Loredo Teaser
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Transcript of Loredo Teaser
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7/27/2019 Loredo Teaser
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Estimating a Normal MeanSuppose we have a sample ofN = 5 values xi,
xi
N(, 1)
We want to estimate , including some quantification ofuncertainty in the estimate: an interval with a probability attached.
Frequentist approaches: method of moments, BLUE,
least-squares/2
, maximum likelihood
Focus on likelihood (equivalent to 2 here); this is closest to Bayes.
L() = p({xi}|)
=
i
1
2 e(xi)
2/22
; = 1
e2()/2
Estimate from maximum likelihood (minimum 2).
Define an interval and its coverage frequency from the L() curve.1 / 7
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Construct an Interval Procedure for Known
Likelihoods for 3 simulated data sets, = 0
-3 -2 -1 0 1 2 31x
-3
-2
-1
0
1
2
3
2x
Sample Space
-3 -2 -1 0 1 2 3
-10
-8
-6
-4
-2
0
2/2=)L(gol
Parameter Space
-3 -2 -1 0 1 2 3
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
)L(gol
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Likelihoods for 100 simulated data sets, = 0
-3 -2 -1 0 1 2 31x
-3
-2
-1
0
1
2
3
2x
Sample Space
-3 -2 -1 0 1 2 3
-10
-8
-6
-4
-2
0
2/2=)L(gol
Parameter Space
-3 -2 -1 0 1 2 3
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
)L(gol
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Explore Dependence on Likelihoods for 100 simulated data sets, = 3
0 1 2 3 4 5 61x
0
1
2
3
4
5
6
2x
Sample Space
0 1 2 3 4 5 6
-10
-8
-6
-4
-2
0
2/2=)L(gol
Parameter Space
0 1 2 3 4 5 6
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
)L(gol
Luckily the logL distribution is the same!
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Apply to Observed Sample
-3 -2 -1 0 1 2 31x
-3
-2
-1
0
1
2
3
2x
Sample Space
-3 -2 -1 0 1 2 3
-10
-8
-6
-4
-2
0
2/2=)L(gol
Parameter Space
-3 -2 -1 0 1 2 3
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
)L(gol
Report the green region, with coverage as calculated for ensemble of hypothetical data(red region, previous slide).
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Bayesian Approach
Normalize the likelihood for the observed sample; report the region that includes68.3% of the normalized likelihood.
-3 -2 -1 0 1 2 31x
-3
-2
-1
0
1
2
3
2x
Sample Space
-3 -2 -1 0 1 2 3
-10
-8
-6
-4
-2
0
2/2=)L(gol
Parameter Space
-3 -2 -1 0 1 2 3
0.0
0.2
0.4
0.6
0.8
1.0
)(Ld
ezilamroN
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When Theyll Differ
Both approaches report [x /N, x+ /N], and assign68.3% to this interval (with different meanings).
This matching is a coincidence!
When might results differ? (F= frequentist, B= Bayes) IfFprocedure doesnt use likelihood directly IfFprocedure properties depend on params (nonlinear models,
pivotal quantities) IfFproperties depend on likelihood shape (conditional inference,
ancillary statistics, recognizable subsets) If there are extra uninteresting parameters (nuisance parameters,
corrected profile likelihood, conditional inference) IfBuses important prior information
Also, for a different taskcomparison of parametric modelstheapproaches are qualitatively different (significance tests & info
criteria vs. Bayes factors)7 / 7
