1.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
Bayesian Core:
A Practical Approach to Computational Bayesian
Statistics
Jean-Michel Marin & Christian P. Robert
QUT, Brisbane, August 4–8 2008
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2.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
Outline
The normal model
1
Regression and variable selection
2
Generalized linear models
3
Capture-recapture experiments
4
Mixture models
5
Dynamic models
6
Image analysis
7
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3.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The normal model
The normal model
1
Normal problems
The Bayesian toolbox
Prior selection
Bayesian estimation
Conﬁdence regions
Testing
Monte Carlo integration
Prediction
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4.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Normal problems
Normal model
Sample
x1 , . . . , xn
from a normal N (µ, σ 2 ) distribution
normal sample
0.6
0.5
0.4
Density
0.3
0.2
0.1
0.0
−2 −1 0 1 2
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5.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Normal problems
Inference on (µ, σ) based on this sample
Estimation of [transforms of] (µ, σ)
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6.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Normal problems
Inference on (µ, σ) based on this sample
Estimation of [transforms of] (µ, σ)
Conﬁdence region [interval] on (µ, σ)
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7.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Normal problems
Inference on (µ, σ) based on this sample
Estimation of [transforms of] (µ, σ)
Conﬁdence region [interval] on (µ, σ)
Test on (µ, σ) and comparison with other samples
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8.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Normal problems
Datasets
Larcenies =normaldata
4
Relative changes in reported
3
larcenies between 1991 and
1995 (relative to 1991) for
2
the 90 most populous US
counties (Source: FBI)
1
0
−0.4 −0.2 0.0 0.2 0.4 0.6
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9.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Normal problems
Cosmological background =CMBdata
Spectral representation of the
“cosmological microwave
background” (CMB),
i.e. electromagnetic radiation
from photons back to 300, 000
years after the Big Bang,
expressed as diﬀerence in
apparent temperature from the
mean temperature
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10.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Normal problems
Cosmological background =CMBdata
Normal estimation
5
4
3
2
1
0
−0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6
CMB
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11.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Normal problems
Cosmological background =CMBdata
Normal estimation
5
4
3
2
1
0
−0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6
CMB
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12.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
The Bayesian toolbox
Bayes theorem = Inversion of probabilities
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13.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
The Bayesian toolbox
Bayes theorem = Inversion of probabilities
If A and E are events such that P (E) = 0, P (A|E) and P (E|A)
are related by
P (E|A)P (A)
P (A|E) =
P (E|A)P (A) + P (E|Ac )P (Ac )
P (E|A)P (A)
=
P (E)
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14.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Who’s Bayes?
Reverend Thomas Bayes (ca. 1702–1761)
Presbyterian minister in
Tunbridge Wells (Kent) from
1731, son of Joshua Bayes,
nonconformist minister. Election
to the Royal Society based on a
tract of 1736 where he defended
the views and philosophy of
Newton.
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15.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Who’s Bayes?
Reverend Thomas Bayes (ca. 1702–1761)
Presbyterian minister in
Tunbridge Wells (Kent) from
1731, son of Joshua Bayes,
nonconformist minister. Election
to the Royal Society based on a
tract of 1736 where he defended
the views and philosophy of
Newton.
Sole probability paper, “Essay Towards Solving a Problem in the
Doctrine of Chances”, published posthumously in 1763 by Pierce
and containing the seeds of Bayes’ Theorem.
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16.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
New perspective
Uncertainty on the parameters θ of a model modeled through
a probability distribution π on Θ, called prior distribution
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17.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
New perspective
Uncertainty on the parameters θ of a model modeled through
a probability distribution π on Θ, called prior distribution
Inference based on the distribution of θ conditional on x,
π(θ|x), called posterior distribution
f (x|θ)π(θ)
π(θ|x) = .
f (x|θ)π(θ) dθ
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18.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Bayesian model
A Bayesian statistical model is made of
a likelihood
1
f (x|θ),
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19.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Bayesian model
A Bayesian statistical model is made of
a likelihood
1
f (x|θ),
and of
a prior distribution on the parameters,
2
π(θ) .
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20.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Justiﬁcations
Semantic drift from unknown θ to random θ
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21.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Justiﬁcations
Semantic drift from unknown θ to random θ
Actualization of information/knowledge on θ by extracting
information/knowledge on θ contained in the observation x
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22.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Justiﬁcations
Semantic drift from unknown θ to random θ
Actualization of information/knowledge on θ by extracting
information/knowledge on θ contained in the observation x
Allows incorporation of imperfect/imprecise information in the
decision process
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23.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Justiﬁcations
Semantic drift from unknown θ to random θ
Actualization of information/knowledge on θ by extracting
information/knowledge on θ contained in the observation x
Allows incorporation of imperfect/imprecise information in the
decision process
Unique mathematical way to condition upon the observations
(conditional perspective)
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24.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Example (Normal illustration (σ 2 = 1))
Assume
π(θ) = exp{−θ} Iθ>0
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25.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Example (Normal illustration (σ 2 = 1))
Assume
π(θ) = exp{−θ} Iθ>0
Then
π(θ|x1 , . . . , xn ) ∝ exp{−θ} exp{−n(θ − x)2 /2} Iθ>0
∝ exp −nθ2 /2 + θ(nx − 1) Iθ>0
∝ exp −n(θ − (x − 1/n))2 /2 Iθ>0
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26.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Example (Normal illustration (2))
Truncated normal distribution
N + ((x − 1/n), 1/n)
mu=−0.08935864
2.5
2.0
1.5
1.0
0.5
0.0
0.0 0.1 0.2 0.3 0.4 0.5
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27.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Prior and posterior distributions
Given f (x|θ) and π(θ), several distributions of interest:
the joint distribution of (θ, x),
1
ϕ(θ, x) = f (x|θ)π(θ) ;
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28.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Prior and posterior distributions
Given f (x|θ) and π(θ), several distributions of interest:
the joint distribution of (θ, x),
1
ϕ(θ, x) = f (x|θ)π(θ) ;
the marginal distribution of x,
2
m(x) = ϕ(θ, x) dθ
= f (x|θ)π(θ) dθ ;
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29.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
the posterior distribution of θ,
3
f (x|θ)π(θ)
π(θ|x) =
f (x|θ)π(θ) dθ
f (x|θ)π(θ)
= ;
m(x)
the predictive distribution of y, when y ∼ g(y|θ, x),
4
g(y|x) = g(y|θ, x)π(θ|x)dθ .
