Abstract
Bayesian decision theory was invented by Leonard Savage, who is on record as saying that it would be “preposterous” and “utterly ridiculous” to apply his theory except in a small world. But modern Bayesians proceed as though Savage’s theory is always the rational way to make choices in all circumstances. What is a small world? Why did Savage restrict his theory to small worlds? Where did Savage think priors come from? What are the implications for behavioral applications? How could Savage’s theory be generalized to apply to at least some large worlds? This paper offers some partial answers to such questions.
Highlights
Bayesian decision theory was invented by Leonard Savage, who is on record as saying that it would be ‘‘preposterous’’ and ‘‘utterly ridiculous’’ to apply his theory except in a small world
What is a small world? Why did Savage restrict his theory to small worlds? Where did Savage think priors come from? What are the implications for behavioral applications? How could Savage’s theory be generalized to apply to at least some large worlds? This paper offers some partial answers to such questions
This paper argues that Savage’s (1954) chose to restrict his theory of rational choice to what he called a small world because he was anxious to avoid this kind of mistake, but that modern Bayesianism has fallen headlong into the errors he sought to evade by proceeding as though his descriptive theory of subjective probability can be reinterpreted as a prescriptive theory of logical probability without any need for further foundational discussion—the same error made by Immanuel Kant when he used Euclidean geometry as his leading example of a synthetic a priori
Summary
This section reviews the theory of probability from the perspective of the preceding section. If Alice’s betting behavior satisfies certain consistency requirements, the theory of subjective probability argues that she will act as though she believes that each relevant state of the world has a probability. These probabilities are said to be subjective, because Alice’s beliefs may be based on little or no objective data, and so there is no particular reason why another person’s subjective probabilities should resemble hers. One may think that she is really playing for the thrill that some people get from gambling, but if Alice’s betting behavior satisfies Savage’s axioms, her subjective probabilities are no less respectable than those of people who consistently play to minimize their objective expected loss. I am skeptical that a viable theory of rational degrees of belief is possible at all when restricted to such a limited tool as a probability measure
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