bounded normal mean minimax estimation
DESCRIPTION
A project for the EuroBayes master presented by Jacopo PrimaveraTRANSCRIPT
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Estimating a Bounded Normal Mean
Jacopo Primavera
TSI-EuroBayes StudentUniversity Paris Dauphine
21 November 2011 / Reading Seminar on Classics
Jacopo Primavera Estimating a Bounded Normal Mean
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"ALL MODELS ARE WRONG, BUT SOME AREUSEFUL"
G. E. P. Box
Jacopo Primavera Estimating a Bounded Normal Mean
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PRESENTING THE PROBLEM
One observation x ∼ N(θ,1)
θ ∈ [−m,m] ⊂ RSquared loss (θ − δ(x))2
R(θ, δ) = MSE(δ) = BIAS(δ) + VAR(δ)
MINIMAX ESTIMATOR δMM = argminδ
[supθ
R(θ, δ)]
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
SUMMARY
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
SUMMARY
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
Outline
1 SECTION 1THE CANDIDATES2-POINTS PRIOR
2 SECTION 2AND WHEN m GETS LARGE?CLOSING THE LOOP
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
Outline
1 SECTION 1THE CANDIDATES2-POINTS PRIOR
2 SECTION 2AND WHEN m GETS LARGE?CLOSING THE LOOP
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SAMPLE MEAN
δSM = x
Main characteristicDOES NOT INVOLVEPRIOR INFORMATION
PropertiesΘ BOUNDED⇒ NOT Minimax
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
MLE
δMLE (x)−m for x ≤ −mx for x ∈ (−m,m)
m for x ≥ m
Main characteristicA SELECTOR ESTIMATOR
PropertiesΘ BOUNDED⇒ δMLE dominates δSM
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
DECISION THEORY
FORMALIZING THE CHOICE
Frequentist approachRESTRICT ∆
U = SET UNBIASED δ
CHOOSE UMVE
Decision-oriented approach(i) K OPTIMAL CRITERIA(ii) CHOOSE δ
MINIMIZING R(θ, δ)W.R.T K
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
BAYES ESTIMATOR
(i) PROBABILITY MEASURE τ ON Θ
(ii) CRITERIA = Eτ
(iii) Eτ [R(θ, δ)] = r(τ, δ)
(iv) minδ
[r(τ, δ)] = r(τ, δB) = r(τ)
(v) δB BAYES RULE
Bayes method and Decision theoryNATURAL ORDERING CRITERIATHE VERSATILITY OF τ MAKES BAYESIAN METHODCOHERENT WITH DECISION THEORYWHICH PRIOR INDUCES MINIMAXITY ?
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
BAYES ESTIMATOR
(i) PROBABILITY MEASURE τ ON Θ
(ii) CRITERIA = Eτ
(iii) Eτ [R(θ, δ)] = r(τ, δ)
(iv) minδ
[r(τ, δ)] = r(τ, δB) = r(τ)
(v) δB BAYES RULE
Bayes method and Decision theoryNATURAL ORDERING CRITERIATHE VERSATILITY OF τ MAKES BAYESIAN METHODCOHERENT WITH DECISION THEORYWHICH PRIOR INDUCES MINIMAXITY ?