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30.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Posterior distribution center of to Bayesian inference
π(θ|x) ∝ f (x|θ) π(θ)
Operates conditional upon the observations
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31.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Posterior distribution center of to Bayesian inference
π(θ|x) ∝ f (x|θ) π(θ)
Operates conditional upon the observations
Integrate simultaneously prior information/knowledge and
information brought by x
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32.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Posterior distribution center of to Bayesian inference
π(θ|x) ∝ f (x|θ) π(θ)
Operates conditional upon the observations
Integrate simultaneously prior information/knowledge and
information brought by x
Avoids averaging over the unobserved values of x
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33.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Posterior distribution center of to Bayesian inference
π(θ|x) ∝ f (x|θ) π(θ)
Operates conditional upon the observations
Integrate simultaneously prior information/knowledge and
information brought by x
Avoids averaging over the unobserved values of x
Coherent updating of the information available on θ,
independent of the order in which i.i.d. observations are
collected
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34.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Posterior distribution center of to Bayesian inference
π(θ|x) ∝ f (x|θ) π(θ)
Operates conditional upon the observations
Integrate simultaneously prior information/knowledge and
information brought by x
Avoids averaging over the unobserved values of x
Coherent updating of the information available on θ,
independent of the order in which i.i.d. observations are
collected
Provides a complete inferential scope and an unique motor of
inference
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35.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Example (Normal-normal case)
Consider x|θ ∼ N (θ, 1) and θ ∼ N (a, 10).
(x − θ)2 (θ − a)2
π(θ|x) ∝ f (x|θ)π(θ) ∝ exp − −
2 20
11θ2
∝ exp − + θ(x + a/10)
20
11
∝ exp − {θ − ((10x + a)/11)}2
20
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36.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
The Bayesian toolbox
Example (Normal-normal case)
Consider x|θ ∼ N (θ, 1) and θ ∼ N (a, 10).
(x − θ)2 (θ − a)2
π(θ|x) ∝ f (x|θ)π(θ) ∝ exp − −
2 20
11θ2
∝ exp − + θ(x + a/10)
20
11
∝ exp − {θ − ((10x + a)/11)}2
20
and
θ|x ∼ N (10x + a) 11, 10 11
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37.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Prior selection
The prior distribution is the key to Bayesian inference
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38.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Prior selection
The prior distribution is the key to Bayesian inference
But...
In practice, it seldom occurs that the available prior information is
precise enough to lead to an exact determination of the prior
distribution
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39.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Prior selection
The prior distribution is the key to Bayesian inference
But...
In practice, it seldom occurs that the available prior information is
precise enough to lead to an exact determination of the prior
distribution
There is no such thing as the prior distribution!
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40.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Strategies for prior determination
Ungrounded prior distributions produce
unjustiﬁed posterior inference.
—Anonymous, ca. 2006
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41.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Strategies for prior determination
Ungrounded prior distributions produce
unjustiﬁed posterior inference.
—Anonymous, ca. 2006
Use a partition of Θ in sets (e.g., intervals), determine the
probability of each set, and approach π by an histogram
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42.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Strategies for prior determination
Ungrounded prior distributions produce
unjustiﬁed posterior inference.
—Anonymous, ca. 2006
Use a partition of Θ in sets (e.g., intervals), determine the
probability of each set, and approach π by an histogram
Select signiﬁcant elements of Θ, evaluate their respective
likelihoods and deduce a likelihood curve proportional to π
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43.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Strategies for prior determination
Ungrounded prior distributions produce
unjustiﬁed posterior inference.
—Anonymous, ca. 2006
Use a partition of Θ in sets (e.g., intervals), determine the
probability of each set, and approach π by an histogram
Select signiﬁcant elements of Θ, evaluate their respective
likelihoods and deduce a likelihood curve proportional to π
Use the marginal distribution of x,
m(x) = f (x|θ)π(θ) dθ
Θ
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44.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Strategies for prior determination
Ungrounded prior distributions produce
unjustiﬁed posterior inference.
—Anonymous, ca. 2006
Use a partition of Θ in sets (e.g., intervals), determine the
probability of each set, and approach π by an histogram
Select signiﬁcant elements of Θ, evaluate their respective
likelihoods and deduce a likelihood curve proportional to π
Use the marginal distribution of x,
m(x) = f (x|θ)π(θ) dθ
Θ
Empirical and hierarchical Bayes techniques
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45.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Conjugate priors
Speciﬁc parametric family with analytical properties
Conjugate prior
A family F of probability distributions on Θ is conjugate for a
likelihood function f (x|θ) if, for every π ∈ F, the posterior
distribution π(θ|x) also belongs to F.
