Leonard J. Savage

Bio: Leonard J. Savage is an academic researcher from Yale University. The author has contributed to research in topics: Statistical inference & Foundations of statistics. The author has an hindex of 24, co-authored 49 publications receiving 20630 citations. Previous affiliations of Leonard J. Savage include University of Michigan & University of Chicago.

The Utility Analysis of Choices Involving Risk

Abstract: 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.

Toeplitz Forms and Their Applications.

Abstract: 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.

Bayesian statistical inference for psychological research.

Abstract: Bayesian statistics, a currently controversial viewpoint concerning statistical inference, is based on a definition of probability as a particular measure of the opinions of ideally consistent people. Statistical inference is modification of these opinions in the light of evidence, and Bayes’ theorem specifies how such modifications should be made. The tools of Bayesian statistics include the theory of specific distributions and the principle of stable estimation, which specifies when actual prior opinions may be satisfactorily approximated by a uniform distribution. A common feature of many classical significance tests is that a sharp null hypothesis is compared with a diffuse alternative hypothesis. Often evidence which, for a Bayesian statistician, strikingly supports the null hypothesis leads to rejection of that hypothesis by standard classical procedures. The likelihood principle emphasized in Bayesian statistics implies, among other things, that the rules governing when data collection stops are irrelevant to data interpretation. It is entirely appropriate to collect data until a point has been proven or disproven, or until the data collector runs out of time, money, or patience.

Elicitation of Personal Probabilities and Expectations

Abstract: Proper scoring rules, i.e., devices of a certain class for eliciting a person's 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.

Prospect theory: an analysis of decision under risk

Abstract: This paper presents a critique of expected utility theory as a descriptive model of decision making under risk, and develops an alternative model, called prospect theory. Choices among risky prospects exhibit several pervasive effects that are inconsistent with the basic tenets of utility theory. In particular, people underweight outcomes that are merely probable in comparison with outcomes that are obtained with certainty. This tendency, called the certainty effect, contributes to risk aversion in choices involving sure gains and to risk seeking in choices involving sure losses. In addition, people generally discard components that are shared by all prospects under consideration. This tendency, called the isolation effect, leads to inconsistent preferences when the same choice is presented in different forms. An alternative theory of choice is developed, in which value is assigned to gains and losses rather than to final assets and in which probabilities are replaced by decision weights. The value function is normally concave for gains, commonly convex for losses, and is generally steeper for losses than for gains. Decision weights are generally lower than the corresponding probabilities, except in the range of low prob- abilities. Overweighting of low probabilities may contribute to the attractiveness of both insurance and gambling. EXPECTED UTILITY THEORY has dominated the analysis of decision making under risk. It has been generally accepted as a normative model of rational choice (24), and widely applied as a descriptive model of economic behavior, e.g. (15, 4). Thus, it is assumed that all reasonable people would wish to obey the axioms of the theory (47, 36), and that most people actually do, most of the time. The present paper describes several classes of choice problems in which preferences systematically violate the axioms of expected utility theory. In the light of these observations we argue that utility theory, as it is commonly interpreted and applied, is not an adequate descriptive model and we propose an alternative account of choice under risk. 2. CRITIQUE

Judgment Under Uncertainty: Heuristics and Biases

Abstract: This article described three heuristics that are employed in making judgements under uncertainty: (i) representativeness, which is usually employed when people are asked to judge the probability that an object or event A belongs to class or process B; (ii) availability of instances or scenarios, which is often employed when people are asked to assess the frequency of a class or the plausibility of a particular development; and (iii) adjustment from an anchor, which is usually employed in numerical prediction when a relevant value is available. These heuristics are highly economical and usually effective, but they lead to systematic and predictable errors. A better understanding of these heuristics and of the biases to which they lead could improve judgements and decisions in situations of uncertainty.

Some methods for classification and analysis of multivariate observations

Abstract: The main purpose of this paper is to describe a process for partitioning an N-dimensional population into k sets on the basis of a sample. The process, which is called 'k-means,' appears to give partitions which are reasonably efficient in the sense of within-class variance. That is, if p is the probability mass function for the population, S = {S1, S2, * *, Sk} is a partition of EN, and ui, i = 1, 2, * , k, is the conditional mean of p over the set Si, then W2(S) = ff=ISi f z u42 dp(z) tends to be low for the partitions S generated by the method. We say 'tends to be low,' primarily because of intuitive considerations, corroborated to some extent by mathematical analysis and practical computational experience. Also, the k-means procedure is easily programmed and is computationally economical, so that it is feasible to process very large samples on a digital computer. Possible applications include methods for similarity grouping, nonlinear prediction, approximating multivariate distributions, and nonparametric tests for independence among several variables. In addition to suggesting practical classification methods, the study of k-means has proved to be theoretically interesting. The k-means concept represents a generalization of the ordinary sample mean, and one is naturally led to study the pertinent asymptotic behavior, the object being to establish some sort of law of large numbers for the k-means. This problem is sufficiently interesting, in fact, for us to devote a good portion of this paper to it. The k-means are defined in section 2.1, and the main results which have been obtained on the asymptotic behavior are given there. The rest of section 2 is devoted to the proofs of these results. Section 3 describes several specific possible applications, and reports some preliminary results from computer experiments conducted to explore the possibilities inherent in the k-means idea. The extension to general metric spaces is indicated briefly in section 4. The original point of departure for the work described here was a series of problems in optimal classification (MacQueen [9]) which represented special

Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference

Abstract: From the Publisher: Probabilistic Reasoning in Intelligent Systems is a complete andaccessible account of the theoretical foundations and computational methods that underlie plausible reasoning under uncertainty. The author provides a coherent explication of probability as a language for reasoning with partial belief and offers a unifying perspective on other AI approaches to uncertainty, such as the Dempster-Shafer formalism, truth maintenance systems, and nonmonotonic logic. The author distinguishes syntactic and semantic approaches to uncertainty—and offers techniques, based on belief networks, that provide a mechanism for making semantics-based systems operational. Specifically, network-propagation techniques serve as a mechanism for combining the theoretical coherence of probability theory with modern demands of reasoning-systems technology: modular declarative inputs, conceptually meaningful inferences, and parallel distributed computation. Application areas include diagnosis, forecasting, image interpretation, multi-sensor fusion, decision support systems, plan recognition, planning, speech recognition—in short, almost every task requiring that conclusions be drawn from uncertain clues and incomplete information. Probabilistic Reasoning in Intelligent Systems will be of special interest to scholars and researchers in AI, decision theory, statistics, logic, philosophy, cognitive psychology, and the management sciences. Professionals in the areas of knowledge-based systems, operations research, engineering, and statistics will find theoretical and computational tools of immediate practical use. The book can also be used as an excellent text for graduate-level courses in AI, operations research, or applied probability.

The Framing of Decisions and the Psychology of Choice

Abstract: The psychological principles that govern the perception of decision problems and the evaluation of probabilities and outcomes produce predictable shifts of preference when the same problem is framed in different ways. Reversals of preference are demonstrated in choices regarding monetary outcomes, both hypothetical and real, and in questions pertaining to the loss of human lives. The effects of frames on preferences are compared to the effects of perspectives on perceptual appearance. The dependence of preferences on the formulation of decision problems is a significant concern for the theory of rational choice.