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
GAME THEORY
Two-person zero-sum game
Θ Set of all possible strategies player 1A Set of all possible strategies player 2L Gain function (pl. 1) and loss function (pl.2)
RATIONAL PLAYER LOOK FOR A GUARANTEEWHATEVER OPPONENT’S MOVEMINIMAX STRATEGY ARISE NATURALLYMINIMAX STRATEGY FOR PLAYER TWO ≡ MAXIMINSTRATEGY FOR PLAYER ONE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
GAME THEORY
Two-person zero-sum game
Θ Set of all possible strategies player 1A Set of all possible strategies player 2L Gain function (pl. 1) and loss function (pl.2)
RATIONAL PLAYER LOOK FOR A GUARANTEEWHATEVER OPPONENT’S MOVEMINIMAX STRATEGY ARISE NATURALLYMINIMAX STRATEGY FOR PLAYER TWO ≡ MAXIMINSTRATEGY FOR PLAYER ONE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
THE LINK
THE LINK
PLAYER II
PLAYER IMAXIMIN STRATEGYGAIN-ORIENTEDRATIONALITY
STATISTICIAN
NATURELEAST FAVORABLESTATE OF NATURELEAST FAVORABLEDISTRIBUTION
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
THE LINK
THE LINK
PLAYER IIPLAYER I
MAXIMIN STRATEGYGAIN-ORIENTEDRATIONALITY
STATISTICIANNATURE
LEAST FAVORABLESTATE OF NATURELEAST FAVORABLEDISTRIBUTION
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
THE LINK
THE LINK
PLAYER IIPLAYER IMAXIMIN STRATEGY
GAIN-ORIENTEDRATIONALITY
STATISTICIANNATURELEAST FAVORABLESTATE OF NATURE
LEAST FAVORABLEDISTRIBUTION
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
THE LINK
THE LINK
PLAYER IIPLAYER IMAXIMIN STRATEGYGAIN-ORIENTEDRATIONALITY
STATISTICIANNATURELEAST FAVORABLESTATE OF NATURELEAST FAVORABLEDISTRIBUTION
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SIMPLE EXAMPLE
UNDER MINIMAXCOMPARE sup
θ[R]
δ1 OPTIMAL
UNDER BAYES
τ p.d.f. on Θ
COMPARE Eτ [R]
τ SUCH THAT δ1 � δ1
SUFFICIENT BIAS TOθ0
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SIMPLE EXAMPLE
UNDER MINIMAXCOMPARE sup
θ[R]
δ1 OPTIMAL
UNDER BAYES
τ p.d.f. on Θ
COMPARE Eτ [R]
τ SUCH THAT δ1 � δ1
SUFFICIENT BIAS TOθ0
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SIMPLE EXAMPLE
UNDER MINIMAXCOMPARE sup
θ[R]
δ1 OPTIMAL
UNDER BAYESτ p.d.f. on Θ
COMPARE Eτ [R]
τ SUCH THAT δ1 � δ1
SUFFICIENT BIAS TOθ0
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
LEAST FAVORABLE PRIOR
LEAST FAVORABLE FOCUSES ON θ’s MAXIMAL RISKPOINTS FOR A GENERIC BAYES RULE
LEAST FAVORABLE MAXIMIZE THE BAYES RISK
Lemmar(τ, δB
τ ) ≥ R(θ, δBτ ) ∀θ ∈ Θ ⇒
δBτ MINIMAX
τ LEAST FAVORABLE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
LEAST FAVORABLE PRIOR
LEAST FAVORABLE FOCUSES ON θ’s MAXIMAL RISKPOINTS FOR A GENERIC BAYES RULE
LEAST FAVORABLE MAXIMIZE THE BAYES RISK
Lemmar(τ, δB
τ ) ≥ R(θ, δBτ ) ∀θ ∈ Θ ⇒
δBτ MINIMAX
τ LEAST FAVORABLE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
LEAST FAVORABLE PRIOR
LEAST FAVORABLE FOCUSES ON θ’s MAXIMAL RISKPOINTS FOR A GENERIC BAYES RULE
LEAST FAVORABLE MAXIMIZE THE BAYES RISK
Lemmar(τ, δB
τ ) ≥ R(θ, δBτ ) ∀θ ∈ Θ ⇒
δBτ MINIMAX
τ LEAST FAVORABLE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
GUESSING MINIMAX DECISION
GUESS MAX. RISK PTS.
Suppose θ = +mLIKELY SAMPLES∈ [m − 1,m + 1]
HIGHLY BIASEDINTERVAL
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
GUESSING MINIMAX DECISION
GUESS MAX. RISK PTS.
Suppose θ = +m
LIKELY SAMPLES∈ [m − 1,m + 1]
HIGHLY BIASEDINTERVAL
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
GUESSING MINIMAX DECISION
GUESS MAX. RISK PTS.
Suppose θ = +mLIKELY SAMPLES∈ [m − 1,m + 1]
HIGHLY BIASEDINTERVAL
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
GUESSING MINIMAX DECISION
GUESS MAX. RISK PTS.