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46.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Conjugate priors
Speciﬁc parametric family with analytical properties
Conjugate prior
A family F of probability distributions on Θ is conjugate for a
likelihood function f (x|θ) if, for every π ∈ F, the posterior
distribution π(θ|x) also belongs to F.
Only of interest when F is parameterised : switching from prior to
posterior distribution is reduced to an updating of the
corresponding parameters.
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47.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Justiﬁcations
Limited/ﬁnite information conveyed by x
Preservation of the structure of π(θ)
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48.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Justiﬁcations
Limited/ﬁnite information conveyed by x
Preservation of the structure of π(θ)
Exchangeability motivations
Device of virtual past observations
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49.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Justiﬁcations
Limited/ﬁnite information conveyed by x
Preservation of the structure of π(θ)
Exchangeability motivations
Device of virtual past observations
Linearity of some estimators
But mostly...
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50.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Justiﬁcations
Limited/ﬁnite information conveyed by x
Preservation of the structure of π(θ)
Exchangeability motivations
Device of virtual past observations
Linearity of some estimators
But mostly... tractability and simplicity
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51.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Justiﬁcations
Limited/ﬁnite information conveyed by x
Preservation of the structure of π(θ)
Exchangeability motivations
Device of virtual past observations
Linearity of some estimators
But mostly... tractability and simplicity
First approximations to adequate priors, backed up by
robustness analysis
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52.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Exponential families
Sampling models of interest
Exponential family
The family of distributions
f (x|θ) = C(θ)h(x) exp{R(θ) · T (x)}
is called an exponential family of dimension k. When Θ ⊂ Rk ,
X ⊂ Rk and
f (x|θ) = h(x) exp{θ · x − Ψ(θ)},
the family is said to be natural.
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53.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Analytical properties of exponential families
Suﬃcient statistics (Pitman–Koopman Lemma)
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54.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Analytical properties of exponential families
Suﬃcient statistics (Pitman–Koopman Lemma)
Common enough structure (normal, Poisson, &tc...)
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55.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Analytical properties of exponential families
Suﬃcient statistics (Pitman–Koopman Lemma)
Common enough structure (normal, Poisson, &tc...)
Analyticity (E[x] = ∇Ψ(θ), ...)
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56.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Analytical properties of exponential families
Suﬃcient statistics (Pitman–Koopman Lemma)
Common enough structure (normal, Poisson, &tc...)
Analyticity (E[x] = ∇Ψ(θ), ...)
Allow for conjugate priors
π(θ|µ, λ) = K(µ, λ) eθ.µ−λΨ(θ) λ>0
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57.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Standard exponential families
f (x|θ) π(θ) π(θ|x)
Normal Normal
N (θ, σ 2 ) N (µ, τ 2 ) N (ρ(σ 2 µ + τ 2 x), ρσ 2 τ 2 )
ρ−1 = σ 2 + τ 2
Poisson Gamma
P(θ) G(α, β) G(α + x, β + 1)
Gamma Gamma
G(ν, θ) G(α, β) G(α + ν, β + x)
Binomial Beta
B(n, θ) Be(α, β) Be(α + x, β + n − x)
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58.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
More...
f (x|θ) π(θ) π(θ|x)
Negative Binomial Beta
N eg(m, θ) Be(α, β) Be(α + m, β + x)
Multinomial Dirichlet
Mk (θ1 , . . . , θk ) D(α1 , . . . , αk ) D(α1 + x1 , . . . , αk + xk )
Normal Gamma
G(α + 0.5, β + (µ − x)2 /2)
N (µ, 1/θ) Ga(α, β)
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59.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Linearity of the posterior mean
If
θ ∼ πλ,µ (θ) ∝ eθ·µ−λΨ(θ)
with µ ∈ X , then
µ
Eπ [∇Ψ(θ)] =
.
λ
where ∇Ψ(θ) = (∂Ψ(θ)/∂θ1 , . . . , ∂Ψ(θ)/∂θp )
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60.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Linearity of the posterior mean
If
θ ∼ πλ,µ (θ) ∝ eθ·µ−λΨ(θ)
with µ ∈ X , then
µ
Eπ [∇Ψ(θ)] =
.
λ
where ∇Ψ(θ) = (∂Ψ(θ)/∂θ1 , . . . , ∂Ψ(θ)/∂θp )
Therefore, if x1 , . . . , xn are i.i.d. f (x|θ),
µ + n¯
x
Eπ [∇Ψ(θ)|x1 , . . . , xn ] = .
