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Savage, Leonard (1917–1971)

SAVAGE, LEONARD
(19171971)

Leonard James Savage was the most influential Bayesian statistician of the second half of the twentieth century. Born November 20, 1917, in Detroit, Michigan, Savage received his PhD in mathematics at the University of Michigan in 1941. He then spent a year serving as John von Neumann's assistant at the Institute for Advanced Study in Princeton, where he was exposed to von Neumann's ideas on game theory and the mathematical modeling of human behavior, topics that became a central focus of Savage's research. In his next position at Columbia University's wartime Statistical Research Groupwhose members included such luminaries as Abraham Wald, Milton Friedman, Harold Hotelling, Fredrick Mostler, and Abraham GirshickSavage developed an interest in statistics and became convinced that the subject should be grounded on a "personalist" conception of probability. After Columbia, Savage went on to hold academic positions at Chicago, Michigan, and Yale.

Savage's research focused on the mathematical analysis of rational belief and desire, and the advancement of Bayesianism in statistics. His masterpiece, The Foundations of Statistics (1954), pursued both these projects by first developing what has come to be the canonical version of subjective expected utility theory, and then attempting to recast all of statistical methodology along subjectivist Bayesian lines.

Savage's Contributions to Decision Theory

Savage's most notable contributions to the study of rational behavior were his construction of a general framework for modeling decisions under uncertainty, his systematic defense of subjective expected utility maximization as the hallmark of rational choice, and his innovative account of the role of "personal" probabilities in decision making.

Savage portrays decision making as being a matter of using beliefs about possible states of the world to choose actions that provide the optimal means of producing desirable consequences. Actions are identified with functions from states to consequences, and the agent is assumed to have a preference ranking over all acts at her disposal. Influenced by the behaviorism that dominated the social sciences of his day, Savage interpreted preferences operationally, so that an agent may be said to prefer one act f to another g if and only if she would be disposed to freely choose f over g. Overt choices thus function as "observables" in decision theory, and talk about the underlying beliefs and desires that cause them is rendered scientifically respectable by showing how they can be operationally defined in terms of preferences. (Savage's behaviorism remains controversial, but some commentators, e.g., Joyce (1999), regard it as inessential to his overall account of rationality.)

Following Frank Ramsey (1931) and Bruno de Finetti (1937), Savage invoked the hypothesis of subjective expected utility maximization to forge a link between empirically measurable preferences and hidden beliefs and desires. Given a probability function P defined over states of the world, and a utility function u defined over consequences, the expected utility of an act f is the probability-weighted average of the utilities of f 's consequences. When there are finitely many states, s 1, s 2, , s n, this expected utility is defined as ExpP,u (f ) = P (s 1)u (f (s 1)) + P (s 2)u (f (s 2)) + + P (sn )u (f (sn )). Savage maintained that an agent's preferences can only be deemed rational to the extent that they can be represented as ranking acts according to increasing subjective expected utility.

To establish this conclusion, Savage proposed that any rational preference ranking should satisfy a specific system of axiomatic constraints. The central axiom is the sure-thing principle, which states, roughly, that for any acts f and g, and any event E, if f is preferred to g both conditional on E and conditional on not-E then f is preferred to g outright. Savage went on to prove that any preference ranking satisfying his axioms implicitly defines a unique subjective probability P, which represents the agent's degrees of confidence in various states, and a utility u, which gauges the strength of her desires for consequences. The agent prefers f to g just in case ExpP,u (f ) > ExpP,u (g ). In this way, the hypothesis of expected utility maximization allows us to extract degrees of belief and desire from rational preferences.

Many objections to Savage's theory misinterpret it as a descriptive account, but it was clearly meant to be prescriptive. The most serious doubts about the theory's normative import concern the status of the sure-thing principle, which some critics see as improperly prohibiting certain sorts of rational aversions to risk or uncertainty. Savage always regarded such worries as misguided, and steadfastly defended the principle's normative credentials. Many people agree with him, as evidenced by the fact that Savage's theory, or its close variants, remain central to treatments of rational decision making across the social sciences.

Savage's Contributions to Statistics

Savage maintained that the subjective or "personal" probabilities that figure into decision making should serve also as the basis for statistical reasoning. He implacably opposed the frequentist paradigm that had come to dominate statistics during the 1930s and 1940s. In Foundations Savage had tried to incorporate the methods of frequentist statisticians, like Ronald A. Fisher and Jerzy Neyman, into his personalist framework, but by the end of his career he had entirely "lost faith in the devices of the frequentist schools" (Savage 1954). In the second edition of Foundations (1972), written six months before his death, he rejects as "ill-founded" such frequentist devices as minimax rules, confidence intervals, tolerance intervals, significance tests, and fiducial probabilities. To take their place he advocated a thoroughgoing Bayesianism in which all question of statistical reasoning boil down to the choice of a prior personal probability and the use of Bayes's rule to alter personal probabilities in light of evidence.

Savage made many contributions to the development of Bayesian statistics, of which the most significant are these: He proved a "washing-out" theorem that shows how, under fairly unrestrictive conditions, Bayesian agents with diverse prior probabilities will eventually converge to the same posterior given a sufficiently long run of shared observations. In a highly influential paper, written with Ward Edwards and Harold Lindeman (1963), he established the principle of stable estimation, which specifics conditions under which the value of a posterior probability will be independent of its prior. In one of his last papers, he developed an elegant general method for eliciting personal probabilities using proper scoring rules (1971). Savage died November 1, 1971, in New Haven, Connecticut, after having made lasting and seminal contributions to statistics, decision theory, psychology, and economics.

See also Bayes, Bayes' Theorem, Bayesian Approach to Philosophy of Science; Decision Theory; Statistics, Foundations of.

Bibliography

de Finetti, Bruno. "La prévision: Ses lois logiques, ses sources subjectives." Annals de l'Institut Henri Poincaré 7 (1937): 168. Translated as "Foresight: Its Logical Laws, Its Subjective Sources." In Studies in Subjective Probability, eds. Henry Kyburg and Howard Smokler. New York: John Wiley, 1964.

Edwards, Ward, Harold Lindeman, and Leonard J. Savage. "Bayesian Statistical Inference for Psychological Research." Psychological Review 70 (1963): 193242.

Lindley, Dennis. "L. J. SavageHis Work in Probability and Statistics." The Annals of Statistics 8 (1980): 124.

Joyce, James M. The Foundations of Causal Decision Theory. Cambridge, U.K.: Cambridge University Press. 1999.

Ramsey, Frank P. "Truth and Probability." In The Foundations of Mathematics and Other Logical Essays, edited by Richard Braithwaite. London: Kegan Paul, 1931.

works by savage

The Foundations of Statistics. New York: Wiley, 1954. 2nd revised edition, New York: Dover, 1972.

"The Elicitation of Personal Probabilities and Expectations." Journal of the American Statistical Association 66 (1971): 783801.

James M. Joyce (2005)

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