Transcript of "Computational methods for Bayesian model choice"
1.
On some computational methods for Bayesian model choice
On some computational methods for Bayesian
model choice
Christian P. Robert
CREST-INSEE and Universit´ Paris Dauphine
e
http://www.ceremade.dauphine.fr/~xian
c Cours OFPR, CREST, Malakoﬀ 2-12 mars 2009
2.
On some computational methods for Bayesian model choice
Outline
Introduction
1
Importance sampling solutions
2
Cross-model solutions
3
Nested sampling
4
ABC model choice
5
3.
On some computational methods for Bayesian model choice
Introduction
Bayes tests
Construction of Bayes tests
Deﬁnition (Test)
Given an hypothesis H0 : θ ∈ Θ0 on the parameter θ ∈ Θ0 of a
statistical model, a test is a statistical procedure that takes its
values in {0, 1}.
Example (Normal mean)
For x ∼ N (θ, 1), decide whether or not θ ≤ 0.
4.
On some computational methods for Bayesian model choice
Introduction
Bayes tests
Construction of Bayes tests
Deﬁnition (Test)
Given an hypothesis H0 : θ ∈ Θ0 on the parameter θ ∈ Θ0 of a
statistical model, a test is a statistical procedure that takes its
values in {0, 1}.
Example (Normal mean)
For x ∼ N (θ, 1), decide whether or not θ ≤ 0.
5.
On some computational methods for Bayesian model choice
Introduction
Bayes tests
The 0 − 1 loss
Neyman–Pearson loss for testing hypotheses
Test of H0 : θ ∈ Θ0 versus H1 : θ ∈ Θ0 .
Then
D = {0, 1}
The 0 − 1 loss
1 − d if θ ∈ Θ0
L(θ, d) =
d otherwise,
6.
On some computational methods for Bayesian model choice
Introduction
Bayes tests
The 0 − 1 loss
Neyman–Pearson loss for testing hypotheses
Test of H0 : θ ∈ Θ0 versus H1 : θ ∈ Θ0 .
Then
D = {0, 1}
The 0 − 1 loss
1 − d if θ ∈ Θ0
L(θ, d) =
d otherwise,
7.
On some computational methods for Bayesian model choice
Introduction
Bayes tests
Type–one and type–two errors
Associated with the risk
R(θ, δ) = Eθ [L(θ, δ(x))]
if θ ∈ Θ0 ,
Pθ (δ(x) = 0)
=
Pθ (δ(x) = 1) otherwise,
Theorem (Bayes test)
The Bayes estimator associated with π and with the 0 − 1 loss is
if π(θ ∈ Θ0 |x) > π(θ ∈ Θ0 |x),
1
δ π (x) =
0 otherwise,
8.
On some computational methods for Bayesian model choice
Introduction
Bayes tests
Type–one and type–two errors
Associated with the risk
R(θ, δ) = Eθ [L(θ, δ(x))]
if θ ∈ Θ0 ,
Pθ (δ(x) = 0)
=
Pθ (δ(x) = 1) otherwise,
Theorem (Bayes test)
The Bayes estimator associated with π and with the 0 − 1 loss is
if π(θ ∈ Θ0 |x) > π(θ ∈ Θ0 |x),
1
δ π (x) =
0 otherwise,
9.
On some computational methods for Bayesian model choice
Introduction
Bayes factor
Bayes factor
Deﬁnition (Bayes factors)
For testing hypotheses H0 : θ ∈ Θ0 vs. Ha : θ ∈ Θ0 , under prior
π(Θ0 )π0 (θ) + π(Θc )π1 (θ) ,
0
central quantity
f (x|θ)π0 (θ)dθ
π(Θ0 |x) π(Θ0 ) Θ0
B01 = =
π(Θc |x) π(Θc )
0 0 f (x|θ)π1 (θ)dθ
Θc
0
[Jeﬀreys, 1939]
10.
On some computational methods for Bayesian model choice
Introduction
Bayes factor
Self-contained concept
Outside decision-theoretic environment:
eliminates impact of π(Θ0 ) but depends on the choice of
(π0 , π1 )
Bayesian/marginal equivalent to the likelihood ratio
Jeﬀreys’ scale of evidence:
π
if log10 (B10 ) between 0 and 0.5, evidence against H0 weak,
π
if log10 (B10 ) 0.5 and 1, evidence substantial,
π
if log10 (B10 ) 1 and 2, evidence strong and
π
if log10 (B10 ) above 2, evidence decisive
Requires the computation of the marginal/evidence under
both hypotheses/models
11.
On some computational methods for Bayesian model choice
Introduction
Bayes factor
Hot hand
Example (Binomial homogeneity)
Consider H0 : yi ∼ B(ni , p) (i = 1, . . . , G) vs. H1 : yi ∼ B(ni , pi ).
Conjugate priors pi ∼ Be(α = ξ/ω, β = (1 − ξ)/ω), with a uniform
prior on E[pi |ξ, ω] = ξ and on p (ω is ﬁxed)
1G 1
pyi (1 − pi )ni −yi pα−1 (1 − pi )β−1 d pi
B10 = i i
0 i=1 0
×Γ(1/ω)/[Γ(ξ/ω)Γ((1 − ξ)/ω)]dξ
1
P P
i (ni −yi )
i yi (1 − p)
0p dp
For instance, log10 (B10 ) = −0.79 for ω = 0.005 and G = 138
slightly favours H0 .
[Kass & Raftery, 1995]
12.
On some computational methods for Bayesian model choice
Introduction
Bayes factor
Hot hand
Example (Binomial homogeneity)
Consider H0 : yi ∼ B(ni , p) (i = 1, . . . , G) vs. H1 : yi ∼ B(ni , pi ).
Conjugate priors pi ∼ Be(α = ξ/ω, β = (1 − ξ)/ω), with a uniform
prior on E[pi |ξ, ω] = ξ and on p (ω is ﬁxed)
1G 1
pyi (1 − pi )ni −yi pα−1 (1 − pi )β−1 d pi
B10 = i i
0 i=1 0
×Γ(1/ω)/[Γ(ξ/ω)Γ((1 − ξ)/ω)]dξ
1
P P
i (ni −yi )
i yi (1 − p)
0p dp
For instance, log10 (B10 ) = −0.79 for ω = 0.005 and G = 138
slightly favours H0 .
[Kass & Raftery, 1995]
13.
On some computational methods for Bayesian model choice
Introduction
Bayes factor
Hot hand
Example (Binomial homogeneity)
Consider H0 : yi ∼ B(ni , p) (i = 1, . . . , G) vs. H1 : yi ∼ B(ni , pi ).