λ+n
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61.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Example (Normal-normal)
In the normal N (θ, σ 2 ) case, conjugate also normal N (µ, τ 2 ) and
Eπ [∇Ψ(θ)|x] = Eπ [θ|x] = ρ(σ 2 µ + τ 2 x)
where
ρ−1 = σ 2 + τ 2
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62.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Example (Full normal)
In the normal N (µ, σ 2 ) case, when both µ and σ are unknown,
there still is a conjugate prior on θ = (µ, σ 2 ), of the form
(σ 2 )−λσ exp − λµ (µ − ξ)2 + α /2σ 2
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63.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Example (Full normal)
In the normal N (µ, σ 2 ) case, when both µ and σ are unknown,
there still is a conjugate prior on θ = (µ, σ 2 ), of the form
(σ 2 )−λσ exp − λµ (µ − ξ)2 + α /2σ 2
since
π(µ, σ 2 |x1 , . . . , xn ) ∝ (σ 2 )−λσ exp − λµ (µ − ξ)2 + α /2σ 2
×(σ 2 )−n/2 exp − n(µ − x)2 + s2 /2σ 2
x
∝ (σ 2 )−λσ −n/2 exp − (λµ + n)(µ − ξx )2
nλµ (x − ξ)2
+α + s2 + /2σ 2
x
n + λµ
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64.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
parameters (0,1,1,1)
2.0
1.5
σ
1.0
0.5
−1.0 −0.5 0.0 0.5 1.0
µ
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65.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Improper prior distribution
Extension from a prior distribution to a prior σ-ﬁnite measure π
such that
π(θ) dθ = +∞
Θ
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66.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Improper prior distribution
Extension from a prior distribution to a prior σ-ﬁnite measure π
such that
π(θ) dθ = +∞
Θ
Formal extension: π cannot be interpreted as a probability any
longer
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67.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Justiﬁcations
Often only way to derive a prior in noninformative/automatic
1
settings
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68.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Justiﬁcations
Often only way to derive a prior in noninformative/automatic
1
settings
Performances of associated estimators usually good
2
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69.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Justiﬁcations
Often only way to derive a prior in noninformative/automatic
1
settings
Performances of associated estimators usually good
2
Often occur as limits of proper distributions
3
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70.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Justiﬁcations
Often only way to derive a prior in noninformative/automatic
1
settings
Performances of associated estimators usually good
2
Often occur as limits of proper distributions
3
More robust answer against possible misspeciﬁcations of the
4
prior
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71.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Justiﬁcations
Often only way to derive a prior in noninformative/automatic
1
settings
Performances of associated estimators usually good
2
Often occur as limits of proper distributions
3
More robust answer against possible misspeciﬁcations of the
4
prior
Improper priors (inﬁnitely!) preferable to vague proper priors
5
such as a N (0, 1002 ) distribution [e.g., BUGS]
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72.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Validation
Extension of the posterior distribution π(θ|x) associated with an
improper prior π given by Bayes’s formula
f (x|θ)π(θ)
π(θ|x) = ,
Θ f (x|θ)π(θ) dθ
when
f (x|θ)π(θ) dθ < ∞
Θ
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73.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Example (Normal+improper)
If x ∼ N (θ, 1) and π(θ) = ̟, constant, the pseudo marginal
distribution is
+∞
1
√ exp −(x − θ)2 /2 dθ = ̟
m(x) = ̟
2π
−∞
and the posterior distribution of θ is
(x − θ)2
1
π(θ | x) = √ exp − ,
2
2π
i.e., corresponds to N (x, 1).
[independent of ̟]
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74.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Meaningless as probability distribution
The mistake is to think of them [the non-informative priors]
as representing ignorance
—Lindley, 1990—
74 / 785
75.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Meaningless as probability distribution
The mistake is to think of them [the non-informative priors]
as representing ignorance
—Lindley, 1990—
Example
Consider a θ ∼ N (0, τ 2 ) prior. Then
P π (θ ∈ [a, b]) −→ 0
when τ → ∞ for any (a, b)
75 / 785
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Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Noninformative prior distributions
What if all we know is that we know “nothing” ?!
76 / 785
77.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Noninformative prior distributions
What if all we know is that we know “nothing” ?!
In the absence of prior information, prior distributions solely
derived from the sample distribution f (x|θ)
77 / 785
78.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Noninformative prior distributions
What if all we know is that we know “nothing” ?!
In the absence of prior information, prior distributions solely
derived from the sample distribution f (x|θ)
Noninformative priors cannot be expected to represent exactly
total ignorance about the problem at hand, but should rather
be taken as reference or default priors, upon which everyone
could fall back when the prior information is missing.
—Kass and Wasserman, 1996—
78 / 785
79.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Laplace’s prior
Principle of Insuﬃcient Reason (Laplace)
Θ = {θ1 , · · · , θp } π(θi ) = 1/p
79 / 785
80.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Laplace’s prior
Principle of Insuﬃcient Reason (Laplace)
Θ = {θ1 , · · · , θp } π(θi ) = 1/p
Extension to continuous spaces
π(θ) ∝ 1
[Lebesgue measure]
80 / 785
81.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Who’s Laplace?
Pierre Simon de Laplace (1749–1827)
French mathematician and
astronomer born in Beaumont en
Auge (Normandie) who formalised
mathematical astronomy in
M´canique C´leste. Survived the
e e
French revolution, the Napoleon
Empire (as a comte!), and the
Bourbon restauration (as a
marquis!!).
81 / 785
82.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Who’s Laplace?
Pierre Simon de Laplace (1749–1827)
French mathematician and
astronomer born in Beaumont en
Auge (Normandie) who formalised
mathematical astronomy in
M´canique C´leste. Survived the
e e
French revolution, the Napoleon
Empire (as a comte!), and the
Bourbon restauration (as a
marquis!!).
In Essai Philosophique sur les Probabilit´s, Laplace set out a
e
mathematical system of inductive reasoning based on probability,
precursor to Bayesian Statistics.
82 / 785
83.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Laplace’s problem
Lack of reparameterization invariance/coherence
1
and ψ = eθ
π(θ) ∝ 1, π(ψ) = = 1 (!!)
ψ
83 / 785
84.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Laplace’s problem
Lack of reparameterization invariance/coherence
1
and ψ = eθ
π(θ) ∝ 1, π(ψ) = = 1 (!!)
ψ
Problems of properness
x ∼ N (µ, σ 2 ), π(µ, σ) = 1
2 2
π(µ, σ|x) ∝ e−(x−µ) /2σ σ −1
⇒ π(σ|x) ∝1 (!!!)
