statistical decision theory - perrywilliams.us · statistical estimation theory should and can be...
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Statistical Decision Theory
Perry Williams
Department of Fish, Wildlife, and Conservation BiologyDepartment of StatisticsColorado State University
26 June 2016
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Motivation, History, and Fundamentals
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Why Statistical Decision Theory?
Decisions are made at every step in scientific investigation
Data collectionModel selectionSummary statisticsManagement
SDT provides a cohesive framework for decision making
Data collection–Dynamic adaptive samplingModel selection–Optimal predictionSummary statistics–Bayes rulesManagement actions–Optimal management
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Why Statistical Decision Theory?
Decisions are made at every step in scientific investigation
Data collectionModel selectionSummary statisticsManagement
SDT provides a cohesive framework for decision making
Data collection–Dynamic adaptive samplingModel selection–Optimal predictionSummary statistics–Bayes rulesManagement actions–Optimal management
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History
Bayes’ Theorem appeared in “AnEssay Towards Solving a Problem inthe Doctrine of Chances”
“Aldrich suggests that we interpret[Bayes’ definition of probability] interms of expected utility, and thus
that Bayes’ result would make senseonly to the extent to which one canbet on its observable consequences.”
-Stephen Fienberg, 2006.
1763 1774 1922 1931 1934 1949 1954 1961
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History
Laplace published “Memoire sur laProbabilite des Causes par lasEvenements”
Elaborate example of inverseprobability
Uniform prior distributions
Methods for choosing estimatorsthat minimize posterior loss
1763 1774 1922 1931 1934 1949 1954 1961
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History
Fisher published “On theMathematical Foundations ofTheoretical Statistics”
Rejected inverse probability
Grounded his theory on frequencyinterpretation of probability
Obviated the need for priordistributions
Introduced likelihood
Tests of significance
1763 1774 1922 1931 1934 1949 1954 1961
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History
Fisher on Probability and Decisions
“We aim, in fact, atmethods of inferencewhich should be equallyconvincing to all rationalminds, irrespective of anyintentions they may havein utilizing the knowledgeinferred.”
1763 1774 1922 1931 1934 1949 1954 1961
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History
1763 1774 1922 1931 1934 1949 1954 1961
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History
Wald published “Statistical DecisionFunctions”
Unified statistical theory bytreating statistical problems asspecial cases of zero-sumtwo-person games
Statistical inference was viewedas a special case of decisiontheory (c.f., Von Neumann andMorgenstern 1944)
1763 1774 1922 1931 1934 1949 1954 1961
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History
“It is well recognized that thestatistical estimation theoryshould and can be organizedwithin the framework of thetheory of statistical decisionfunctions (Wald 1950)”
Akaike, H. 1973. Informationtheory and an extension of themaximum likelihood principle.
1763 1774 1922 1931 1934 1949 1954 1961
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History
Savage published “The Foundationsof Statistics”
Set the stage for Bayesianrevival
1763 1774 1922 1931 1934 1949 1954 1961
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History
“Decision theory is the bestand most stimulating, if notthe only, systematic model ofstatistics.”
1763 1774 1922 1931 1934 1949 1954 1961
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History
Raiffa and Schlaifer published“Applied Statistical DecisionTheory”
Methods of Fisher, Neyman,and Pearson did not address themain problem of a businessman:how to make decisions underuncertainty
Developed Bayesian decisiontheory
1763 1774 1922 1931 1934 1949 1954 1961
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F.P. Ramsey
B. De Finetti
J.M. Keynes
H. Jeffreys
D.V. Lindley
D.R. Cox
J.W. Tukey
A. Birnbaum
M. Kendall
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Fundamentals of SDT
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Inferential Steps
1 Identify possible states of nature (support)
2 Assign prior probabilities
3 Assign model processes (for potentially many models)
4 Apply Bayes theorem to obtain posterior probabilities from data
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Decision Steps
1 Identify possible states of nature (support)
2 Assign prior probabilities
3 Assign model processes
4 Apply Bayes theorem to obtain posterior probabilities from data
5 Enumerate possible decisions
6 Assign a loss function
7 Choose decision that minimizes expected loss
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Data (y)
Conceptual Model
Potential Actions
a
Loss L(θ, a)
Likelihood [y| θ]
Prior [θ]
Optimal Decision a*=argmina {∫ ∫L(θ, a)[y|θ]dy[θ]dθ}
Williams, P.J., and M.B. Hooten. 2016. Combining statistical inference and decisions inecology. Ecological Applications.