Conjugate priors pi ∼ Be(α = ξ/ω, β = (1 − ξ)/ω), with a uniform
prior on E[pi |ξ, ω] = ξ and on p (ω is ﬁxed)
1G 1
pyi (1 − pi )ni −yi pα−1 (1 − pi )β−1 d pi
B10 = i i
0 i=1 0
×Γ(1/ω)/[Γ(ξ/ω)Γ((1 − ξ)/ω)]dξ
1
P P
i (ni −yi )
i yi (1 − p)
0p dp
For instance, log10 (B10 ) = −0.79 for ω = 0.005 and G = 138
slightly favours H0 .
[Kass & Raftery, 1995]
14.
On some computational methods for Bayesian model choice
Introduction
Model choice
Model choice and model comparison
Choice between models
Several models available for the same observation
Mi : x ∼ fi (x|θi ), i∈I
where I can be ﬁnite or inﬁnite
Replace hypotheses with models but keep marginal likelihoods and
Bayes factors
15.
On some computational methods for Bayesian model choice
Introduction
Model choice
Model choice and model comparison
Choice between models
Several models available for the same observation
Mi : x ∼ fi (x|θi ), i∈I
where I can be ﬁnite or inﬁnite
Replace hypotheses with models but keep marginal likelihoods and
Bayes factors
16.
On some computational methods for Bayesian model choice
Introduction
Model choice
Bayesian model choice
Probabilise the entire model/parameter space
allocate probabilities pi to all models Mi
deﬁne priors πi (θi ) for each parameter space Θi
compute
pi fi (x|θi )πi (θi )dθi
Θi
π(Mi |x) =
pj fj (x|θj )πj (θj )dθj
Θj
j
take largest π(Mi |x) to determine “best” model,
or use averaged predictive
π(Mj |x) fj (x |θj )πj (θj |x)dθj
Θj
j
17.
On some computational methods for Bayesian model choice
Introduction
Model choice
Bayesian model choice
Probabilise the entire model/parameter space
allocate probabilities pi to all models Mi
deﬁne priors πi (θi ) for each parameter space Θi
compute
pi fi (x|θi )πi (θi )dθi
Θi
π(Mi |x) =
pj fj (x|θj )πj (θj )dθj
Θj
j
take largest π(Mi |x) to determine “best” model,
or use averaged predictive
π(Mj |x) fj (x |θj )πj (θj |x)dθj
Θj
j
18.
On some computational methods for Bayesian model choice
Introduction
Model choice
Bayesian model choice
Probabilise the entire model/parameter space
allocate probabilities pi to all models Mi
deﬁne priors πi (θi ) for each parameter space Θi
compute
pi fi (x|θi )πi (θi )dθi
Θi
π(Mi |x) =
pj fj (x|θj )πj (θj )dθj
Θj
j
take largest π(Mi |x) to determine “best” model,
or use averaged predictive
π(Mj |x) fj (x |θj )πj (θj |x)dθj
Θj
j
19.
On some computational methods for Bayesian model choice
Introduction
Model choice
Bayesian model choice
Probabilise the entire model/parameter space
allocate probabilities pi to all models Mi
deﬁne priors πi (θi ) for each parameter space Θi
compute
pi fi (x|θi )πi (θi )dθi
Θi
π(Mi |x) =
pj fj (x|θj )πj (θj )dθj
Θj
j
take largest π(Mi |x) to determine “best” model,
or use averaged predictive
π(Mj |x) fj (x |θj )πj (θj |x)dθj
Θj
j
20.
On some computational methods for Bayesian model choice
Introduction
Evidence
Evidence
All these problems end up with a similar quantity, the evidence
Zk = πk (θk )Lk (θk ) dθk ,
Θk
aka the marginal likelihood.
21.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Importance sampling
Paradox
Simulation from f (the true density) is not necessarily optimal
Alternative to direct sampling from f is importance sampling,
based on the alternative representation
f (x)
Ef [h(X)] = h(x) g(x) dx .
g(x)
X
which allows us to use other distributions than f
22.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Importance sampling
Paradox
Simulation from f (the true density) is not necessarily optimal
Alternative to direct sampling from f is importance sampling,
based on the alternative representation
f (x)
Ef [h(X)] = h(x) g(x) dx .
g(x)
X
which allows us to use other distributions than f
23.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Importance sampling algorithm
Evaluation of
Ef [h(X)] = h(x) f (x) dx
X
by
Generate a sample X1 , . . . , Xn from a distribution g
1
Use the approximation
2
m
f (Xj )
1
h(Xj )
m g(Xj )
j=1
24.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Bayes factor approximation
When approximating the Bayes factor
f0 (x|θ0 )π0 (θ0 )dθ0
Θ0
B01 =
f1 (x|θ1 )π1 (θ1 )dθ1
Θ1
use of importance functions and and
0 1
n−1 n0 i i i
i=1 f0 (x|θ0 )π0 (θ0 )/ 0 (θ0 )
0
B01 =
n−1 n1 i i i
i=1 f1 (x|θ1 )π1 (θ1 )/ 1 (θ1 )
1
25.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Bridge sampling
Special case:
If
π1 (θ1 |x) ∝ π1 (θ1 |x)
˜
π2 (θ2 |x) ∝ π2 (θ2 |x)
˜
live on the same space (Θ1 = Θ2 ), then
n
π1 (θi |x)
1 ˜
B12 ≈ θi ∼ π2 (θ|x)
π2 (θi |x)
n ˜
i=1
[Gelman & Meng, 1998; Chen, Shao & Ibrahim, 2000]
26.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Bridge sampling variance
The bridge sampling estimator does poorly if
2
π1 (θ) − π2 (θ)
var(B12 ) 1
=E
2 n π2 (θ)
B12
is large, i.e. if π1 and π2 have little overlap...
27.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Bridge sampling variance
The bridge sampling estimator does poorly if
2
π1 (θ) − π2 (θ)
var(B12 ) 1
=E
2 n π2 (θ)
B12
is large, i.e. if π1 and π2 have little overlap...
29.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
An infamous example
When
1
α(θ) =
π1 (θ)˜2 (θ)
˜ π
harmonic mean approximation to B12
n1
1
1/˜ (θ1i |x)
π
n1
i=1 1
θji ∼ πj (θ|x)
B12 = n2
1
1/˜2 (θ2i |x)
π
n2
i=1
[Newton & Raftery, 1994]
Infamous: Most often leads to an inﬁnite variance!!!