Suppose θ = +mLIKELY SAMPLES∈ [m − 1,m + 1]
HIGHLY BIASEDINTERVAL
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
Outline
1 SECTION 1THE CANDIDATES2-POINTS PRIOR
2 SECTION 2AND WHEN m GETS LARGE?CLOSING THE LOOP
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
A BAYES RULE
CONCENTRATING ONTHE BOUNDS
τ◦m TWO-POINTS
PRIORδ◦m(x) =
m × tanh(m × x)
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
A BAYES RULE
CONCENTRATING ONTHE BOUNDSτ◦m TWO-POINTS
PRIOR
δ◦m(x) =
m × tanh(m × x)
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
A BAYES RULE
CONCENTRATING ONTHE BOUNDSτ◦m TWO-POINTS
PRIORδ◦m(x) =
m × tanh(m × x)
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SHRINKING TO THE BOUNDS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SHRINKING TO THE BOUNDS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SHRINKING TO THE BOUNDS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SHRINKING TO THE BOUNDS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
CONDITIONS FOR MINIMAXITY
minimaxity of δ◦m depends on the interval width
Theorem
x ∼ N(θ,1)
θ ∼ [−m,m]
m ≤ m0
L Gaussian loss
δ◦m minimaxτ◦m least favorable
Numerical solution for m0
1.056742
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
CONDITIONS FOR MINIMAXITY
minimaxity of δ◦m depends on the interval width
Theorem
x ∼ N(θ,1)
θ ∼ [−m,m]
m ≤ m0
L Gaussian loss
δ◦m minimaxτ◦m least favorable
Numerical solution for m0
1.056742
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
CONDITIONS FOR MINIMAXITY
minimaxity of δ◦m depends on the interval width
Theorem
x ∼ N(θ,1)
θ ∼ [−m,m]
m ≤ m0
L Gaussian loss
δ◦m minimaxτ◦m least favorable
Numerical solution for m0
1.056742
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
CONDITIONS FOR MINIMAXITY
minimaxity of δ◦m depends on the interval width
Theorem
x ∼ N(θ,1)
θ ∼ [−m,m]
m ≤ m0
L Gaussian loss
δ◦m minimaxτ◦m least favorable
Numerical solution for m0
1.056742
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
NUMERICAL EVIDENCE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
NUMERICAL EVIDENCE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
NUMERICAL EVIDENCE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
NUMERICAL EVIDENCE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - 1ST STEP
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
R′ At most 3 sign chg(−+)(+−)(−+)
R′(0) = 0
R′(θ) = −R′(−θ)
Extremum for θ > 0 is(−+)
R even function⇒ Maximum attainedat 0 or at the bounds
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - 1ST STEP
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
R′ At most 3 sign chg
(−+)(+−)(−+)
R′(0) = 0
R′(θ) = −R′(−θ)
Extremum for θ > 0 is(−+)
R even function⇒ Maximum attainedat 0 or at the bounds
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - 1ST STEP
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
R′ At most 3 sign chg(−+)(+−)(−+)
R′(0) = 0
R′(θ) = −R′(−θ)
Extremum for θ > 0 is(−+)
R even function⇒ Maximum attainedat 0 or at the bounds
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - 1ST STEP
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
R′ At most 3 sign chg(−+)(+−)(−+)
R′(0) = 0
R′(θ) = −R′(−θ)
Extremum for θ > 0 is(−+)
R even function⇒ Maximum attainedat 0 or at the bounds
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - 1ST STEP
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
R′ At most 3 sign chg(−+)(+−)(−+)
R′(0) = 0
R′(θ) = −R′(−θ)
Extremum for θ > 0 is(−+)
R even function⇒ Maximum attainedat 0 or at the bounds