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Basic Elements
Θ: potential states of nature
θ: true state of nature
A : potential actions
a: a specific action, possibly a function of data (i.e., δ(y))
L : Θ×A 7→ R: loss function
L(θ, a): loss incurred if action a is made and θ is true
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Example: Measurement With Uncertainty
0 1 2 3 4 5θ
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Example: Measurement With Uncertainty
0 1 2 3 4 5θ
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Example: Measurement With Uncertainty
0 1 2 3 4 5θ
MeanMedian2.5
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Example: Measurement With Uncertainty
0 1 2 3 4 5θ
MeanMedian2.5
01
23
45
Loss
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Example: Measurement With Uncertainty
0 1 2 3 4 5θ
MeanMedian2.5
01
23
45
Loss
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Example: Measurement With Uncertainty
0 1 2 3 4 5θ
MeanMedian2.5
01
23
45
Loss
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Types of Loss
Squared Error Loss:L(θ, a) = (θ − a)2
Linear Loss:
L(θ, a) =
{c1(θ − a) if θ > a
c2(a− θ) if θ < a
0–1 Loss:
L(θ, a) =
{0 if θ = a
1 if θ 6= a
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Risk
Frequentist Risk
Bayesian Expected Loss
Bayesian Risk
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Decision Rule δ(y)
Y : a random variable that depends on θ
Y : the sample space of Y
y : a realization from Y
δ : Y 7→ A
(for any possible realization y ∈ Y , δ describes which action to take)
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Decision Rule Example
H0 : θ ≥ 0
Ha : θ < 0
δ1(y) =
{a1 = Reject H0 y < −0.3
a2 = Fail to reject H0 y ≥ −0.3
δ2(y) =
{a1 = Reject H0 y < −0.5
a2 = Fail to reject H0 y ≥ −0.5
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Frequentist Risk
How much you expect to lose when using a decision rule ∀y ∈ Y
R(θ, δ) =E [L(θ, δ(y))]
=
∫YL(θ, δ(y))[y |θ]dy
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Frequentist Risk and p-values
δ1(y) =
{a1 = Reject H0 y < −0.3
a2 = Fail to reject H0 y ≥ −0.3 L(θ, δ(y)) =
1 if reject H0 and θ ≥ 0
1 if fail to reject H0 and θ < 0
0 otherwise
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Frequentist Risk
−3 −2 −1 0 1 2 3
0.0
0.2
0.4
0.6
0.8
1.0
θ
Ris
k
δ1δ2
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Decision Theory is Inherently Bayesian
Goal of Decision Theory: Make a decision based on our belief in theprobability of an unknown state
Frequentist Probability: The limit of a state’s relative frequency in alarge number of trials
Bayesian Probability: Degree of rational belief to which a state isentitled in light of the given evidence
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Bayesian Expected Loss
Probability distribution assigned to θ
Prior probability distribution: [θ]
Posterior probability distribution: [θ|y ]
Bayesian Expected Loss: the loss averaged over thedistribution of θ.
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Bayesian Expected Loss
ρ(a) = EθL(θ, a) =∫
Θ L(θ, a)[θ]dθ
ρ(a) = Eθ|yL(θ, a) =∫
Θ L(θ, a)[θ|y ]dθ
Bayes Rule: a∗ = argmina(ρ(a))
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Bayesian Expected Loss
0 1 2 3 4 5θ
MeanMedian2.5
01
23
45
Loss
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Bayesian Expected Loss
0 1 2 3 4 5θ
MeanMedian2.5
L(θ,
a)[θ
|y]
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Bayes Risk
Mathematical Relationship Between Frequentist Risk and Bayes ExpectedLoss:
r(a) =
∫Θ
{∫YL(θ, a)[y |θ]dy
}[θ]dθ
Note: [y |θ][θ] = [θ|y ][y ]
r(a) =
∫Y
{∫θ
L(θ, a)[θ|y ]dθ
}[y ]dy
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Bayesian Expected Loss vs. Frequentist Risk
R(θ, δ) integrates over y , but y is known (i.e., data)
R(θ, δ) is a function of unknown θ
R(θ, δ) doesn’t make use of auxiliary information (e.g., priorknowledge)
δ that minimizes ρ(δ) also minimizes r(δ) (don’t need to integrateover hypothetical replicates of y)
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Bayesian Point Estimation
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Bayesian Point Estimation
Suppose we want to summarize a posterior distribution [θ|y ] with apoint estimate
We want to minimize the information lost using the point estimate tosummarize [θ|y ]
Is the choice of point estimate arbitrary?
Bayes Estimator: Bayes Rule for point estimation
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Bayesian Point Estimation
Suppose we want to summarize a posterior distribution [θ|y ] with apoint estimate
We want to minimize the information lost using the point estimate tosummarize [θ|y ]
Is the choice of point estimate arbitrary?