[Radford Neal’s blog, 2008]
30.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
An infamous example
When
1
α(θ) =
π1 (θ)˜2 (θ)
˜ π
harmonic mean approximation to B12
n1
1
1/˜ (θ1i |x)
π
n1
i=1 1
θji ∼ πj (θ|x)
B12 = n2
1
1/˜2 (θ2i |x)
π
n2
i=1
[Newton & Raftery, 1994]
Infamous: Most often leads to an inﬁnite variance!!!
[Radford Neal’s blog, 2008]
31.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
“The Worst Monte Carlo Method Ever”
“The good news is that the Law of Large Numbers guarantees that
this estimator is consistent ie, it will very likely be very close to the
correct answer if you use a suﬃciently large number of points from
the posterior distribution.
The bad news is that the number of points required for this
estimator to get close to the right answer will often be greater
than the number of atoms in the observable universe. The even
worse news is that itws easy for people to not realize this, and to
naively accept estimates that are nowhere close to the correct
value of the marginal likelihood.”
[Radford Neal’s blog, Aug. 23, 2008]
32.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
“The Worst Monte Carlo Method Ever”
“The good news is that the Law of Large Numbers guarantees that
this estimator is consistent ie, it will very likely be very close to the
correct answer if you use a suﬃciently large number of points from
the posterior distribution.
The bad news is that the number of points required for this
estimator to get close to the right answer will often be greater
than the number of atoms in the observable universe. The even
worse news is that itws easy for people to not realize this, and to
naively accept estimates that are nowhere close to the correct
value of the marginal likelihood.”
[Radford Neal’s blog, Aug. 23, 2008]
33.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Optimal bridge sampling
The optimal choice of auxiliary function is
n1 + n2
α=
n1 π1 (θ|x) + n2 π2 (θ|x)
leading to
n1
π2 (θ1i |x)
1 ˜
n1 π1 (θ1i |x) + n2 π2 (θ1i |x)
n1
i=1
B12 ≈ n2
π1 (θ2i |x)
1 ˜
n1 π1 (θ2i |x) + n2 π2 (θ2i |x)
n2
i=1
Back later!
34.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Optimal bridge sampling (2)
Reason:
π1 (θ)π2 (θ)[n1 π1 (θ) + n2 π2 (θ)]α(θ)2 dθ
Var(B12 ) 1
≈ −1
2 2
n1 n2
B12 π1 (θ)π2 (θ)α(θ) dθ
(by the δ method)
Dependence on the unknown normalising constants solved
iteratively
35.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Optimal bridge sampling (2)
Reason:
π1 (θ)π2 (θ)[n1 π1 (θ) + n2 π2 (θ)]α(θ)2 dθ
Var(B12 ) 1
≈ −1
2 2
n1 n2
B12 π1 (θ)π2 (θ)α(θ) dθ
(by the δ method)
Dependence on the unknown normalising constants solved
iteratively
36.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Ratio importance sampling
Another identity:
Eϕ [˜1 (θ)/ϕ(θ)]
π
B12 =
Eϕ [˜2 (θ)/ϕ(θ)]
π
for any density ϕ with suﬃciently large support
[Torrie & Valleau, 1977]
Use of a single sample θ1 , . . . , θn from ϕ
i=1 π1 (θi )/ϕ(θi )
˜
B12 =
i=1 π2 (θi )/ϕ(θi )
˜
37.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Ratio importance sampling
Another identity:
Eϕ [˜1 (θ)/ϕ(θ)]
π
B12 =
Eϕ [˜2 (θ)/ϕ(θ)]
π
for any density ϕ with suﬃciently large support
[Torrie & Valleau, 1977]
Use of a single sample θ1 , . . . , θn from ϕ
i=1 π1 (θi )/ϕ(θi )
˜
B12 =
i=1 π2 (θi )/ϕ(θi )
˜
38.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Ratio importance sampling (2)
Approximate variance:
2
(π1 (θ) − π2 (θ))2
var(B12 ) 1
= Eϕ
2 ϕ(θ)2
n
B12
Optimal choice:
| π1 (θ) − π2 (θ) |
ϕ∗ (θ) =
| π1 (η) − π2 (η) | dη
[Chen, Shao & Ibrahim, 2000]
39.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Ratio importance sampling (2)
Approximate variance:
2
(π1 (θ) − π2 (θ))2
var(B12 ) 1
= Eϕ
2 ϕ(θ)2
n
B12
Optimal choice:
| π1 (θ) − π2 (θ) |
ϕ∗ (θ) =
| π1 (η) − π2 (η) | dη
[Chen, Shao & Ibrahim, 2000]
40.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Improving upon bridge sampler
Theorem 5.5.3: The asymptotic variance of the optimal ratio
importance sampling estimator is smaller than the asymptotic
variance of the optimal bridge sampling estimator
[Chen, Shao, & Ibrahim, 2000]
Does not require the normalising constant
| π1 (η) − π2 (η) | dη
but a simulation from
ϕ∗ (θ) ∝| π1 (θ) − π2 (θ) | .
41.
On some computational methods for Bayesian model choice
Importance sampling solutions
Regular importance
Improving upon bridge sampler
Theorem 5.5.3: The asymptotic variance of the optimal ratio
importance sampling estimator is smaller than the asymptotic
variance of the optimal bridge sampling estimator
[Chen, Shao, & Ibrahim, 2000]
Does not require the normalising constant
| π1 (η) − π2 (η) | dη
but a simulation from
ϕ∗ (θ) ∝| π1 (θ) − π2 (θ) | .
42.
On some computational methods for Bayesian model choice
Importance sampling solutions
Varying dimensions
Generalisation to point null situations
When
π1 (θ1 )dθ1
˜
Θ1
B12 =
π2 (θ2 )dθ2
˜
Θ2
and Θ2 = Θ1 × Ψ, we get θ2 = (θ1 , ψ) and
π1 (θ1 )ω(ψ|θ1 )
˜
B12 = Eπ2
π2 (θ1 , ψ)
˜
holds for any conditional density ω(ψ|θ1 ).
43.
On some computational methods for Bayesian model choice
Importance sampling solutions
Varying dimensions
X-dimen’al bridge sampling
Generalisation of the previous identity:
For any α,
Eπ [˜1 (θ1 )ω(ψ|θ1 )α(θ1 , ψ)]
π
B12 = 2
Eπ1 ×ω [˜2 (θ1 , ψ)α(θ1 , ψ)]
π
and, for any density ϕ,
Eϕ [˜1 (θ1 )ω(ψ|θ1 )/ϕ(θ1 , ψ)]
π
B12 =
Eϕ [˜2 (θ1 , ψ)/ϕ(θ1 , ψ)]
π
[Chen, Shao, & Ibrahim, 2000]
Optimal choice: ω(ψ|θ1 ) = π2 (ψ|θ1 )
[Theorem 5.8.2]
44.
