Lecture+6+Bayesian+Statistics+III

Lecture+6+Bayesian+Statistics+III - ECON 123A, Fall 2011,...

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ECON 123A, Fall 2011, Lecture 6 Dale J. Poirier 6-1 (5.1) 5 Interval Estimation From the Bayesian standpoint, given a region ÷ d 1 and the data y, it is meaningful to ask what is the probability that 2 lies in ÷ ? The answer is direct: where " (0 < " < 1) is defined implicitly in (5.1). is known as a 1 - " Bayesian credible region . C There is no need to introduce the additional frequentist concept of “confidence.”
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ECON 123A, Fall 2011, Lecture 6 Dale J. Poirier 6-2 This chapter focuses on the case in which the posterior probability content is first set at some preassigned " (say, " = .10, .05, or .01), and then the “smallest” credible region which attains posterior probability content of 1 - " is sought. This leads to the following definition. Definition 5.1: Let p( 2 * y) be a posterior density function. Let 1 * d 1 satisfy (a) Pr( 2 0 1 * * y) = 1 - " , (b) For all 2 1 0 1 * and 2 2 ó 1 * , p( 2 1 * y) $ p( 2 2 * y). Then 1 * is defined to be a highest posterior density (HPD) region of content (1 - " ) for 2 .
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ECON 123A, Fall 2011, Lecture 6 Dale J. Poirier 6-3 Given a probability content of 1- " , the HPD region 1 * has the smallest possible volume in the parameter space 1 of any 1 - " Bayesian credible region. C If p( 2 * y) is not uniform over 1 , then the HPD region of content 1- " is unique. C Hereafter, we focus on the case in which there is a single parameter of interest and all other parameters have been integrated-out of the posterior, i.e, interval estimation.
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ECON 123A, Fall 2011, Lecture 6 Dale J. Poirier 6-4 Constructing a HPD interval is conceptually straightforward. Part (b) of Definition 5.1 implies that if [a, b] is a HPD interval, then p(a * y) = p(b * y). C This suggests a graphical approach in which a horizontal line is gradually moved vertically downward across the posterior density, and where it intersects the posterior density, the corresponding abscissa values are noted, and the posterior density is integrated between these points. C Once the desired posterior probability is reached the process stops. C If the posterior density is symmetric and unimodal with mode 2 m , then the resulting 1- " HPD interval is of the form [ 2 m - * , 2 m + * ], for suitable * , and it cuts off equal probability " /2 in each tail.
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ECON 123A, Fall 2011, Lecture 6 Dale J. Poirier 6-5 C In (5.1) we started with the region and then found its posterior probability content. C In Definition 5.1, we started with the posterior probability content and then found the smallest region with that content.
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ECON 123A, Fall 2011, Lecture 6 Dale J. Poirier 6-6 Exercise 5.1 (Graphical determination of an HPD interval): C Consider the posterior density in Figure 5.1. Use the graphical described earlier to obtain three sets of HPD intervals.
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ECON 123A, Fall 2011, Lecture 6 Dale J. Poirier 6-7
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ECON 123A, Fall 2011, Lecture 6 Dale J. Poirier 6-8 Exercise 5.3 (HPD intervals under normal sampling with known variances): Consider a random sample y 1 , y 2 , . .., y T from a N( 2 1 , 2 2 -1 ) distribution with known variance 2 2 -1 , and a N( : , 2 2 -1 / ) prior for the unknown mean 2 1 .
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This note was uploaded on 12/13/2011 for the course ECON 123a taught by Professor Staff during the Fall '08 term at UC Irvine.

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Lecture+6+Bayesian+Statistics+III - ECON 123A, Fall 2011,...

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