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - 1ST STEP
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
R′ At most 3 sign chg(−+)(+−)(−+)
R′(0) = 0
R′(θ) = −R′(−θ)
Extremum for θ > 0 is(−+)
R even function⇒ Maximum attainedat 0 or at the bounds
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - 1ST STEP
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
R′ At most 3 sign chg(−+)(+−)(−+)
R′(0) = 0
R′(θ) = −R′(−θ)
Extremum for θ > 0 is(−+)
R even function
⇒ Maximum attainedat 0 or at the bounds
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - 1ST STEP
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
R′ At most 3 sign chg(−+)(+−)(−+)
R′(0) = 0
R′(θ) = −R′(−θ)
Extremum for θ > 0 is(−+)
R even function⇒ Maximum attainedat 0 or at the bounds
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - CONCLUSION
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
∃m0 such that R(m)≥ R(0) ∀m ≤ m0
r(τ◦m, δ
◦m) = 1
2R(−m) +12R(m) = R(m, δ
◦m)
= implies ≥⇒ Theorem is proved
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - CONCLUSION
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
∃m0 such that R(m)≥ R(0) ∀m ≤ m0
r(τ◦m, δ
◦m) = 1
2R(−m) +12R(m) = R(m, δ
◦m)
= implies ≥⇒ Theorem is proved
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - CONCLUSION
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
∃m0 such that R(m)≥ R(0) ∀m ≤ m0
r(τ◦m, δ
◦m) = 1
2R(−m) +12R(m) = R(m, δ
◦m)
= implies ≥
⇒ Theorem is proved
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
THE CANDIDATES2-POINTS PRIOR
SKETCHING THE PROOF - CONCLUSION
Prove that r(τ◦m, δ
◦m) ≥ R(θ, δ
◦m) ∀θ ∈ Θ
∃m0 such that R(m)≥ R(0) ∀m ≤ m0
r(τ◦m, δ
◦m) = 1
2R(−m) +12R(m) = R(m, δ
◦m)
= implies ≥⇒ Theorem is proved
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
Outline
1 SECTION 1THE CANDIDATES2-POINTS PRIOR
2 SECTION 2AND WHEN m GETS LARGE?CLOSING THE LOOP
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
HOW TO PROCEED
AS LONG AS Θ IS COMPACTR(θ, δτ ) analytic and 6= cost
⇓
THE NUMBER OF MAXIMAL RISK θ’s ISFINITESτ =SUPPORT OF τLS FINITE
IN GENERALcard{Sτ} INCREASES AS m INCREASES
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
HOW TO PROCEED
AS LONG AS Θ IS COMPACTR(θ, δτ ) analytic and 6= cost
⇓THE NUMBER OF MAXIMAL RISK θ’s ISFINITE
Sτ =SUPPORT OF τLS FINITE
IN GENERALcard{Sτ} INCREASES AS m INCREASES
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
HOW TO PROCEED
AS LONG AS Θ IS COMPACTR(θ, δτ ) analytic and 6= cost
⇓THE NUMBER OF MAXIMAL RISK θ’s ISFINITESτ =SUPPORT OF τLS FINITE
IN GENERALcard{Sτ} INCREASES AS m INCREASES
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
HOW TO PROCEED
AS LONG AS Θ IS COMPACTR(θ, δτ ) analytic and 6= cost
⇓THE NUMBER OF MAXIMAL RISK θ’s ISFINITESτ =SUPPORT OF τLS FINITE
IN GENERALcard{Sτ} INCREASES AS m INCREASES
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GUESSING MINIMAX DECISION m>1
GUESS THE NEXT MAX.RISK PT.
Suppose θ = 0LIKELY SAMPLES∈ [−1,1]
LARGE RANGE (= 2)suppose θ 6= 0 and6= ±mSMALLER RANGE(≤ 2)
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GUESSING MINIMAX DECISION m>1
GUESS THE NEXT MAX.RISK PT.
Suppose θ = 0
LIKELY SAMPLES∈ [−1,1]
LARGE RANGE (= 2)suppose θ 6= 0 and6= ±mSMALLER RANGE(≤ 2)
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GUESSING MINIMAX DECISION m>1
GUESS THE NEXT MAX.RISK PT.
Suppose θ = 0LIKELY SAMPLES∈ [−1,1]
LARGE RANGE (= 2)suppose θ 6= 0 and6= ±mSMALLER RANGE(≤ 2)
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GUESSING MINIMAX DECISION m>1
GUESS THE NEXT MAX.RISK PT.