84 / 785
85.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Jeﬀreys’ prior
Based on Fisher information
∂ log ℓ ∂ log ℓ
I F (θ) = Eθ
∂θt ∂θ
Ron Fisher (1890–1962)
85 / 785
86.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Jeﬀreys’ prior
Based on Fisher information
∂ log ℓ ∂ log ℓ
I F (θ) = Eθ
∂θt ∂θ
Ron Fisher (1890–1962)
the Jeﬀreys prior distribution is
π J (θ) ∝ |I F (θ)|1/2
86 / 785
87.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Who’s Jeﬀreys?
Sir Harold Jeﬀreys (1891–1989)
English mathematician,
statistician, geophysicist, and
astronomer. Founder of English
Geophysics & originator of the
theory that the Earth core is
liquid.
87 / 785
88.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Who’s Jeﬀreys?
Sir Harold Jeﬀreys (1891–1989)
English mathematician,
statistician, geophysicist, and
astronomer. Founder of English
Geophysics & originator of the
theory that the Earth core is
liquid.
Formalised Bayesian methods for
the analysis of geophysical data
and ended up writing Theory of
Probability
88 / 785
89.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Pros & Cons
Relates to information theory
89 / 785
90.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Pros & Cons
Relates to information theory
Agrees with most invariant priors
90 / 785
91.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Pros & Cons
Relates to information theory
Agrees with most invariant priors
Parameterization invariant
91 / 785
92.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Prior selection
Pros & Cons
Relates to information theory
Agrees with most invariant priors
Parameterization invariant
Suﬀers from dimensionality curse
92 / 785
93.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Evaluating estimators
Purpose of most inferential studies: to provide the
statistician/client with a decision d ∈ D
93 / 785
94.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Evaluating estimators
Purpose of most inferential studies: to provide the
statistician/client with a decision d ∈ D
Requires an evaluation criterion/loss function for decisions and
estimators
L(θ, d)
94 / 785
95.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Evaluating estimators
Purpose of most inferential studies: to provide the
statistician/client with a decision d ∈ D
Requires an evaluation criterion/loss function for decisions and
estimators
L(θ, d)
There exists an axiomatic derivation of the existence of a
loss function.
[DeGroot, 1970]
95 / 785
96.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Loss functions
Decision procedure δ π usually called estimator
(while its value δ π (x) is called estimate of θ)
96 / 785
97.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Loss functions
Decision procedure δ π usually called estimator
(while its value δ π (x) is called estimate of θ)
Impossible to uniformly minimize (in d) the loss function
L(θ, d)
when θ is unknown
97 / 785
98.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Bayesian estimation
Principle Integrate over the space Θ to get the posterior expected
loss
= Eπ [L(θ, d)|x]
= L(θ, d)π(θ|x) dθ,
Θ
and minimise in d
98 / 785
99.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Bayes estimates
Bayes estimator
A Bayes estimate associated with a prior distribution π and a loss
function L is
arg min Eπ [L(θ, d)|x] .
d
99 / 785
100.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
The quadratic loss
Historically, ﬁrst loss function
(Legendre, Gauss, Laplace)
L(θ, d) = (θ − d)2
100 / 785
101.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
The quadratic loss
Historically, ﬁrst loss function
(Legendre, Gauss, Laplace)
L(θ, d) = (θ − d)2
The Bayes estimate δ π (x) associated with the prior π and with the
quadratic loss is the posterior expectation
Θ θf (x|θ)π(θ) dθ
δ π (x) = Eπ [θ|x] = .
Θ f (x|θ)π(θ) dθ
101 / 785
102.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
The absolute error loss
Alternatives to the quadratic loss:
L(θ, d) =| θ − d |,
or
k2 (θ − d) if θ > d,
Lk1 ,k2 (θ, d) =
k1 (d − θ) otherwise.
102 / 785
103.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
The absolute error loss
Alternatives to the quadratic loss:
L(θ, d) =| θ − d |,
or
k2 (θ − d) if θ > d,
Lk1 ,k2 (θ, d) =
k1 (d − θ) otherwise.
Associated Bayes estimate is (k2 /(k1 + k2 )) fractile of π(θ|x)
103 / 785
104.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
MAP estimator
With no loss function, consider using the maximum a posteriori
(MAP) estimator
arg max ℓ(θ|x)π(θ)
θ
104 / 785
105.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
MAP estimator
With no loss function, consider using the maximum a posteriori
(MAP) estimator
arg max ℓ(θ|x)π(θ)
θ
Penalized likelihood estimator
105 / 785
106.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
MAP estimator
With no loss function, consider using the maximum a posteriori
(MAP) estimator
arg max ℓ(θ|x)π(θ)
θ
Penalized likelihood estimator
Further appeal in restricted parameter spaces
106 / 785
107.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Example (Binomial probability)
Consider x|θ ∼ B(n, θ).
Possible priors:
1
π J (θ) = θ−1/2 (1 − θ)−1/2 ,
B(1/2, 1/2)
π2 (θ) = θ−1 (1 − θ)−1 .
π1 (θ) = 1 and
107 / 785
108.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Example (Binomial probability)
Consider x|θ ∼ B(n, θ).
Possible priors:
1
π J (θ) = θ−1/2 (1 − θ)−1/2 ,
B(1/2, 1/2)
π2 (θ) = θ−1 (1 − θ)−1 .
