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bayes estimation - 2.6 Bayes estimation The denition of...

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� � March 12, 2003 2.6 Bayes estimation . The definition of Bayes estimator is a special case of the general definition of Bayes decision rule given in Sec. 1.3. Given a family { P θ , θ Θ } of laws, where , T ) is a measurable space, a loss function L ( θ, y ), the risk for an estimator U at θ defined by r ( θ, U ) := E θ L ( θ, U ), and a prior π defined on , T ), an estimator T is Bayes for π iff the Bayes risk r ( π, U ) := r ( θ, U ) ( θ ) has its minimum for all statistics U when U = T . Recall that by Theorem 1.3.8, if a decision problem for a measurable family and a given prior has a decision rule with finite risk and some decision rule a ( · ) minimizes the posterior risk for almost all x , then it is Bayes. Recall also that if a family { P θ , θ Θ } is dominated by a σ -finite measure v , we can choose v equivalent to the family by Lemma 2.1.6. For squared-error loss, Bayes estimates are just expectations for the posterior: 2.6.1 Theorem . Let { P θ , θ Θ } be a measurable family equivalent to a σ -finite measure v . Let π be a prior on Θ and g a measurable function from Θ into some R d . Then for squared-error loss, there exists a Bayes estimator for g ( θ ) if and only if there exists an estimator U for g ( θ ) with finite risk, r ( π, U ) = | U ( x ) g ( θ ) 2 dP θ ( x ) ( θ ) < . | Then a Bayes estimator is given by T ( x ) := g ( θ ) x ( θ ) where the integral with respect to the posterior π x exists and is finite for v -almost all x . T is the unique Bayes estimator up to equality v -almost everywhere. Thus T is an admissible estimator of g . Proof. Since | · 2 is the sum of squares of coordinates, we can assume d = 1. By | Propositions 1.3.5 and 1.3.13, the posterior distributions π x have the properties of regular conditional probabilities of θ given x as defined in RAP, Section 10.2. “Only if” holds since by definition, a Bayes estimator has finite risk. To prove “if,” let U have finite risk, r ( π, U ) < . Let dQ ( θ, x ) := dP θ ( x ) ( θ ) be the usual joint distribution of θ and x . Then the function ( θ, x ) U ( x ) g ( θ ) is in L 2 ( Q ), even though possibly neither x U ( x ) nor θ g ( θ ) is. Thus U ( x ) g ( θ ) ∈ L 1 ( Q ), and we have the conditional expectation (by RAP, Theorem 10.2.5) E ( U ( x ) g ( θ ) x ) = U ( x ) g ( θ ) x ( θ ) = U ( x ) g ( θ ) x ( θ ) | for v -almost all x , since U ( x ) doesn’t depend on θ . Thus T ( x ) is well-defined for v -almost all x . Now x U ( x ) T ( x ) is the orthogonal projection in L 2 ( Q ) of U ( x ) g ( θ ) into the space H of square-integrable functions of x for Q (RAP, Theorem 10.2.9), which is unique up to a.s. equality (RAP, Theorem 5.3.8). Thus ( U ( x ) g ( θ ) f ( x )) 2 dQ ( θ, x ) is minimized over all square-integrable functions f of x when and only when f ( x ) = U ( x ) T ( x ) for v -almost all x . For any other estimator V ( x ) of g ( θ ) with finite risk, U V H . Thus ( V ( x ) g ( θ )) 2 dQ ( θ, x ) is minimized among all estimators V ( x ) of g ( θ ) when V = T , in other words, T is a Bayes estimator of g ( θ ), unique up to v -almost everywhere equality.
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