On some computational methods for Bayesian model choice
Importance sampling solutions
Varying dimensions
X-dimen’al bridge sampling
Generalisation of the previous identity:
For any α,
Eπ [˜1 (θ1 )ω(ψ|θ1 )α(θ1 , ψ)]
π
B12 = 2
Eπ1 ×ω [˜2 (θ1 , ψ)α(θ1 , ψ)]
π
and, for any density ϕ,
Eϕ [˜1 (θ1 )ω(ψ|θ1 )/ϕ(θ1 , ψ)]
π
B12 =
Eϕ [˜2 (θ1 , ψ)/ϕ(θ1 , ψ)]
π
[Chen, Shao, & Ibrahim, 2000]
Optimal choice: ω(ψ|θ1 ) = π2 (ψ|θ1 )
[Theorem 5.8.2]
45.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Approximating Zk from a posterior sample
Use of the [harmonic mean] identity
ϕ(θk ) ϕ(θk ) πk (θk )Lk (θk ) 1
Eπk x= dθk =
πk (θk )Lk (θk ) πk (θk )Lk (θk ) Zk Zk
no matter what the proposal ϕ(·) is.
[Gelfand & Dey, 1994; Bartolucci et al., 2006]
Direct exploitation of the MCMC output
RB-RJ
46.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Approximating Zk from a posterior sample
Use of the [harmonic mean] identity
ϕ(θk ) ϕ(θk ) πk (θk )Lk (θk ) 1
Eπk x= dθk =
πk (θk )Lk (θk ) πk (θk )Lk (θk ) Zk Zk
no matter what the proposal ϕ(·) is.
[Gelfand & Dey, 1994; Bartolucci et al., 2006]
Direct exploitation of the MCMC output
RB-RJ
47.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Comparison with regular importance sampling
Harmonic mean: Constraint opposed to usual importance sampling
constraints: ϕ(θ) must have lighter (rather than fatter) tails than
πk (θk )Lk (θk ) for the approximation
T (t)
ϕ(θk )
1
Z1k = 1 (t) (t)
T πk (θk )Lk (θk )
t=1
to have a ﬁnite variance.
E.g., use ﬁnite support kernels (like Epanechnikov’s kernel) for ϕ
48.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Comparison with regular importance sampling
Harmonic mean: Constraint opposed to usual importance sampling
constraints: ϕ(θ) must have lighter (rather than fatter) tails than
πk (θk )Lk (θk ) for the approximation
T (t)
ϕ(θk )
1
Z1k = 1 (t) (t)
T πk (θk )Lk (θk )
t=1
to have a ﬁnite variance.
E.g., use ﬁnite support kernels (like Epanechnikov’s kernel) for ϕ
49.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Comparison with regular importance sampling (cont’d)
Compare Z1k with a standard importance sampling approximation
T (t) (t)
πk (θk )Lk (θk )
1
=
Z2k (t)
T ϕ(θk )
t=1
(t)
where the θk ’s are generated from the density ϕ(·) (with fatter
tails like t’s)
50.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Approximating Zk using a mixture representation
Bridge sampling redux
Design a speciﬁc mixture for simulation [importance sampling]
purposes, with density
ϕk (θk ) ∝ ω1 πk (θk )Lk (θk ) + ϕ(θk ) ,
where ϕ(·) is arbitrary (but normalised)
Note: ω1 is not a probability weight
51.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Approximating Zk using a mixture representation
Bridge sampling redux
Design a speciﬁc mixture for simulation [importance sampling]
purposes, with density
ϕk (θk ) ∝ ω1 πk (θk )Lk (θk ) + ϕ(θk ) ,
where ϕ(·) is arbitrary (but normalised)
Note: ω1 is not a probability weight
52.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Approximating Z using a mixture representation (cont’d)
Corresponding MCMC (=Gibbs) sampler
At iteration t
Take δ (t) = 1 with probability
1
(t−1) (t−1) (t−1) (t−1) (t−1)
ω1 πk (θk )Lk (θk ) ω1 πk (θk )Lk (θk ) + ϕ(θk )
and δ (t) = 2 otherwise;
(t) (t−1)
If δ (t) = 1, generate θk ∼ MCMC(θk , θk ) where
2
MCMC(θk , θk ) denotes an arbitrary MCMC kernel associated
with the posterior πk (θk |x) ∝ πk (θk )Lk (θk );
(t)
If δ (t) = 2, generate θk ∼ ϕ(θk ) independently
3
53.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Approximating Z using a mixture representation (cont’d)
Corresponding MCMC (=Gibbs) sampler
At iteration t
Take δ (t) = 1 with probability
1
(t−1) (t−1) (t−1) (t−1) (t−1)
ω1 πk (θk )Lk (θk ) ω1 πk (θk )Lk (θk ) + ϕ(θk )
and δ (t) = 2 otherwise;
(t) (t−1)
If δ (t) = 1, generate θk ∼ MCMC(θk , θk ) where
2
MCMC(θk , θk ) denotes an arbitrary MCMC kernel associated
with the posterior πk (θk |x) ∝ πk (θk )Lk (θk );
(t)
If δ (t) = 2, generate θk ∼ ϕ(θk ) independently
3
54.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Approximating Z using a mixture representation (cont’d)
Corresponding MCMC (=Gibbs) sampler
At iteration t
Take δ (t) = 1 with probability
1
(t−1) (t−1) (t−1) (t−1) (t−1)
ω1 πk (θk )Lk (θk ) ω1 πk (θk )Lk (θk ) + ϕ(θk )
and δ (t) = 2 otherwise;
(t) (t−1)
If δ (t) = 1, generate θk ∼ MCMC(θk , θk ) where
2
MCMC(θk , θk ) denotes an arbitrary MCMC kernel associated
with the posterior πk (θk |x) ∝ πk (θk )Lk (θk );
(t)
If δ (t) = 2, generate θk ∼ ϕ(θk ) independently
3
55.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Evidence approximation by mixtures
Rao-Blackwellised estimate
T
ˆ1 (t) (t) (t) (t) (t)
ξ= ω1 πk (θk )Lk (θk ) ω1 πk (θk )Lk (θk ) + ϕ(θk ) ,
T
t=1
converges to ω1 Zk /{ω1 Zk + 1}
ˆ
Deduce Zˆ from ω1 Z3k /{ω1 Z3k + 1} = ξ ie
ˆ ˆ
3k
(t) (t) (t) (t) (t)
T
t=1 ω1 πk (θk )Lk (θk ) ω1 π(θk )Lk (θk ) + ϕ(θk )
ˆ
Z3k =
(t) (t) (t) (t)
T
t=1 ϕ(θk ) ω1 πk (θk )Lk (θk ) + ϕ(θk )
[Bridge sampler]
56.
