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Week8Student2009

# Week8Student2009 - Week 8 Lecture 15 Model y i =& i ¡z...

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Unformatted text preview: Week 8 Lecture 15 Model : y i = & i + ¡z i , z i i:i:d: & N (0 ; 1) , & 2 & M where & M is an ellipsoid in l 2 ( N ) : & = ( & : X i a 2 i & 2 i ¡ M ) . Pinsker&s Theorem: Let & = & & : P a 2 i & 2 i ¡ M ¡ and a i ! 1 , then R N (& ;¡ ) & R L (& ;¡ ) as ¡ ! . We will only prove this result for the following Sobolev ball, & M = ( & : X i a 2 i & 2 i ¡ M , a 2 k = a 2 k +1 = (2 ¢k ) m ) : Review of Linear Minimaxity : Recall that R L (& ;¡ ) = inf c sup & X P ( c i y i ¢ & ) 2 = X i =1 ¡ 2 £ ¡ 2 ( £ & =a i ¢ 1) + ¡ 2 + ¡ 2 ( £ & =a i ¢ 1) + = ¡ 2 X i =1 (1 ¢ a i =£ & ) + where £ & is determined by the following equation ¡ 2 X a i ( £ & ¢ a i ) + = M: This suggests the least favorable prior would be & i & N ¢ ;¤ 2 i £ with ¤ 2 i = ¡ 2 ( £ & =a i ¢ 1) + for which the Bayes risk is r ( G ¡ ;¥ G & ) = X i =1 ¡ 2 ¤ 2 i = ¢ ¡ 2 + ¤ 2 i £ = ¡ 2 X i =1 (1 ¢ a i =£ & ) + = R L (& ;¡ ) . Strategy : It is natural to de&ne G ¡ = Y i N ¢ ;¡ 2 ( £ & =a i ¢ 1) + £ . But E X a 2 i & 2 i = X ¡ 2 a 2 i ( £ & =a i ¢ 1) + = ¡ 2 X a i ( £ & ¢ a i ) + = M , 1 then G & is not supported in &! We need to &nd a sequence of priors Q & supported in & such that R L (& ;& ) = (1 + o (1)) r ( Q & ;¡ Q & ) : Then R N (& ;& ) & R L (& ;& ) since r ( Q & ;¡ Q & ) ¡ R N (& ;& ) ¡ R L (& ;& ) ....
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