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eco331-03-BayesianInference - Northwestern University Fall...

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Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 BAYESIAN INFERENCE AND THE FREQUENTIST VIEW 1. Introduction This lecture has two main objectives. First, it will briefly introduce you to Bayes’ Rule, which plays an important role in the subjectivist approach to statistical inference, as well as in economics. Second, it will illustrate that, in a precise sense, the subjectivist approach generalizes the frequentist view that probability equals long-run frequency; while you might not care so much about this fact per se, it is handy to know that it is still OK, in a precise sense, to say that “the probability of Heads in a coin toss is 1 2 ”, even if we subscribe to the subjectivist view. 2. Bayes’ Rule and the Frequency View Bayes’ Rule is, loosely speaking, a (rather obvious) way of flipping the events E and F in the conditional probability P ( E | F ). Formally, suppose that P ( E ) > 0 and P ( F ) (0 , 1): then (1) P ( F | E ) = P ( E | F ) P ( F ) P ( E | F ) P ( F ) + P ( E | F c ) P ( F c ) . To prove Eq. (1), and to understand why it’s useful, note that, by definition, P ( F | E ) = P ( F E ) P ( E ) (so we need P ( E ) > 0 for this to make sense). Bayes’ Rule is useful precisely when P ( F E ) and P ( E ) are not known. We know that, by definition, P ( E | F ) = P ( E F ) P ( F ) (so we need P ( F ) > 0); hence, P ( F E ) = P ( E F ) = P ( E | F ) P ( F ), which explains what’s in the numerator of Eq. (1). For the denominator, note that E = ( E F ) ( E F c ), so P ( E ) = P ( E F ) + P ( E F c ). We already know that P ( E F ) = P ( E | F ) P ( F ): similarly, P ( E F c ) = P ( E | F c ) P ( F c ), which explains why we also need P ( F c ) > 0, or P ( F ) < 1. Bayes’ Rule is used in virtually all interesting economic applications where agents make decisions at different points in time, so we will have ample time to appreciate its significance. Moreover, it suggests a rather natural approach to statistical inference—one that is intimately connected with the subjective view of probability. 1 We discuss this briefly in the next section. Perhaps most importantly, this approach provides a way to reconcile the frequentist and Bayesian views of probability . This can be shown formally, but it requires rather formidable mathematics. Thus, I’ll try to provide you with an intuition of what’s going on, by relying on an analogy. Consider an urn that contains N black balls and 100 - N white balls, with 0 N 100. The urn is sampled with replacement: that is, a ball is drawn, its color is observed, then the ball is replaced in the urn. Since we can draw any number of balls from the urn (with replacement), this is precisely the kind of situation where the frequentist approach makes sense: the probability of drawing a black ball is the long-run frequency of black draws, as we keep sampling from the urn. 1 It is possible to extend our definition of forecast systems and admissibility to derive the updating formula, and hence Bayes’ Rule, as a consequence of the assumption that people avoid inadmissible forecasts.
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