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Unformatted text preview: ISyE8843A, Brani Vidakovic Handout 7 1 Estimation and Beyond in the Bayes Universe. 1.1 Estimation No Bayes estimate can be unbiased but Bayesians are not upset! No Bayes estimate with respect to the squared error loss can be unbiased, except in a trivial case when its Bayes’ risk is 0. Suppose that for a proper prior π the Bayes estimator δ π ( X ) is unbiased, ( ∀ θ ) E X | θ δ π ( X ) = θ. This implies that the Bayes risk is 0. The Bayes risk of δ π ( X ) can be calculated as repeated expectation in two ways, r ( π,δ π ) = E θ E X | θ ( θ- δ π ( X )) 2 = E X E θ | X ( θ- δ π ( X )) 2 . Thus, conveniently choosing either E θ E X | θ or E X E θ | X and using the properties of conditional expectation we have, r ( π,δ π ) = E θ E X | θ θ 2- E θ E X | θ θδ π ( X )- E X E θ | X θδ π ( X ) + E X E θ | X δ 2 π ( X ) = E θ E X | θ θ 2- E θ θ [ E X | θ δ π ( X )]- E X δ π ( X ) E θ | X θ + E X E θ | X δ 2 π ( X ) = E θ E X | θ θ 2- E θ θ · θ- E X δ π ( X ) δ π ( X ) + E X E θ | X δ 2 π ( X ) = 0 . Bayesians are not upset. To check for its unbiasedness, the Bayes estimator is averaged with respect to the model measure ( X | θ ), and one of the Bayesian commandments is: Thou shall not average with respect to sample space, unless you have Bayesian design in mind. Even frequentist agree that insisting on unbiasedness can lead to bad estimators, and that in their quest to minimize the risk by trading off between variance and bias-squared a small dosage of bias can help. The relationship between Bayes estimators and unbiasedness is discussed in Lehmann (1951), Girshick (1954), Bickel and Blackwell (1967), Noorbaloochi and Meeden (1983) and OHagan (1994). Here is an interesting Bayes estimation problem. For the solution I admittedly used Wolfram’s MATHE- MATICA software since my operational knowledge of special functions is not to write home about. Binomial n from a single observation! Two graduate students from GaTech were conducting a survey of what percentage p of Atlanteans will vote for reelection of the President in November 2004. The student who did the actual survey left the value X = 10 on the answering machine of the other student but did not say what sample size n was used, and left for China, while the project was due in a few days. What n and p should be reported? The problem can be formalized to estimation of Binomial proportion and sample size on the basis of a single measurement. This is a problem where one wishes to be a Bayesian since the frequentist solutions involve lots of hand waiving! Let X be a single observation from B ( n,p ) and let p ∼ Beta ( α,β ) and n ∼ Poi ( λ ) ....
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This note was uploaded on 10/23/2011 for the course ISYE 8843 taught by Professor Vidakovic during the Spring '11 term at Georgia Tech.
- Spring '11