talk on Stopping Rules presented at the UCLA Interdisciplinary Colloquium on Mathematics in Behavioral Sciences. Other participants: K. Borch, H. Chernoif, R. Dorfman, W. Edwards, T. S. Ferguson, G. Graves, K. Miyasawa, P. Randolph, L. J. Savage, R. Schlaifer, R. L. Winkler. Attention is also drawn to K. Borchs article in this issue.

Comments on a Weakened Principle of Conditionality

Abstract Continuity suggests that the conditioning variable not be required to depend only on the values of the minimal sufficient statistic.

Reading Suggestions for the Foundations of Statistics

These notes are thrown together to help you and stimulate you to find reading in connection with this course. The available literature is large and enormously heterogeneous in point of view, style, and subject matter. The course will be greatly amplified if you do a considerable amount of reading according to your own taste, background, and interest. You are each to make at least two brief, but thoroughly prepared, reading reports in class. But the reading you do for yourself should go considerably beyond that. Here I briefly describe the principal kinds of literature bearing on the foundations of statistics, make a few comments on relevant bibliographies, and give a short annotated bibliography. For your convenience, there is a growing and changing shelf of reserved books for this course in the library. A few items that are rare or fragile will be on three-hour reserve, the others on three-day reserve. Typically, you will in fact be able to keep a book out for much more than three days but must surrender it promptly to someone else who wants to look at it.

How to Gamble If You Must (Inequalities for Stochastic Processes)

PDF : How To Gamble If You Must: Inequalities For Stochastic Processes (Dover Books On Mathematics) By Lester E. Dubins;Leonard J. Savage Doc : How To Gamble If You Must: Inequalities For Stochastic Processes (Dover Books On Mathematics) By Lester E. Dubins;Leonard J. Savage ePub : How To Gamble If You Must: Inequalities For Stochastic Processes (Dover Books On Mathematics) By Lester E. Dubins;Leonard J. Savage

Toeplitz Forms and Their Applications.

Part I: Toeplitz Forms: Preliminaries Orthogonal polynomials. Algebraic properties Orthogonal polynomials. Limit properties The trigonometric moment problem Eigenvalues of Toeplitz forms Generalizations and analogs of Toeplitz forms Further generalizations Certain matrices and integral equations of the Toeplitz type Part II: Applications of Toeplitz Forms: Applications to analytic functions Applications to probability theory Applications to statistics Appendix: Notes and references Bibliography Index.