eco331-02-PenaltySchemes

# eco331-02-PenaltySchemes - Northwestern University Fall...

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Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 THE BAYESIAN APPROACH TO PROBABILITY 1. Introduction Last week we focused on the formal properties of probability. We now turn to its interpretation. By and large, there are three main “schools of thought”: Classical: Probability describes the “physical” chance that an experiment (a coin toss, a die roll) produces some outcome. For instance, the probability that a die roll results in an odd number equals the number of “favorable” cases (3) divided by the number of “possible” cases (6). “Cases” are what we called “states” last time, and they are thought of as being “equally likely” in some physically obvious sense (which is why this is not much of a definition, of course!) Frequentist: Probability corresponds to the long-run frequency of an event in an experiment or phenomenon that can be repeated (or occur) many, many times. If we roll a die many, many times, and record the outcome of each roll, then approximately half of the times we get an odd number. This suggests that the probability of getting an odd number is 1 2 . As long as it makes sense to repeat the exact same experiment however many times we wish (as least conceptually), this definition makes a lot more sense than the classical one. Subjectivist: Probability is a numerical representation of the individual’s subjective beliefs about the relative likelihood of events, as revealed by the individual’s behavior . This the main topic of today’s class. It is important to realize that economists just have to rely on the subjectivist interpretation, because virtually all economic phenomena of interest concern specific, non-repeatable situations. Furthermore, the subjectivist viewpoint almost automatically suggests a way to carry out sta- tistical inference that, in many ways, may strike you as being more “natural” than the frequentist approach. Both the interpretation of probability and what we can do with it are derived within a cohesive framework that emphasizes decisions . We will discuss decision theory in some detail towards the end of this course. Strictly speaking, only at that point will we be able to gain a full understanding of the subjective view of probability. However, it’s important to at least get acquainted with these ideas soon, because they have had a tremendous influence on the way economists think about uncertainty. 2. The Subjectivist Interpretation of Probability Suppose we toss a coin and record the outcome: it is natural to define the set of states as S = { H, T } , in obvious notation. Recall our definition of probability for finite state spaces: Definition 1. A probability over a finite set S of states is a function P : P ( S ) R that associates, to each event E S , a real number, with the following restrictions: (1) P ( E ) 0 ; (2) P ( ) = 0 ; P ( S ) = 1 .

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