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# Bounded normal mean minimax estimation

## by Christian Robert, Mathematician at University of Warwick on Nov 22, 2011

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A project for the EuroBayes master presented by Jacopo Primavera

A project for the EuroBayes master presented by Jacopo Primavera

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## Bounded normal mean minimax estimationPresentation Transcript

• SECTION 1 SECTION 2 Estimating a Bounded Normal Mean Jacopo Primavera TSI-EuroBayes Student University Paris Dauphine21 November 2011 / Reading Seminar on Classics Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 SECTION 2 "ALL MODELS ARE WRONG, BUT SOME ARE USEFUL"G. E. P. Box Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 SECTION 2PRESENTING THE PROBLEM One observation x ∼ N(θ, 1) θ ∈ [−m, m] ⊂ R Squared loss (θ − δ(x))2 R(θ, δ) = MSE(δ) = BIAS(δ) + VAR(δ) MINIMAX ESTIMATOR δMM = argmin[supR(θ, δ)] δ θ Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 SECTION 2SUMMARY Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 SECTION 2SUMMARY Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 SECTION 2Outline 1 SECTION 1 THE CANDIDATES 2-POINTS PRIOR 2 SECTION 2 AND WHEN m GETS LARGE? CLOSING THE LOOP Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIOROutline 1 SECTION 1 THE CANDIDATES 2-POINTS PRIOR 2 SECTION 2 AND WHEN m GETS LARGE? CLOSING THE LOOP Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSAMPLE MEAN δSM = x Main characteristic DOES NOT INVOLVE PRIOR INFORMATION Properties Θ BOUNDED ⇒ NOT Minimax Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORMLE δMLE (x)  −m for x ≤ −m  x for x ∈ (−m, m)  m for x ≥ m  Main characteristic A SELECTOR ESTIMATOR Properties Θ BOUNDED ⇒ δMLE dominates δSM Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORDECISION THEORY FORMALIZING THE CHOICE Frequentist approach Decision-oriented approach RESTRICT ∆ (i) K OPTIMAL CRITERIA U = SET UNBIASED δ (ii) CHOOSE δ MINIMIZING R(θ, δ) CHOOSE UMVE W.R.T K Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORBAYES ESTIMATOR (i) PROBABILITY MEASURE τ ON Θ (ii) CRITERIA = Eτ (iii) Eτ [R(θ, δ)] = r (τ, δ) (iv) min[r (τ, δ)] = r (τ, δB ) = r (τ ) δ (v) δB BAYES RULE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORBAYES ESTIMATOR (i) PROBABILITY MEASURE τ ON Θ (ii) CRITERIA = Eτ (iii) Eτ [R(θ, δ)] = r (τ, δ) (iv) min[r (τ, δ)] = r (τ, δB ) = r (τ ) δ (v) δB BAYES RULE Bayes method and Decision theory NATURAL ORDERING CRITERIA THE VERSATILITY OF τ MAKES BAYESIAN METHOD COHERENT WITH DECISION THEORY WHICH PRIOR INDUCES MINIMAXITY ? Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORGAME THEORY Two-person zero-sum game Θ Set of all possible strategies player 1 A Set of all possible strategies player 2 L Gain function (pl. 1) and loss function (pl.2) Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORGAME THEORY Two-person zero-sum game Θ Set of all possible strategies player 1 A Set of all possible strategies player 2 L Gain function (pl. 1) and loss function (pl.2) RATIONAL PLAYER LOOK FOR A GUARANTEE WHATEVER OPPONENT’S MOVE MINIMAX STRATEGY ARISE NATURALLY MINIMAX STRATEGY FOR PLAYER TWO ≡ MAXIMIN STRATEGY FOR PLAYER ONE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORTHE LINK THE LINK PLAYER II STATISTICIAN Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORTHE LINK THE LINK PLAYER II STATISTICIAN PLAYER I NATURE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORTHE LINK THE