Bayes Estimator: Bayes Rule for point estimation
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Bayes rule for squared-error loss
ρ(a) =
∫θ(θ − a)2[θ|y ]dθ
d
daρ(a) = 2E [θ|y ]− 2a
2E [θ|y ]− 2aset= 0
a∗ = E [θ|y ]
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Implied Loss of Posterior Mean
a = E [θ|y ]
f ′′(a)(a−E [θ|y ]) = 0
∫f ′′(a)(a− E [θ|y ])da =
∫(f ′(a)(a− θ)− f (a) + g(θ))[θ|y ]dθ
L(θ, a) =f ′(a)(a− θ)− f (a) + g(θ)
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Bayes rule for absolute-error loss
ρ(a) =
∫Θ|θ − a|[θ|y ]dθ
d
daρ(a) = −P(θ ≥ [a|y ]) + P(θ ≤ [a|y ])
− P(θ ≥ [a|y ]) + P(θ ≤ [a|y ])set= 0
a∗ = median([θ|y ])
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Bayes rule for 0–1 loss
ρ(a) =
∫ a2
a1
0[θ|y ]dθ +
∫ a1
−∞1[θ|y ]dθ +
∫ ∞a2
1[θ|y ]dθ
d
daρ(a) =[a1|y ]− [a2|y ], as a1 → a← a2
a∗ = mode([θ|y ])
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Asymmetric Distributions
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
φ
Den
sity
MeanMedianMode
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Asymmetric Distributions
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
φ
Den
sity
MeanMedianModeTruth
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Bayesian Point Estimation
Bayesian point estimation is NOT arbitrary!
Different loss functions yield different point estimates.
SEL: Posterior meanAbsolute Loss: Posterior median0–1 Loss: Posterior mode
A choice of point estimator implies decision maker’s choice of lossfunction (or class of functions).
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Management Decisions
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Henslow’s Sparrow
Williams, P.J., and M.B. Hooten. 2016. Combining statistical inference and decisions in ecology. Ecological Applications.
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Big Oaks National Wildlife Refuge
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Optimal Prescribed Fire Return Interval?
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Non-Optimal Solution
●
1 2 3 4
0.0
0.5
1.0
1.5
2.0
2.5
Year since burn
HE
SP
den
sity
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Elements for SDT
Data: Density estimates
Prior Information: Mean henslow’s sparrow densities at other sites(Herkert and Glass 1999)
Loss Function related to cost and management objectives
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Response-to-fire model
yj ,t ∼Poisson(Ajλj ,t),
log(λj ,t) = x ′j ,tβ + ηj ,
β ∼ Normal(µ, σ2I ),
ηj ∼ Normal(0, σ2η).
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Cumulative abundance as a derived parameter
Na = limT→∞
20∑8
j=1
∑Tt=T+1 Aiλj ,t
T − T
j=1, a=3 A1λ1,1 A1λ1,2 A1λ1,3 A1λ1,1 A1λ1,2 A1λ1,3 … A1λ1,1 A1λ1,2 A1λ1,3
t=1 t=2 t=3 t=4 t=5 t=6 … t=
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Posterior Distributions
0 500 1000 1500
N
Pro
babi
lity
Burn interval
1 year2 years3 years4 years
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Axioms of Henslow’s Sparrow Management
1 Frequent fire intervals are more expensive
2 More birds are better
3 The relative importance of cost decreases as abundance increases
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Henslow’s sparrow loss function
L(θ, a) =
{α0(a) + α1(a)Na,θ, Na,θ < 1835
0, Na,θ ≥ 1835
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0 500 1000 1500
Loss
Loss
01
N
Burn interval
1 year2 years3 years4 years
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Den
sity
Pro
babi
lity
0 500 1000 1500
Burn interval
1 year2 years3 years4 years
0 500 1000 1500
Loss
Loss
01
N
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Convolution of Loss Function and Posterior Distribution
0 500 1000 1500
Loss
L(θ,
a)[θ
|y]
N
Burn interval
1 year2 years3 years4 years
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Bayes Expected Loss
0 500 1000 1500
Loss
L(θ,
a)[θ
|y]
N
Burn interval
1 year2 years3 years4 years
0.65
0.340.26
0.27
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Model Selection Example
Posterior predictive distribution
[y|y,m] =
∫[y|y,θ(m)][θ(m)|y]dθ(m)
Posterior predictive loss (Gelfand and Ghosh 1998)
L(y, y, a) =L(y, a) + kL(y, a)
Posterior predictive expectation of loss for model m, and action a.
Ey|y,mL(y, y, a)
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Summary
SDT combines statistical analyses and decision theory
Implications for data collection, point estimation, and model selection
Ecological/Management applications
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