π1 (θ) = 1 and
Corresponding MAP estimators:
x − 1/2
δ πJ (x) = max ,0 ,
n−1
δ π1 (x) = x/n,
x−1
δ π2 (x) = max ,0 .
n−2
108 / 785
109.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Not always appropriate
Example (Fixed MAP)
Consider
1 −1
1 + (x − θ)2
f (x|θ) = ,
π
1
and π(θ) = 2 e−|θ| .
109 / 785
110.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Bayesian estimation
Not always appropriate
Example (Fixed MAP)
Consider
1 −1
1 + (x − θ)2
f (x|θ) = ,
π
1
and π(θ) = 2 e−|θ| . Then the MAP estimate of θ is always
δ π (x) = 0
110 / 785
111.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Conﬁdence regions
Credible regions
Natural conﬁdence region: Highest posterior density (HPD) region
π
Cα = {θ; π(θ|x) > kα }
111 / 785
112.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Conﬁdence regions
Credible regions
Natural conﬁdence region: Highest posterior density (HPD) region
π
Cα = {θ; π(θ|x) > kα }
Optimality
0.15
The HPD regions give the highest
probabilities of containing θ for a
0.10
given volume
0.05
0.00
−10 −5 0 5 10
µ
112 / 785
113.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Conﬁdence regions
Example
If the posterior distribution of θ is N (µ(x), ω −2 ) with
ω 2 = τ −2 + σ −2 and µ(x) = τ 2 x/(τ 2 + σ 2 ), then
Cα = µ(x) − kα ω −1 , µ(x) + kα ω −1 ,
π
where kα is the α/2-quantile of N (0, 1).
113 / 785
114.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Conﬁdence regions
Example
If the posterior distribution of θ is N (µ(x), ω −2 ) with
ω 2 = τ −2 + σ −2 and µ(x) = τ 2 x/(τ 2 + σ 2 ), then
Cα = µ(x) − kα ω −1 , µ(x) + kα ω −1 ,
π
where kα is the α/2-quantile of N (0, 1).
If τ goes to +∞,
π
Cα = [x − kα σ, x + kα σ] ,
the “usual” (classical) conﬁdence interval
114 / 785
115.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Conﬁdence regions
Full normal
Under [almost!] Jeﬀreys prior prior
π(µ, σ 2 ) = 1/σ 2 ,
posterior distribution of (µ, σ)
σ2
µ|σ, x, s2
¯x ∼ N x,
¯ ,
n
n − 1 s2
,x
σ 2 |¯, sx
x2 ∼ IG .
2 2
115 / 785
116.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Conﬁdence regions
Full normal
Under [almost!] Jeﬀreys prior prior
π(µ, σ 2 ) = 1/σ 2 ,
posterior distribution of (µ, σ)
σ2
µ|σ, x, s2
¯x ∼ N x,
¯ ,
n
n − 1 s2
,x
σ 2 |¯, sx
x2 ∼ IG .
2 2
Then
n(¯ − µ)2 (n−3)/2
x
π(µ|¯, s2 ) ∝ ω 1/2 exp −ω exp{−ωs2 /2} dω
xx ω x
2
−n/2
s2 + n(¯ − µ)2
∝ x
x
[Tn−1 distribution]
116 / 785
117.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Conﬁdence regions
Normal credible interval
Derived credible interval on µ
[¯ − tα/2,n−1 sx
x n(n − 1), x + tα/2,n−1 sx
¯ n(n − 1)]
117 / 785
118.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Conﬁdence regions
Normal credible interval
Derived credible interval on µ
[¯ − tα/2,n−1 sx
x n(n − 1), x + tα/2,n−1 sx
¯ n(n − 1)]
normaldata
Corresponding 95% conﬁdence region for µ
[−0.070, −0.013, ]
Since 0 does not belong to this interval, reporting a signiﬁcant
decrease in the number of larcenies between 1991 and 1995 is
acceptable
118 / 785
119.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Testing hypotheses
Deciding about validity of assumptions or restrictions on the
parameter θ from the data, represented as
H0 : θ ∈ Θ0 versus H1 : θ ∈ Θ0
119 / 785
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Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Testing hypotheses
Deciding about validity of assumptions or restrictions on the
parameter θ from the data, represented as
H0 : θ ∈ Θ0 versus H1 : θ ∈ Θ0
Binary outcome of the decision process: accept [coded by 1] or
reject [coded by 0]
D = {0, 1}
120 / 785
121.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Testing hypotheses
Deciding about validity of assumptions or restrictions on the
parameter θ from the data, represented as
H0 : θ ∈ Θ0 versus H1 : θ ∈ Θ0
Binary outcome of the decision process: accept [coded by 1] or
reject [coded by 0]
D = {0, 1}
Bayesian solution formally very close from a likelihood ratio test
statistic, but numerical values often strongly diﬀer from classical
solutions
121 / 785
122.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
The 0 − 1 loss
Rudimentary loss function
1−d if θ ∈ Θ0
L(θ, d) =
d otherwise,
Associated Bayes estimate
1 if P π (θ ∈ Θ |x) > 1 ,
0
π
δ (x) = 2
0 otherwise.
122 / 785
123.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
The 0 − 1 loss
Rudimentary loss function
1−d if θ ∈ Θ0
L(θ, d) =
d otherwise,
Associated Bayes estimate
1 if P π (θ ∈ Θ |x) > 1 ,
0
π
δ (x) = 2
0 otherwise.
