2016-09-12(Chap1b).pdf - Where are the randomized rules...

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𝑹𝑹 𝜽𝜽 𝟏𝟏 , 𝒅𝒅 𝑹𝑹 𝜽𝜽 𝟐𝟐 , 𝒅𝒅 𝒅𝒅 𝟏𝟏 𝒅𝒅 𝟔𝟔 𝒅𝒅 𝟕𝟕 𝒅𝒅 𝟑𝟑 𝒅𝒅 𝟓𝟓 𝒅𝒅 𝟐𝟐 𝒅𝒅 𝟒𝟒
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Where are the randomized rules?
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Where are the randomized rules? forming the “convex hull” of points 𝒅𝒅 𝟏𝟏 , , 𝒅𝒅 𝟕𝟕
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In general, we can describe a set of rules (including both “deterministic” rules & their randomized mixtures) by a RISK SET A risk set must be convex 𝒓𝒓 𝒔𝒔 𝝀𝝀𝒓𝒓 + 𝟏𝟏 − 𝝀𝝀 𝒔𝒔
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In general, we can describe a set of rules (including both “deterministic” rules & their randomized mixtures) by a RISK SET outside set 𝒓𝒓 𝒔𝒔 𝝀𝝀𝒓𝒓 + 𝟏𝟏 − 𝝀𝝀 𝒔𝒔 𝒓𝒓 𝒔𝒔 not convex
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A risk set must be convex Proof
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𝑟𝑟 1 , 𝑟𝑟 2 𝒙𝒙 < 𝒓𝒓 𝟏𝟏 𝒚𝒚 < 𝒓𝒓 𝟐𝟐 𝒙𝒙 < 𝒓𝒓 𝟏𝟏 𝒚𝒚 < 𝒓𝒓 𝟐𝟐 𝑟𝑟 1 , 𝑟𝑟 2 admissible inadmissible strictly dominating
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