Suppose θ = 0LIKELY SAMPLES∈ [−1,1]
LARGE RANGE (= 2)
suppose θ 6= 0 and6= ±mSMALLER RANGE(≤ 2)
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GUESSING MINIMAX DECISION m>1
GUESS THE NEXT MAX.RISK PT.
Suppose θ = 0LIKELY SAMPLES∈ [−1,1]
LARGE RANGE (= 2)suppose θ 6= 0 and6= ±m
SMALLER RANGE(≤ 2)
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GUESSING MINIMAX DECISION m>1
GUESS THE NEXT MAX.RISK PT.
Suppose θ = 0LIKELY SAMPLES∈ [−1,1]
LARGE RANGE (= 2)suppose θ 6= 0 and6= ±mSMALLER RANGE(≤ 2)
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
3-POINTS PRIOR
the expression
δαm(x) = (1−α)mtanh(mx)
1−α+αexp(m22 )sech(mx)
when 1.4 ≤ m ≤ 1.6∃α∗ such that δαm(x) is minimax and ταm is least favorable
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
3-POINTS PRIOR
the expression
δαm(x) = (1−α)mtanh(mx)
1−α+αexp(m22 )sech(mx)
when 1.4 ≤ m ≤ 1.6∃α∗ such that δαm(x) is minimax and ταm is least favorable
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
NUMERICAL EVIDENCE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
NUMERICAL EVIDENCE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
NUMERICAL EVIDENCE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
4-PTS PRIOR
θ’s max risk pts w.r.t a generic bayes rule
MAX BIAS → BOUNDSMAX VARIANCE → PTS FAR FROM BOUNDS
Due to normal symmetryθMS ’s pop up pairwise around zero
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
4-PTS PRIOR
θ’s max risk pts w.r.t a generic bayes rule
MAX BIAS → BOUNDSMAX VARIANCE → PTS FAR FROM BOUNDS
Due to normal symmetryθMS ’s pop up pairwise around zero
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
4-PTS PRIOR
θ’s max risk pts w.r.t a generic bayes rule
MAX BIAS → BOUNDSMAX VARIANCE → PTS FAR FROM BOUNDS
Due to normal symmetryθMS ’s pop up pairwise around zero
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
Outline
1 SECTION 1THE CANDIDATES2-POINTS PRIOR
2 SECTION 2AND WHEN m GETS LARGE?CLOSING THE LOOP
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GENERALIZED BAYES RULE
Generalizing the approach
(τn) SEQUENCE OFPROPER PRIORSlim−→ δn = δ0
δ0 GENERALIZEDBAYES RULE
Lemma
δ0 Generalized bayesrulelim−→[rn] ≥ Rδ0(θ) ∀θ<∞
δ0 MINIMAX RULE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GENERALIZED BAYES RULE
Generalizing the approach
(τn) SEQUENCE OFPROPER PRIORS
lim−→ δn = δ0
δ0 GENERALIZEDBAYES RULE
Lemma
δ0 Generalized bayesrulelim−→[rn] ≥ Rδ0(θ) ∀θ<∞
δ0 MINIMAX RULE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GENERALIZED BAYES RULE
Generalizing the approach
(τn) SEQUENCE OFPROPER PRIORSlim−→ δn = δ0
δ0 GENERALIZEDBAYES RULE
Lemma
δ0 Generalized bayesrulelim−→[rn] ≥ Rδ0(θ) ∀θ<∞
δ0 MINIMAX RULE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GENERALIZED BAYES RULE
Generalizing the approach
(τn) SEQUENCE OFPROPER PRIORSlim−→ δn = δ0
δ0 GENERALIZEDBAYES RULE
Lemma
δ0 Generalized bayesrulelim−→[rn] ≥ Rδ0(θ) ∀θ<∞
δ0 MINIMAX RULE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GENERALIZED BAYES RULE