On some computational methods for Bayesian model choice
Importance sampling solutions
Harmonic means
Evidence approximation by mixtures
Rao-Blackwellised estimate
T
ˆ1 (t) (t) (t) (t) (t)
ξ= ω1 πk (θk )Lk (θk ) ω1 πk (θk )Lk (θk ) + ϕ(θk ) ,
T
t=1
converges to ω1 Zk /{ω1 Zk + 1}
ˆ
Deduce Zˆ from ω1 Z3k /{ω1 Z3k + 1} = ξ ie
ˆ ˆ
3k
(t) (t) (t) (t) (t)
T
t=1 ω1 πk (θk )Lk (θk ) ω1 π(θk )Lk (θk ) + ϕ(θk )
ˆ
Z3k =
(t) (t) (t) (t)
T
t=1 ϕ(θk ) ω1 πk (θk )Lk (θk ) + ϕ(θk )
[Bridge sampler]
57.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Chib’s representation
Direct application of Bayes’ theorem: given x ∼ fk (x|θk ) and
θk ∼ πk (θk ),
fk (x|θk ) πk (θk )
Zk = mk (x) =
πk (θk |x)
Use of an approximation to the posterior
∗ ∗
fk (x|θk ) πk (θk )
Zk = mk (x) = .
ˆ∗
πk (θk |x)
58.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Chib’s representation
Direct application of Bayes’ theorem: given x ∼ fk (x|θk ) and
θk ∼ πk (θk ),
fk (x|θk ) πk (θk )
Zk = mk (x) =
πk (θk |x)
Use of an approximation to the posterior
∗ ∗
fk (x|θk ) πk (θk )
Zk = mk (x) = .
ˆ∗
πk (θk |x)
59.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Case of latent variables
For missing variable z as in mixture models, natural Rao-Blackwell
estimate
T
1 (t)
∗ ∗
πk (θk |x) = πk (θk |x, zk ) ,
T
t=1
(t)
where the zk ’s are Gibbs sampled latent variables
60.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Label switching
A mixture model [special case of missing variable model] is
invariant under permutations of the indices of the components.
E.g., mixtures
0.3N (0, 1) + 0.7N (2.3, 1)
and
0.7N (2.3, 1) + 0.3N (0, 1)
are exactly the same!
c The component parameters θi are not identiﬁable
marginally since they are exchangeable
61.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Label switching
A mixture model [special case of missing variable model] is
invariant under permutations of the indices of the components.
E.g., mixtures
0.3N (0, 1) + 0.7N (2.3, 1)
and
0.7N (2.3, 1) + 0.3N (0, 1)
are exactly the same!
c The component parameters θi are not identiﬁable
marginally since they are exchangeable
62.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Connected diﬃculties
Number of modes of the likelihood of order O(k!):
1
c Maximization and even [MCMC] exploration of the
posterior surface harder
Under exchangeable priors on (θ, p) [prior invariant under
2
permutation of the indices], all posterior marginals are
identical:
c Posterior expectation of θ1 equal to posterior expectation
of θ2
63.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Connected diﬃculties
Number of modes of the likelihood of order O(k!):
1
c Maximization and even [MCMC] exploration of the
posterior surface harder
Under exchangeable priors on (θ, p) [prior invariant under
2
permutation of the indices], all posterior marginals are
identical:
c Posterior expectation of θ1 equal to posterior expectation
of θ2
64.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
License
Since Gibbs output does not produce exchangeability, the Gibbs
sampler has not explored the whole parameter space: it lacks
energy to switch simultaneously enough component allocations at
once
0.2 0.3 0.4 0.5
−1 0 1 2 3
µi
pi −1 0 1 2 3
0 100 200 300 400 500
µ
n i
0.4 0.6 0.8 1.0
0.2 0.3 0.4 0.5
σi
pi
0 100 200 300 400 500 0.2 0.3 0.4 0.5
n pi
0.4 0.6 0.8 1.0
−1 0 1 2 3
σi
µi
0 100 200 300 400 500 0.4 0.6 0.8 1.0
σ
n i
65.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Label switching paradox
We should observe the exchangeability of the components [label
switching] to conclude about convergence of the Gibbs sampler.
If we observe it, then we do not know how to estimate the
parameters.
If we do not, then we are uncertain about the convergence!!!
66.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Label switching paradox
We should observe the exchangeability of the components [label
switching] to conclude about convergence of the Gibbs sampler.
If we observe it, then we do not know how to estimate the
parameters.
If we do not, then we are uncertain about the convergence!!!
67.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Label switching paradox
We should observe the exchangeability of the components [label
switching] to conclude about convergence of the Gibbs sampler.
If we observe it, then we do not know how to estimate the
parameters.
If we do not, then we are uncertain about the convergence!!!
68.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Compensation for label switching
(t)
For mixture models, zk usually fails to visit all conﬁgurations in a
balanced way, despite the symmetry predicted by the theory
1
πk (θk |x) = πk (σ(θk )|x) = πk (σ(θk )|x)
k!
σ∈S
for all σ’s in Sk , set of all permutations of {1, . . . , k}.
Consequences on numerical approximation, biased by an order k!
Recover the theoretical symmetry by using
T
1 (t)
∗ ∗
πk (θk |x) = πk (σ(θk )|x, zk ) .
T k!
σ∈Sk t=1
[Berkhof, Mechelen, & Gelman, 2003]
69.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Compensation for label switching
(t)
For mixture models, zk usually fails to visit all conﬁgurations in a
balanced way, despite the symmetry predicted by the theory
1
πk (θk |x) = πk (σ(θk )|x) = πk (σ(θk )|x)
k!
σ∈S
for all σ’s in Sk , set of all permutations of {1, . . . , k}.
Consequences on numerical approximation, biased by an order k!
Recover the theoretical symmetry by using
T
1 (t)
∗ ∗
πk (θk |x) = πk (σ(θk )|x, zk ) .
T k!
σ∈Sk t=1
[Berkhof, Mechelen, & Gelman, 2003]
70.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Galaxy dataset
n = 82 galaxies as a mixture of k normal distributions with both
mean and variance unknown.
