ch15problem - 15.4 Concluding Remarks 597 say that G is a...

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15.4 Concluding Remarks 597 say that G is a family of conjugate priors to H. Thus, the beta distribution is a conjugate prior to the binomial, and the normal is self-conjugate. Conjugate priors may not exist; when they do, selecting a member of the conjugate family as a prior is done mostly for mathematical convenience, since the posterior can be evaluated very simply. More generally, numerical methods of integration would have to be used to evaluate the posterior. ~ ncluding Remarks This chapter has presented a brief introduction to two areas of mathematical statis- tics, decision theory and Bayesian inference. The analysis has been somewhat more abstract and theoretical than that of earlier chapters. In order for decision theory to be relevant to a practical problem, the problem must be such that a choice is to be made from a set of specified actions and a definite measure of loss is associated with each possible action. Critics of decision theory claim that most scientific investigations do not fit this paradigm; some have gone so far as to call such a point of view totalitarian! In fact, experimental scientists very rarely use decision theoretic methods in analyzing their data, but decision theory has been used more often in business and economics, where it seems to be easier to specify the relevant loss functions. The subjectivist, or Bayesian, point of view toward statistics is somewhat contro- versial. Arguments between frequentists and Bayesians have often been quite heated, and it is probably fair to say that the Bayesian approach has not been adopted by most practicing statisticians. Critics of the Bayesian paradigm look with disfavor on the introduction of a prior distribution. Some dislike the subjective element thereby introduced, maintaining that statistics should be "objective." Others, although not to- tally unsympathetic, question how prior distributions can be determined in practice. This book has been relatively unconcerned with philosophical or foundational matters and has made no attempt to develop a consistent "theory" of statistics. The subject matter under investigation and the role that statistics plays in the investigation should effectively determine whether it is more appropriate for the user of statistics to take a Bayesian, decision-theoretic, frequentist, or purely data-analytic point of view, or some combination of these. ) )blems The losses of actions ai' a 2 , and a 3 , depending on t1 1 , t1 2 , and t1 3 , are given in the following table: 1
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59B Chapter 15: Decision Theory and Bayesian Inference 2 1 5 o 4 6 4 o 2 An observation X has the following distributions depending on e: - .4 .5 .3 .6 '.5 .7 a Pick three decision rules and evaluate their risks. b
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This note was uploaded on 03/11/2011 for the course STA 506 taught by Professor Lisa during the Spring '11 term at West Chester.

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ch15problem - 15.4 Concluding Remarks 597 say that G is a...

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