Generalizing the approach
(τn) SEQUENCE OFPROPER PRIORSlim−→ δn = δ0
δ0 GENERALIZEDBAYES RULE
Lemma
δ0 Generalized bayesrulelim−→[rn] ≥ Rδ0(θ) ∀θ<∞
δ0 MINIMAX RULE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GENERALIZED BAYES RULE
Generalizing the approach
(τn) SEQUENCE OFPROPER PRIORSlim−→ δn = δ0
δ0 GENERALIZEDBAYES RULE
Lemma
δ0 Generalized bayesrule
lim−→[rn] ≥ Rδ0(θ) ∀θ<∞
δ0 MINIMAX RULE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GENERALIZED BAYES RULE
Generalizing the approach
(τn) SEQUENCE OFPROPER PRIORSlim−→ δn = δ0
δ0 GENERALIZEDBAYES RULE
Lemma
δ0 Generalized bayesrulelim−→[rn] ≥ Rδ0(θ) ∀θ
<∞
δ0 MINIMAX RULE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GENERALIZED BAYES RULE
Generalizing the approach
(τn) SEQUENCE OFPROPER PRIORSlim−→ δn = δ0
δ0 GENERALIZEDBAYES RULE
Lemma
δ0 Generalized bayesrulelim−→[rn] ≥ Rδ0(θ) ∀θ<∞
δ0 MINIMAX RULE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
GENERALIZED BAYES RULE
Generalizing the approach
(τn) SEQUENCE OFPROPER PRIORSlim−→ δn = δ0
δ0 GENERALIZEDBAYES RULE
Lemma
δ0 Generalized bayesrulelim−→[rn] ≥ Rδ0(θ) ∀θ<∞
δ0 MINIMAX RULE
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
NO MORE BOUNDS
Generalized bayesian approach
τn ∼ N(0,n)τn(θ|x) ∼ N( xn
n+1 ,n
n+1)
δn = xnn+1 → x = δ0
r(τn) =n
n+1 → 1R(θ, δ0) = 1⇒ δ0 MINIMAX
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
NO MORE BOUNDS
Generalized bayesian approach
τn ∼ N(0,n)
τn(θ|x) ∼ N( xnn+1 ,
nn+1)
δn = xnn+1 → x = δ0
r(τn) =n
n+1 → 1R(θ, δ0) = 1⇒ δ0 MINIMAX
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
NO MORE BOUNDS
Generalized bayesian approach
τn ∼ N(0,n)τn(θ|x) ∼ N( xn
n+1 ,n
n+1)
δn = xnn+1 → x = δ0
r(τn) =n
n+1 → 1R(θ, δ0) = 1⇒ δ0 MINIMAX
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
NO MORE BOUNDS
Generalized bayesian approach
τn ∼ N(0,n)τn(θ|x) ∼ N( xn
n+1 ,n
n+1)
δn = xnn+1 → x = δ0
r(τn) =n
n+1 → 1R(θ, δ0) = 1⇒ δ0 MINIMAX
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
NO MORE BOUNDS
Generalized bayesian approach
τn ∼ N(0,n)τn(θ|x) ∼ N( xn
n+1 ,n
n+1)
δn = xnn+1 → x = δ0
r(τn) =n
n+1 → 1
R(θ, δ0) = 1⇒ δ0 MINIMAX
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
NO MORE BOUNDS
Generalized bayesian approach
τn ∼ N(0,n)τn(θ|x) ∼ N( xn
n+1 ,n
n+1)
δn = xnn+1 → x = δ0
r(τn) =n
n+1 → 1R(θ, δ0) = 1
⇒ δ0 MINIMAX
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
NO MORE BOUNDS
Generalized bayesian approach
τn ∼ N(0,n)τn(θ|x) ∼ N( xn
n+1 ,n
n+1)
δn = xnn+1 → x = δ0
r(τn) =n
n+1 → 1R(θ, δ0) = 1⇒ δ0 MINIMAX
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
EVERY GOOD ESTIMATOR IS BAYES
Berger and Srinivasan 1978
θ natural parameter exponential familyquadratic lossEVERY ADMISSIBLE ESTIMATOR = GENERALIZEDBAYES ESTIMATOR
Wald 1950
Θ COMPACTRISK SET R CONVEXALL ESTIMATORS HAVE CONTINUOUS RISKFUNCTION⇒ BAYES ESTIMATORS’ SET = COMPLETE CLASS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
EVERY GOOD ESTIMATOR IS BAYES
Berger and Srinivasan 1978