[Roeder, 1992]
Average density
0.8
0.6
Relative Frequency
0.4
0.2
0.0
−2 −1 0 1 2 3
data
71.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Galaxy dataset (k)
∗
Using only the original estimate, with θk as the MAP estimator,
log(mk (x)) = −105.1396
ˆ
for k = 3 (based on 103 simulations), while introducing the
permutations leads to
log(mk (x)) = −103.3479
ˆ
Note that
−105.1396 + log(3!) = −103.3479
k 2 3 4 5 6 7 8
mk (x) -115.68 -103.35 -102.66 -101.93 -102.88 -105.48 -108.44
Estimations of the marginal likelihoods by the symmetrised Chib’s
approximation (based on 105 Gibbs iterations and, for k > 5, 100
permutations selected at random in Sk ).
[Lee, Marin, Mengersen & Robert, 2008]
72.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Galaxy dataset (k)
∗
Using only the original estimate, with θk as the MAP estimator,
log(mk (x)) = −105.1396
ˆ
for k = 3 (based on 103 simulations), while introducing the
permutations leads to
log(mk (x)) = −103.3479
ˆ
Note that
−105.1396 + log(3!) = −103.3479
k 2 3 4 5 6 7 8
mk (x) -115.68 -103.35 -102.66 -101.93 -102.88 -105.48 -108.44
Estimations of the marginal likelihoods by the symmetrised Chib’s
approximation (based on 105 Gibbs iterations and, for k > 5, 100
permutations selected at random in Sk ).
[Lee, Marin, Mengersen & Robert, 2008]
73.
On some computational methods for Bayesian model choice
Importance sampling solutions
Chib’s solution
Galaxy dataset (k)
∗
Using only the original estimate, with θk as the MAP estimator,
log(mk (x)) = −105.1396
ˆ
for k = 3 (based on 103 simulations), while introducing the
permutations leads to
log(mk (x)) = −103.3479
ˆ
Note that
−105.1396 + log(3!) = −103.3479
k 2 3 4 5 6 7 8
mk (x) -115.68 -103.35 -102.66 -101.93 -102.88 -105.48 -108.44
Estimations of the marginal likelihoods by the symmetrised Chib’s
approximation (based on 105 Gibbs iterations and, for k > 5, 100
permutations selected at random in Sk ).
[Lee, Marin, Mengersen & Robert, 2008]
74.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Bayesian variable selection
Regression setting: one dependent random variable y and a set
{x1 , . . . , xk } of k explanatory variables.
Question: Are all xi ’s involved in the regression?
Assumption: every subset {i1 , . . . , iq } of q (0 ≤ q ≤ k)
explanatory variables, {1n , xi1 , . . . , xiq }, is a proper set of
explanatory variables for the regression of y [intercept included in
every corresponding model]
Computational issue
2k models in competition...
75.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Bayesian variable selection
Regression setting: one dependent random variable y and a set
{x1 , . . . , xk } of k explanatory variables.
Question: Are all xi ’s involved in the regression?
Assumption: every subset {i1 , . . . , iq } of q (0 ≤ q ≤ k)
explanatory variables, {1n , xi1 , . . . , xiq }, is a proper set of
explanatory variables for the regression of y [intercept included in
every corresponding model]
Computational issue
2k models in competition...
76.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Bayesian variable selection
Regression setting: one dependent random variable y and a set
{x1 , . . . , xk } of k explanatory variables.
Question: Are all xi ’s involved in the regression?
Assumption: every subset {i1 , . . . , iq } of q (0 ≤ q ≤ k)
explanatory variables, {1n , xi1 , . . . , xiq }, is a proper set of
explanatory variables for the regression of y [intercept included in
every corresponding model]
Computational issue
2k models in competition...
77.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Bayesian variable selection
Regression setting: one dependent random variable y and a set
{x1 , . . . , xk } of k explanatory variables.
Question: Are all xi ’s involved in the regression?
Assumption: every subset {i1 , . . . , iq } of q (0 ≤ q ≤ k)
explanatory variables, {1n , xi1 , . . . , xiq }, is a proper set of
explanatory variables for the regression of y [intercept included in
every corresponding model]
Computational issue
2k models in competition...
78.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Model notations
1
X = 1n x 1 · · · xk
is the matrix containing 1n and all the k potential predictor
variables
Each model Mγ associated with binary indicator vector
2
γ ∈ Γ = {0, 1}k where γi = 1 means that the variable xi is
included in the model Mγ
qγ = 1T γ number of variables included in the model Mγ
3
n
t1 (γ) and t0 (γ) indices of variables included in the model and
4
indices of variables not included in the model
79.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Model indicators
For β ∈ Rk+1 and X, we deﬁne βγ as the subvector
βγ = β0 , (βi )i∈t1 (γ)
and Xγ as the submatrix of X where only the column 1n and the
columns in t1 (γ) have been left.
80.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Models in competition
The model Mγ is thus deﬁned as
y|γ, βγ , σ 2 , X ∼ Nn Xγ βγ , σ 2 In
where βγ ∈ Rqγ +1 and σ 2 ∈ R∗ are the unknown parameters.
+
Warning
σ 2 is common to all models and thus uses the same prior for all
models
81.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Models in competition
The model Mγ is thus deﬁned as
y|γ, βγ , σ 2 , X ∼ Nn Xγ βγ , σ 2 In
where βγ ∈ Rqγ +1 and σ 2 ∈ R∗ are the unknown parameters.
+
Warning
σ 2 is common to all models and thus uses the same prior for all
models
82.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Informative G-prior
Many (2k ) models in competition: we cannot expect a practitioner
to specify a prior on every Mγ in a completely subjective and
autonomous manner.
Shortcut: We derive all priors from a single global prior associated
with the so-called full model that corresponds to γ = (1, . . . , 1).
83.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Prior deﬁnitions
For the full model, Zellner’s G-prior:
(i)
β|σ 2 , X ∼ Nk+1 (β, cσ 2 (X T X)−1 ) and σ 2 ∼ π(σ 2 |X) = σ −2
˜
For each model Mγ , the prior distribution of βγ conditional
(ii)
on σ 2 is ﬁxed as
−1
˜
βγ |γ, σ 2 ∼ Nqγ +1 βγ , cσ 2 Xγ Xγ
T
,
−1
˜ T˜
T Xγ β and same prior on σ 2 .
where βγ = Xγ Xγ
84.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Prior completion
The joint prior for model Mγ is the improper prior
1 T
−(qγ +1)/2−1 ˜
π(βγ , σ 2 |γ) ∝ σ2 exp − βγ − βγ
2(cσ 2 )
˜
T
(Xγ Xγ ) βγ − βγ .