LINK PLAYER II STATISTICIAN PLAYER I NATURE MAXIMIN STRATEGY LEAST FAVORABLE STATE OF NATURE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORTHE LINK THE LINK PLAYER II STATISTICIAN PLAYER I NATURE MAXIMIN STRATEGY LEAST FAVORABLE GAIN-ORIENTED STATE OF NATURE RATIONALITY LEAST FAVORABLE DISTRIBUTION Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSIMPLE EXAMPLE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSIMPLE EXAMPLE UNDER MINIMAX COMPARE sup[R] θ δ1 OPTIMAL Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSIMPLE EXAMPLE UNDER MINIMAX COMPARE sup[R] θ δ1 OPTIMAL UNDER BAYES τ p.d.f. on Θ COMPARE Eτ [R] τ SUCH THAT δ1 δ1 SUFFICIENT BIAS TO θ0 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORLEAST FAVORABLE PRIOR LEAST FAVORABLE FOCUSES ON θ’s MAXIMAL RISK POINTS FOR A GENERIC BAYES RULE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORLEAST FAVORABLE PRIOR LEAST FAVORABLE FOCUSES ON θ’s MAXIMAL RISK POINTS FOR A GENERIC BAYES RULE LEAST FAVORABLE MAXIMIZE THE BAYES RISK Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORLEAST FAVORABLE PRIOR LEAST FAVORABLE FOCUSES ON θ’s MAXIMAL RISK POINTS FOR A GENERIC BAYES RULE LEAST FAVORABLE MAXIMIZE THE BAYES RISK Lemma r (τ, δB τ ) ≥ R(θ, δB τ ) ∀θ ∈ Θ ⇒ δB τ MINIMAX τ LEAST FAVORABLE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORGUESSING MINIMAX DECISION GUESS MAX. RISK PTS. Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORGUESSING MINIMAX DECISION GUESS MAX. RISK PTS. Suppose θ = +m Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORGUESSING MINIMAX DECISION GUESS MAX. RISK PTS. Suppose θ = +m LIKELY SAMPLES ∈ [m − 1, m + 1] Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORGUESSING MINIMAX DECISION GUESS MAX. RISK PTS. Suppose θ = +m LIKELY SAMPLES ∈ [m − 1, m + 1] HIGHLY BIASED INTERVAL Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIOROutline 1 SECTION 1 THE CANDIDATES 2-POINTS PRIOR 2 SECTION 2 AND WHEN m GETS LARGE? CLOSING THE LOOP Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORA BAYES RULE CONCENTRATING ON THE BOUNDS Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORA BAYES RULE CONCENTRATING ON THE BOUNDS ◦ τm TWO-POINTS PRIOR Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORA BAYES RULE CONCENTRATING ON THE BOUNDS ◦ τm TWO-POINTS PRIOR ◦ δm (x) = m × tanh(m × x) Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSHRINKING TO THE BOUNDS Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSHRINKING TO THE BOUNDS Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSHRINKING TO THE BOUNDS Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSHRINKING TO THE BOUNDS Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORCONDITIONS FOR MINIMAXITY ◦ minimaxity of δm depends on the interval width Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORCONDITIONS FOR MINIMAXITY ◦ minimaxity of δm depends on the interval width Theorem x ∼ N(θ, 1) θ ∼ [−m, m] m ≤ m0 L Gaussian loss Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORCONDITIONS FOR MINIMAXITY ◦ minimaxity of δm depends on the interval width Theorem ◦ x ∼ N(θ, 1) δm minimax ◦ θ ∼ [−m, m] τm least favorable m ≤ m0 L Gaussian loss Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORCONDITIONS FOR MINIMAXITY ◦ minimaxity of δm depends on the interval width Theorem ◦ x ∼ N(θ, 1) δm minimax ◦ θ ∼ [−m, m] τm least favorable m ≤ m0 L Gaussian loss Numerical solution for m0 1.056742 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORNUMERICAL EVIDENCE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORNUMERICAL EVIDENCE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORNUMERICAL EVIDENCE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORNUMERICAL EVIDENCE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - 1ST STEP ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - 1ST STEP ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ R At most 3 sign chg Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - 1ST STEP ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ R At most 3 sign chg (−+)(+−)(−+) Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - 1ST STEP ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ R At most 3 sign chg (−+)(+−)(−+) R (0) = 0 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - 1ST STEP ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ R At most 3 sign chg (−+)(+−)(−+) R (0) = 0 R (θ) = −R (−θ) Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - 1ST STEP ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ R At most 3 sign chg (−+)(+−)(−+) R (0) = 0 R (θ) = −R (−θ) Extremum for θ > 0 is (−+) Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - 1ST STEP ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ R At most 3 sign chg (−+)(+−)(−+) R (0) = 0 R (θ) = −R (−θ) Extremum for θ > 0 is (−+) R even function Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - 1ST STEP ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ R At most 3 sign chg (−+)(+−)(−+) R (0) = 0 R (θ) = −R (−θ) Extremum for θ > 0 is (−+) R even function ⇒ Maximum attained at 0 or at the bounds Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - CONCLUSION ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ ∃m0 such that R(m) ≥ R(0) ∀m ≤ m0 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - CONCLUSION ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ ∃m0 such that R(m) ≥ R(0) ∀m ≤ m0 ◦ ◦ r (τm , δm ) = 1 R(−m) + 2 1 ◦ 2 R(m) = R(m, δm ) Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - CONCLUSION ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ ∃m0 such that R(m) ≥ R(0) ∀m ≤ m0 ◦ ◦ r (τm , δm ) = 1 R(−m) + 2 1 ◦ 2 R(m) = R(m, δm ) = implies ≥ Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 THE CANDIDATES SECTION 2 2-POINTS PRIORSKETCHING THE PROOF - CONCLUSION ◦ ◦ ◦ Prove that r (τm , δm ) ≥ R(θ, δm ) ∀θ ∈ Θ ∃m0 such that R(m) ≥ R(0) ∀m ≤ m0 ◦ ◦ r (τm , δm ) = 1 R(−m) + 2 1 ◦ 2 R(m) = R(m, δm ) = implies ≥ ⇒ Theorem is proved Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPOutline 1 SECTION 1 THE CANDIDATES 2-POINTS PRIOR 2 SECTION 2 AND WHEN m GETS LARGE? CLOSING THE LOOP Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPHOW TO PROCEED AS LONG AS Θ IS COMPACT R(θ, δ τ ) analytic and = cost ⇓ Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPHOW TO PROCEED AS LONG AS Θ IS COMPACT R(θ, δ τ ) analytic and = cost ⇓ THE NUMBER OF MAXIMAL RISK θ’s IS FINITE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPHOW TO PROCEED AS LONG AS Θ IS COMPACT R(θ, δ τ ) analytic and = cost ⇓ THE NUMBER OF MAXIMAL RISK θ’s IS FINITE Sτ =SUPPORT OF τLS FINITE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPHOW TO PROCEED AS LONG AS Θ IS COMPACT R(θ, δ τ ) analytic and = cost ⇓ THE NUMBER OF MAXIMAL RISK θ’s IS FINITE Sτ =SUPPORT OF τLS FINITE IN GENERAL card{Sτ } INCREASES AS m INCREASES Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGUESSING MINIMAX DECISION m>1 GUESS THE NEXT MAX. RISK PT. Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGUESSING MINIMAX DECISION m>1 GUESS THE NEXT MAX. RISK PT. Suppose θ = 0 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGUESSING MINIMAX DECISION m>1 GUESS THE NEXT MAX. RISK PT. Suppose θ = 0 LIKELY SAMPLES ∈ [−1, 1] Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGUESSING MINIMAX DECISION m>1 GUESS THE NEXT MAX. RISK PT. Suppose θ = 0 LIKELY SAMPLES ∈ [−1, 1] LARGE RANGE (= 2) Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGUESSING MINIMAX DECISION m>1 GUESS THE NEXT MAX. RISK PT. Suppose θ = 0 LIKELY SAMPLES ∈ [−1, 1] LARGE RANGE (= 2) suppose θ = 0 and = ±m Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGUESSING MINIMAX DECISION m>1 GUESS THE NEXT MAX. RISK PT. Suppose θ = 0 LIKELY SAMPLES ∈ [−1, 1] LARGE RANGE (= 2) suppose θ = 0 and = ±m SMALLER RANGE (≤ 2) Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOP3-POINTS PRIOR the expression α (1−α)mtanh(mx) δm (x) = 2 1−α+αexp( m )sech(mx) 2 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOP3-POINTS PRIOR the expression α (1−α)mtanh(mx) δm (x) = 2 1−α+αexp( m )sech(mx) 2 when 1.4 ≤ m ≤ 1.6 ∃α∗ such that α δm (x) α is minimax and τm is least favorable Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPNUMERICAL EVIDENCE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPNUMERICAL EVIDENCE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPNUMERICAL EVIDENCE Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOP4-PTS PRIOR Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOP4-PTS PRIOR θ’s max risk pts w.r.t a generic bayes rule MAX BIAS → BOUNDS MAX VARIANCE → PTS FAR FROM BOUNDS Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOP4-PTS PRIOR θ’s max risk pts w.r.t a generic bayes rule MAX BIAS → BOUNDS MAX VARIANCE → PTS FAR FROM BOUNDS Due to normal symmetry θMS ’s pop up pairwise around zero Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPOutline 1 SECTION 1 THE CANDIDATES 2-POINTS PRIOR 2 SECTION 2 AND WHEN m GETS LARGE? CLOSING THE LOOP Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGENERALIZED BAYES RULE Generalizing the approach Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGENERALIZED BAYES RULE Generalizing the approach (τn ) SEQUENCE OF PROPER PRIORS Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGENERALIZED BAYES RULE Generalizing the approach (τn ) SEQUENCE OF PROPER PRIORS lim δn = δ0 −→ Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGENERALIZED BAYES RULE Generalizing the approach (τn ) SEQUENCE OF δ0 GENERALIZED PROPER PRIORS BAYES RULE lim δn = δ0 −→ Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGENERALIZED BAYES RULE Generalizing the approach (τn ) SEQUENCE OF δ0 GENERALIZED PROPER PRIORS BAYES RULE lim δn = δ0 −→ Lemma Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGENERALIZED BAYES RULE Generalizing the approach (τn ) SEQUENCE OF δ0 GENERALIZED PROPER PRIORS BAYES RULE lim δn = δ0 −→ Lemma δ0 Generalized bayes rule Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGENERALIZED BAYES RULE Generalizing the approach (τn ) SEQUENCE OF δ0 GENERALIZED PROPER PRIORS BAYES RULE lim δn = δ0 −→ Lemma δ0 Generalized bayes rule lim[rn ] ≥ R δ0 (θ) ∀θ −→ Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGENERALIZED BAYES RULE Generalizing the approach (τn ) SEQUENCE OF δ0 GENERALIZED PROPER PRIORS BAYES RULE lim δn = δ0 −→ Lemma δ0 Generalized bayes rule lim[rn ] ≥ R δ0 (θ) ∀θ −→ <∞ Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPGENERALIZED BAYES RULE Generalizing the approach (τn ) SEQUENCE OF δ0 GENERALIZED PROPER PRIORS BAYES RULE lim δn = δ0 −→ Lemma δ0 Generalized bayes rule δ0 MINIMAX RULE lim[rn ] ≥ R δ0 (θ) ∀θ −→ <∞ Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPNO MORE BOUNDS Generalized bayesian approach Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPNO MORE BOUNDS Generalized bayesian approach τn ∼ N(0, n) Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPNO MORE BOUNDS Generalized bayesian approach τn ∼ N(0, n) xn n τn (θ|x) ∼ N( n+1 , n+1 ) Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPNO MORE BOUNDS Generalized bayesian approach τn ∼ N(0, n) xn n τn (θ|x) ∼ N( n+1 , n+1 ) xn δn = n+1 → x = δ0 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPNO MORE BOUNDS Generalized bayesian approach τn ∼ N(0, n) xn n τn (θ|x) ∼ N( n+1 , n+1 ) xn δn = n+1 → x = δ0 n r (τn ) = n+1 → 1 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPNO MORE BOUNDS Generalized bayesian approach τn ∼ N(0, n) xn n τn (θ|x) ∼ N( n+1 , n+1 ) xn δn = n+1 → x = δ0 n r (τn ) = n+1 → 1 R(θ, δ0 ) = 1 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPNO MORE BOUNDS Generalized bayesian approach τn ∼ N(0, n) xn n τn (θ|x) ∼ N( n+1 , n+1 ) xn δn = n+1 → x = δ0 n r (τn ) = n+1 → 1 R(θ, δ0 ) = 1 ⇒ δ0 MINIMAX Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPEVERY GOOD ESTIMATOR IS BAYES Berger and Srinivasan 1978 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPEVERY GOOD ESTIMATOR IS BAYES Berger and Srinivasan 1978 θ natural parameter exponential family Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPEVERY GOOD ESTIMATOR IS BAYES Berger and Srinivasan 1978 θ natural parameter exponential family quadratic loss Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPEVERY GOOD ESTIMATOR IS BAYES Berger and Srinivasan 1978 θ natural parameter exponential family quadratic loss EVERY ADMISSIBLE ESTIMATOR = GENERALIZED BAYES ESTIMATOR Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPEVERY GOOD ESTIMATOR IS BAYES Berger and Srinivasan 1978 θ natural parameter exponential family quadratic loss EVERY ADMISSIBLE ESTIMATOR = GENERALIZED BAYES ESTIMATOR Wald 1950 Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPEVERY GOOD ESTIMATOR IS BAYES Berger and Srinivasan 1978 θ natural parameter exponential family quadratic loss EVERY ADMISSIBLE ESTIMATOR = GENERALIZED BAYES ESTIMATOR Wald 1950 Θ COMPACT Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPEVERY GOOD ESTIMATOR IS BAYES Berger and Srinivasan 1978 θ natural parameter exponential family quadratic loss EVERY ADMISSIBLE ESTIMATOR = GENERALIZED BAYES ESTIMATOR Wald 1950 Θ COMPACT RISK SET R CONVEX Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPEVERY GOOD ESTIMATOR IS BAYES Berger and Srinivasan 1978 θ natural parameter exponential family quadratic loss EVERY ADMISSIBLE ESTIMATOR = GENERALIZED BAYES ESTIMATOR Wald 1950 Θ COMPACT RISK SET R CONVEX ALL ESTIMATORS HAVE CONTINUOUS RISK FUNCTION Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPEVERY GOOD ESTIMATOR IS BAYES Berger and Srinivasan 1978 θ natural parameter exponential family quadratic loss EVERY ADMISSIBLE ESTIMATOR = GENERALIZED BAYES ESTIMATOR Wald 1950 Θ COMPACT RISK SET R CONVEX ALL ESTIMATORS HAVE CONTINUOUS RISK FUNCTION ⇒ BAYES ESTIMATORS’ SET = COMPLETE CLASS Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOPAS m GOES TO INFINITY Bickel 1981 Minimize Fisher information w.r.t. any prior cos2 (π/2) × x, |x| ≤ 1 π2 ρ(m) = 1 − m2 + o(m−2 ) as m → ∞ Jacopo Primavera Estimating a Bounded Normal Mean
• SECTION 1 AND WHEN m GETS LARGE? SECTION 2 CLOSING THE LOOP THANK YOU FOR YOUR ATTENTIONJacopo Primavera Estimating a Bounded Normal Mean