Intuitive structure
123 / 785
124.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Extension
Weighted 0 − 1 (or a0 − a1 ) loss
0 if d = IΘ0 (θ),
L(θ, d) = a0 if θ ∈ Θ0 and d = 0,
a1 if θ ∈ Θ0 and d = 1,
124 / 785
125.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Extension
Weighted 0 − 1 (or a0 − a1 ) loss
0 if d = IΘ0 (θ),
L(θ, d) = a0 if θ ∈ Θ0 and d = 0,
a1 if θ ∈ Θ0 and d = 1,
Associated Bayes estimator
a1
1 if P π (θ ∈ Θ0 |x) > ,
a0 + a1
π
δ (x) =
0 otherwise.
125 / 785
126.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Example (Normal-normal)
For x ∼ N (θ, σ 2 ) and θ ∼ N (µ, τ 2 ), π(θ|x) is N (µ(x), ω 2 ) with
σ2µ + τ 2x σ2τ 2
ω2 =
µ(x) = and .
σ2 + τ 2 σ2 + τ 2
126 / 785
127.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Example (Normal-normal)
For x ∼ N (θ, σ 2 ) and θ ∼ N (µ, τ 2 ), π(θ|x) is N (µ(x), ω 2 ) with
σ2µ + τ 2x σ2τ 2
ω2 =
µ(x) = and .
σ2 + τ 2 σ2 + τ 2
To test H0 : θ < 0, we compute
θ − µ(x) −µ(x)
P π (θ < 0|x) = P π <
ω ω
= Φ (−µ(x)/ω) .
127 / 785
128.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Example (Normal-normal (2))
If za0 ,a1 is the a1 /(a0 + a1 ) quantile, i.e.,
Φ(za0 ,a1 ) = a1 /(a0 + a1 ) ,
H0 is accepted when
−µ(x) > za0 ,a1 ω,
the upper acceptance bound then being
σ2 σ2
x≤− µ − (1 + 2 )ωza0 ,a1 .
τ2 τ
128 / 785
129.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Bayes factor
Bayesian testing procedure depends on P π (θ ∈ Θ0 |x) or
alternatively on the Bayes factor
{P π (θ ∈ Θ1 |x)/P π (θ ∈ Θ0 |x)}
π
B10 =
{P π (θ ∈ Θ1 )/P π (θ ∈ Θ0 )}
in the absence of loss function parameters a0 and a1
129 / 785
130.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Associated reparameterisations
Corresponding models M1 vs. M0 compared via
P π (M1 |x) P π (M1 )
π
B10 =
P π (M0 |x) P π (M0 )
130 / 785
131.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Associated reparameterisations
Corresponding models M1 vs. M0 compared via
P π (M1 |x) P π (M1 )
π
B10 =
P π (M0 |x) P π (M0 )
If we rewrite the prior as
π(θ) = Pr(θ ∈ Θ1 ) × π1 (θ) + Pr(θ ∈ Θ0 ) × π0 (θ)
then
π
B10 = f (x|θ1 )π1 (θ1 )dθ1 f (x|θ0 )π0 (θ0 )dθ0 = m1 (x)/m0 (x)
[Akin to likelihood ratio]
131 / 785
132.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Jeﬀreys’ scale
π
if log10 (B10 ) varies between 0 and 0.5,
1
the evidence against H0 is poor,
if it is between 0.5 and 1, it is is
2
substantial,
if it is between 1 and 2, it is strong,
3
and
if it is above 2 it is decisive.
4
132 / 785
133.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Point null diﬃculties
If π absolutely continuous,
P π (θ = θ0 ) = 0 . . .
133 / 785
134.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Point null diﬃculties
If π absolutely continuous,
P π (θ = θ0 ) = 0 . . .
How can we test H0 : θ = θ0 ?!
134 / 785
135.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
New prior for new hypothesis
Testing point null diﬃculties requires a modiﬁcation of the prior
distribution so that
π(Θ0 ) > 0 and π(Θ1 ) > 0
(hidden information) or
π(θ) = P π (θ ∈ Θ0 ) × π0 (θ) + P π (θ ∈ Θ1 ) × π1 (θ)
135 / 785
136.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
New prior for new hypothesis
Testing point null diﬃculties requires a modiﬁcation of the prior
distribution so that
π(Θ0 ) > 0 and π(Θ1 ) > 0
(hidden information) or
π(θ) = P π (θ ∈ Θ0 ) × π0 (θ) + P π (θ ∈ Θ1 ) × π1 (θ)
[E.g., when Θ0 = {θ0 }, π0 is Dirac mass at θ0 ]
136 / 785
137.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Posteriors with Dirac masses
If H0 : θ = θ0 (= Θ0 ),
ρ = P π (θ = θ0 ) and π(θ) = ρIθ0 (θ) + (1 − ρ)π1 (θ)
then
f (x|θ0 )ρ
π(Θ0 |x) =
f (x|θ)π(θ) dθ
f (x|θ0 )ρ
=
f (x|θ0 )ρ + (1 − ρ)m1 (x)
with
m1 (x) = f (x|θ)π1 (θ) dθ.
Θ1
137 / 785
138.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Example (Normal-normal)
For x ∼ N (θ, σ 2 ) and θ ∼ N (0, τ 2 ), to test H0 : θ = 0 requires a
modiﬁcation of the prior, with
2 /2τ 2
π1 (θ) ∝ e−θ Iθ=0
and π0 (θ) the Dirac mass in 0
138 / 785
139.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Example (Normal-normal)
For x ∼ N (θ, σ 2 ) and θ ∼ N (0, τ 2 ), to test H0 : θ = 0 requires a
modiﬁcation of the prior, with
2 /2τ 2
π1 (θ) ∝ e−θ Iθ=0
and π0 (θ) the Dirac mass in 0
Then
2 2 2
e−x /2(σ +τ )
m1 (x) σ
√
= 2 2
f (x|0) σ 2 + τ 2 e−x /2σ
τ 2 x2
σ2
= exp ,
σ2 + τ 2 2σ 2 (σ 2 + τ 2 )
139 / 785
140.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Example (cont’d)
and
−1
τ 2 x2
σ2
1−ρ
π(θ = 0|x) = 1 + exp .