85.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Prior competition (2)
Inﬁnitely many ways of deﬁning a prior on the model index γ:
choice of uniform prior π(γ|X) = 2−k .
Posterior distribution of γ central to variable selection since it is
proportional to marginal density of y on Mγ (or evidence of Mγ )
π(γ|y, X) ∝ f (y|γ, X)π(γ|X) ∝ f (y|γ, X)
f (y|γ, β, σ 2 , X)π(β|γ, σ 2 , X) dβ π(σ 2 |X) dσ 2 .
=
86.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
f (y|γ, σ 2 , X) = f (y|γ, β, σ 2 )π(β|γ, σ 2 ) dβ
1T
−n/2
= (c + 1)−(qγ +1)/2 (2π)−n/2 σ 2 exp − yy
2σ 2
1 −1
˜T T ˜
cy T Xγ Xγ Xγ
T T
Xγ y − βγ Xγ Xγ βγ
+ ,
2σ 2 (c + 1)
this posterior density satisﬁes
−1
cT
π(γ|y, X) ∝ (c + 1)−(qγ +1)/2 y T y − T T
y Xγ Xγ Xγ Xγ y
c+1
−n/2
1 ˜T T ˜
− β X Xγ βγ .
c+1 γ γ
88.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Pine processionary caterpillars (cont’d)
Interpretation
Model Mγ with the highest posterior probability is
t1 (γ) = (1, 2, 4, 5), which corresponds to the variables
- altitude,
- slope,
- height of the tree sampled in the center of the area, and
- diameter of the tree sampled in the center of the area.
Corresponds to the ﬁve variables identiﬁed in the R regression
output
89.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Pine processionary caterpillars (cont’d)
Interpretation
Model Mγ with the highest posterior probability is
t1 (γ) = (1, 2, 4, 5), which corresponds to the variables
- altitude,
- slope,
- height of the tree sampled in the center of the area, and
- diameter of the tree sampled in the center of the area.
Corresponds to the ﬁve variables identiﬁed in the R regression
output
90.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Noninformative extension
For Zellner noninformative prior with π(c) = 1/c, we have
∞
c−1 (c + 1)−(qγ +1)/2 y T y−
π(γ|y, X) ∝
c=1
−n/2
−1
cT T T
y Xγ Xγ Xγ Xγ y .
c+1
92.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Stochastic search for the most likely model
When k gets large, impossible to compute the posterior
probabilities of the 2k models.
Need of a tailored algorithm that samples from π(γ|y, X) and
selects the most likely models.
Can be done by Gibbs sampling, given the availability of the full
conditional posterior probabilities of the γi ’s.
If γ−i = (γ1 , . . . , γi−1 , γi+1 , . . . , γk ) (1 ≤ i ≤ k)
π(γi |y, γ−i , X) ∝ π(γ|y, X)
(to be evaluated in both γi = 0 and γi = 1)
93.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Stochastic search for the most likely model
When k gets large, impossible to compute the posterior
probabilities of the 2k models.
Need of a tailored algorithm that samples from π(γ|y, X) and
selects the most likely models.
Can be done by Gibbs sampling, given the availability of the full
conditional posterior probabilities of the γi ’s.
If γ−i = (γ1 , . . . , γi−1 , γi+1 , . . . , γk ) (1 ≤ i ≤ k)
π(γi |y, γ−i , X) ∝ π(γ|y, X)
(to be evaluated in both γi = 0 and γi = 1)
94.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Gibbs sampling for variable selection
Initialization: Draw γ 0 from the uniform
distribution on Γ
(t−1) (t−1)
Iteration t: Given (γ1 , . . . , γk ), generate
(t) (t−1) (t−1)
1. γ1 according to π(γ1 |y, γ2 , . . . , γk , X)
(t)
2. γ2 according to
(t) (t−1) (t−1)
π(γ2 |y, γ1 , γ3 , . . . , γk , X)
.
.
.
(t) (t) (t)
p. γk according to π(γk |y, γ1 , . . . , γk−1 , X)
95.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
MCMC interpretation
After T 1 MCMC iterations, output used to approximate the
posterior probabilities π(γ|y, X) by empirical averages
T
1
π(γ|y, X) = Iγ (t) =γ .
T − T0 + 1
t=T0
where the T0 ﬁrst values are eliminated as burnin.
And approximation of the probability to include i-th variable,
T
1
P π (γi = 1|y, X) = Iγ (t) =1 .
T − T0 + 1 i
t=T0
96.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
MCMC interpretation
After T 1 MCMC iterations, output used to approximate the
posterior probabilities π(γ|y, X) by empirical averages
T
1
π(γ|y, X) = Iγ (t) =γ .
T − T0 + 1
t=T0
where the T0 ﬁrst values are eliminated as burnin.
And approximation of the probability to include i-th variable,
T
1
P π (γi = 1|y, X) = Iγ (t) =1 .
T − T0 + 1 i
t=T0
97.
On some computational methods for Bayesian model choice
Cross-model solutions
Variable selection
Pine processionary caterpillars
P π (γi = 1|y, X) P π (γi = 1|y, X)
γi
γ1 0.8624 0.8844
γ2 0.7060 0.7716
γ3 0.1482 0.2978
γ4 0.6671 0.7261
γ5 0.6515 0.7006
γ6 0.1678 0.3115
γ7 0.1371 0.2880
γ8 0.1555 0.2876
γ9 0.4039 0.5168
γ10 0.1151 0.2609
˜
Probabilities of inclusion with both informative (β = 011 , c = 100)
and noninformative Zellner’s priors
98.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Reversible jump
Idea: Set up a proper measure–theoretic framework for designing
moves between models Mk
[Green, 1995]
Create a reversible kernel K on H = k {k} × Θk such that
K(x, dy)π(x)dx = K(y, dx)π(y)dy
A B B A
for the invariant density π [x is of the form (k, θ(k) )]
99.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Reversible jump
Idea: Set up a proper measure–theoretic framework for designing
moves between models Mk
[Green, 1995]
Create a reversible kernel K on H = k {k} × Θk such that
K(x, dy)π(x)dx = K(y, dx)π(y)dy
A B B A
for the invariant density π [x is of the form (k, θ(k) )]
100.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Local moves
For a move between two models, M1 and M2 , the Markov chain
being in state θ1 ∈ M1 , denote by K1→2 (θ1 , dθ) and K2→1 (θ2 , dθ)
the corresponding kernels, under the detailed balance condition
π(dθ1 ) K1→2 (θ1 , dθ) = π(dθ2 ) K2→1 (θ2 , dθ) ,
and take, wlog, dim(M2 ) > dim(M1 ).