θ natural parameter exponential family
quadratic lossEVERY ADMISSIBLE ESTIMATOR = GENERALIZEDBAYES ESTIMATOR
Wald 1950
Θ COMPACTRISK SET R CONVEXALL ESTIMATORS HAVE CONTINUOUS RISKFUNCTION⇒ BAYES ESTIMATORS’ SET = COMPLETE CLASS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
EVERY GOOD ESTIMATOR IS BAYES
Berger and Srinivasan 1978
θ natural parameter exponential familyquadratic loss
EVERY ADMISSIBLE ESTIMATOR = GENERALIZEDBAYES ESTIMATOR
Wald 1950
Θ COMPACTRISK SET R CONVEXALL ESTIMATORS HAVE CONTINUOUS RISKFUNCTION⇒ BAYES ESTIMATORS’ SET = COMPLETE CLASS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
EVERY GOOD ESTIMATOR IS BAYES
Berger and Srinivasan 1978
θ natural parameter exponential familyquadratic lossEVERY ADMISSIBLE ESTIMATOR = GENERALIZEDBAYES ESTIMATOR
Wald 1950
Θ COMPACTRISK SET R CONVEXALL ESTIMATORS HAVE CONTINUOUS RISKFUNCTION⇒ BAYES ESTIMATORS’ SET = COMPLETE CLASS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
EVERY GOOD ESTIMATOR IS BAYES
Berger and Srinivasan 1978
θ natural parameter exponential familyquadratic lossEVERY ADMISSIBLE ESTIMATOR = GENERALIZEDBAYES ESTIMATOR
Wald 1950
Θ COMPACTRISK SET R CONVEXALL ESTIMATORS HAVE CONTINUOUS RISKFUNCTION⇒ BAYES ESTIMATORS’ SET = COMPLETE CLASS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
EVERY GOOD ESTIMATOR IS BAYES
Berger and Srinivasan 1978
θ natural parameter exponential familyquadratic lossEVERY ADMISSIBLE ESTIMATOR = GENERALIZEDBAYES ESTIMATOR
Wald 1950Θ COMPACT
RISK SET R CONVEXALL ESTIMATORS HAVE CONTINUOUS RISKFUNCTION⇒ BAYES ESTIMATORS’ SET = COMPLETE CLASS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
EVERY GOOD ESTIMATOR IS BAYES
Berger and Srinivasan 1978
θ natural parameter exponential familyquadratic lossEVERY ADMISSIBLE ESTIMATOR = GENERALIZEDBAYES ESTIMATOR
Wald 1950Θ COMPACTRISK SET R CONVEX
ALL ESTIMATORS HAVE CONTINUOUS RISKFUNCTION⇒ BAYES ESTIMATORS’ SET = COMPLETE CLASS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
EVERY GOOD ESTIMATOR IS BAYES
Berger and Srinivasan 1978
θ natural parameter exponential familyquadratic lossEVERY ADMISSIBLE ESTIMATOR = GENERALIZEDBAYES ESTIMATOR
Wald 1950Θ COMPACTRISK SET R CONVEXALL ESTIMATORS HAVE CONTINUOUS RISKFUNCTION
⇒ BAYES ESTIMATORS’ SET = COMPLETE CLASS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
EVERY GOOD ESTIMATOR IS BAYES
Berger and Srinivasan 1978
θ natural parameter exponential familyquadratic lossEVERY ADMISSIBLE ESTIMATOR = GENERALIZEDBAYES ESTIMATOR
Wald 1950Θ COMPACTRISK SET R CONVEXALL ESTIMATORS HAVE CONTINUOUS RISKFUNCTION⇒ BAYES ESTIMATORS’ SET = COMPLETE CLASS
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
AS m GOES TO INFINITY
Bickel 1981Minimize Fisherinformation w.r.t. anypriorcos2(π/2)× x , |x | ≤ 1
ρ(m) = 1− π2
m2 + o(m−2)
as m→∞
Jacopo Primavera Estimating a Bounded Normal Mean
SECTION 1SECTION 2
AND WHEN m GETS LARGE?CLOSING THE LOOP
THANK YOU
FOR
YOUR ATTENTION
Jacopo Primavera Estimating a Bounded Normal Mean