σ2 + τ 2 2σ 2 (σ 2 + τ 2 )
ρ
For z = x/σ and ρ = 1/2:
z 0 0.68 1.28 1.96
π(θ = 0|z, τ = σ) 0.586 0.557 0.484 0.351
π(θ = 0|z, τ = 3.3σ) 0.768 0.729 0.612 0.366
140 / 785
141.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Banning improper priors
Impossibility of using improper priors for testing!
141 / 785
142.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Banning improper priors
Impossibility of using improper priors for testing!
Reason: When using the representation
π(θ) = P π (θ ∈ Θ1 ) × π1 (θ) + P π (θ ∈ Θ0 ) × π0 (θ)
π1 and π0 must be normalised
142 / 785
143.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Example (Normal point null)
When x ∼ N (θ, 1) and H0 : θ = 0, for the improper prior
π(θ) = 1, the prior is transformed as
1 1
π(θ) = I0 (θ) + · Iθ=0 ,
2 2
and
2 /2
e−x
π(θ = 0|x) = +∞
e−x2 /2 + −∞ e−(x−θ)2 /2 dθ
1
√
= .
1 + 2πex2 /2
143 / 785
144.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Example (Normal point null (2))
Consequence: probability of H0 is bounded from above by
√
π(θ = 0|x) ≤ 1/(1 + 2π) = 0.285
x 0.0 1.0 1.65 1.96 2.58
π(θ = 0|x) 0.285 0.195 0.089 0.055 0.014
Regular tests: Agreement with the classical p-value (but...)
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145.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Example (Normal one-sided)
For x ∼ N (θ, 1), π(θ) = 1, and H0 : θ ≤ 0 to test versus
H1 : θ > 0
0
1 2 /2
e−(x−θ)
π(θ ≤ 0|x) = √ dθ = Φ(−x).
2π −∞
The generalized Bayes answer is also the p-value
145 / 785
146.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Example (Normal one-sided)
For x ∼ N (θ, 1), π(θ) = 1, and H0 : θ ≤ 0 to test versus
H1 : θ > 0
0
1 2 /2
e−(x−θ)
π(θ ≤ 0|x) = √ dθ = Φ(−x).
2π −∞
The generalized Bayes answer is also the p-value
normaldata
If π(µ, σ 2 ) = 1/σ 2 ,
π(µ ≥ 0|x) = 0.0021
since µ|x ∼ T89 (−0.0144, 0.000206).
146 / 785
147.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Jeﬀreys–Lindley paradox
Limiting arguments not valid in testing settings: Under a
conjugate prior
−1
τ 2 x2
σ2
1−ρ
π(θ = 0|x) = 1+ exp ,
σ2 + τ 2 2σ 2 (σ 2 + τ 2 )
ρ
which converges to 1 when τ goes to +∞, for every x
147 / 785
148.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Jeﬀreys–Lindley paradox
Limiting arguments not valid in testing settings: Under a
conjugate prior
−1
τ 2 x2
σ2
1−ρ
π(θ = 0|x) = 1+ exp ,
σ2 + τ 2 2σ 2 (σ 2 + τ 2 )
ρ
which converges to 1 when τ goes to +∞, for every x
Diﬀerence with the “noninformative” answer
√
[1 + 2π exp(x2 /2)]−1
[ Invalid answer]
148 / 785
149.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Normalisation diﬃculties
If g0 and g1 are σ-ﬁnite measures on the subspaces Θ0 and Θ1 , the
choice of the normalizing constants inﬂuences the Bayes factor:
If gi replaced by ci gi (i = 0, 1), Bayes factor multiplied by
c0 /c1
149 / 785
150.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Testing
Normalisation diﬃculties
If g0 and g1 are σ-ﬁnite measures on the subspaces Θ0 and Θ1 , the
choice of the normalizing constants inﬂuences the Bayes factor:
If gi replaced by ci gi (i = 0, 1), Bayes factor multiplied by
c0 /c1
Example
If the Jeﬀreys prior is uniform and g0 = c0 , g1 = c1 ,
ρc0 f (x|θ) dθ
Θ0
π(θ ∈ Θ0 |x) =
ρc0 f (x|θ) dθ + (1 − ρ)c1 f (x|θ) dθ
Θ0 Θ1
ρ f (x|θ) dθ
Θ0
=
ρ f (x|θ) dθ + (1 − ρ)[c1 /c0 ] f (x|θ) dθ
Θ0 Θ1
150 / 785
151.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Monte Carlo integration
Monte Carlo integration
Generic problem of evaluating an integral
I = Ef [h(X)] = h(x) f (x) dx
X
where X is uni- or multidimensional, f is a closed form, partly
closed form, or implicit density, and h is a function
151 / 785
152.
Bayesian Core:A Practical Approach to Computational Bayesian Statistics
The normal model
Monte Carlo integration
Monte Carlo Principle
Use a sample (x1 , . . . , xm ) from the density f to approximate the
integral I by the empirical average
m
1
hm = h(xj )
m
j=1
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