Proposal expressed as
θ2 = Ψ1→2 (θ1 , v1→2 )
where v1→2 is a random variable of dimension
dim(M2 ) − dim(M1 ), generated as
v1→2 ∼ ϕ1→2 (v1→2 ) .
101.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Local moves
For a move between two models, M1 and M2 , the Markov chain
being in state θ1 ∈ M1 , denote by K1→2 (θ1 , dθ) and K2→1 (θ2 , dθ)
the corresponding kernels, under the detailed balance condition
π(dθ1 ) K1→2 (θ1 , dθ) = π(dθ2 ) K2→1 (θ2 , dθ) ,
and take, wlog, dim(M2 ) > dim(M1 ).
Proposal expressed as
θ2 = Ψ1→2 (θ1 , v1→2 )
where v1→2 is a random variable of dimension
dim(M2 ) − dim(M1 ), generated as
v1→2 ∼ ϕ1→2 (v1→2 ) .
102.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Local moves (2)
In this case, q1→2 (θ1 , dθ2 ) has density
−1
∂Ψ1→2 (θ1 , v1→2 )
ϕ1→2 (v1→2 ) ,
∂(θ1 , v1→2 )
by the Jacobian rule.
Reverse importance link
If probability 1→2 of choosing move to M2 while in M1 ,
acceptance probability reduces to
π(M2 , θ2 ) 2→1 ∂Ψ1→2 (θ1 , v1→2 )
α(θ1 , v1→2 ) = 1∧ .
π(M1 , θ1 ) 1→2 ϕ1→2 (v1→2 ) ∂(θ1 , v1→2 )
c Diﬃcult calibration
103.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Local moves (2)
In this case, q1→2 (θ1 , dθ2 ) has density
−1
∂Ψ1→2 (θ1 , v1→2 )
ϕ1→2 (v1→2 ) ,
∂(θ1 , v1→2 )
by the Jacobian rule.
Reverse importance link
If probability 1→2 of choosing move to M2 while in M1 ,
acceptance probability reduces to
π(M2 , θ2 ) 2→1 ∂Ψ1→2 (θ1 , v1→2 )
α(θ1 , v1→2 ) = 1∧ .
π(M1 , θ1 ) 1→2 ϕ1→2 (v1→2 ) ∂(θ1 , v1→2 )
c Diﬃcult calibration
104.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Local moves (2)
In this case, q1→2 (θ1 , dθ2 ) has density
−1
∂Ψ1→2 (θ1 , v1→2 )
ϕ1→2 (v1→2 ) ,
∂(θ1 , v1→2 )
by the Jacobian rule.
Reverse importance link
If probability 1→2 of choosing move to M2 while in M1 ,
acceptance probability reduces to
π(M2 , θ2 ) 2→1 ∂Ψ1→2 (θ1 , v1→2 )
α(θ1 , v1→2 ) = 1∧ .
π(M1 , θ1 ) 1→2 ϕ1→2 (v1→2 ) ∂(θ1 , v1→2 )
c Diﬃcult calibration
105.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Interpretation
The representation puts us back in a ﬁxed dimension setting:
M1 × V1→2 and M2 in one-to-one relation.
reversibility imposes that θ1 is derived as
(θ1 , v1→2 ) = Ψ−1 (θ2 )
1→2
appears like a regular Metropolis–Hastings move from the
couple (θ1 , v1→2 ) to θ2 when stationary distributions are
π(M1 , θ1 ) × ϕ1→2 (v1→2 ) and π(M2 , θ2 ), and when proposal
distribution is deterministic (??)
106.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Interpretation
The representation puts us back in a ﬁxed dimension setting:
M1 × V1→2 and M2 in one-to-one relation.
reversibility imposes that θ1 is derived as
(θ1 , v1→2 ) = Ψ−1 (θ2 )
1→2
appears like a regular Metropolis–Hastings move from the
couple (θ1 , v1→2 ) to θ2 when stationary distributions are
π(M1 , θ1 ) × ϕ1→2 (v1→2 ) and π(M2 , θ2 ), and when proposal
distribution is deterministic (??)
107.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Pseudo-deterministic reasoning
Consider the proposals
θ2 ∼ N (Ψ1→2 (θ1 , v1→2 ), ε) Ψ1→2 (θ1 , v1→2 ) ∼ N (θ2 , ε)
and
Reciprocal proposal has density
exp −(θ2 − Ψ1→2 (θ1 , v1→2 ))2 /2ε ∂Ψ1→2 (θ1 , v1→2 )
√ ×
∂(θ1 , v1→2 )
2πε
by the Jacobian rule.
Thus Metropolis–Hastings acceptance probability is
π(M2 , θ2 ) ∂Ψ1→2 (θ1 , v1→2 )
1∧
π(M1 , θ1 ) ϕ1→2 (v1→2 ) ∂(θ1 , v1→2 )
Does not depend on ε: Let ε go to 0
108.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Pseudo-deterministic reasoning
Consider the proposals
θ2 ∼ N (Ψ1→2 (θ1 , v1→2 ), ε) Ψ1→2 (θ1 , v1→2 ) ∼ N (θ2 , ε)
and
Reciprocal proposal has density
exp −(θ2 − Ψ1→2 (θ1 , v1→2 ))2 /2ε ∂Ψ1→2 (θ1 , v1→2 )
√ ×
∂(θ1 , v1→2 )
2πε
by the Jacobian rule.
Thus Metropolis–Hastings acceptance probability is
π(M2 , θ2 ) ∂Ψ1→2 (θ1 , v1→2 )
1∧
π(M1 , θ1 ) ϕ1→2 (v1→2 ) ∂(θ1 , v1→2 )
Does not depend on ε: Let ε go to 0
109.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Generic reversible jump acceptance probability
If several models are considered simultaneously, with probability
1→2 of choosing move to M2 while in M1 , as in
∞
XZ
K(x, B) = ρm (x, y)qm (x, dy) + ω(x)IB (x)
m=1
acceptance probability of θ2 = Ψ1→2 (θ1 , v1→2 ) is
π(M2 , θ2 ) 2→1 ∂Ψ1→2 (θ1 , v1→2 )
α(θ1 , v1→2 ) = 1 ∧
π(M1 , θ1 ) 1→2 ϕ1→2 (v1→2 ) ∂(θ1 , v1→2 )
while acceptance probability of θ1 with (θ1 , v1→2 ) = Ψ−1 (θ2 ) is
1→2
−1
π(M1 , θ1 ) 1→2 ϕ1→2 (v1→2 ) ∂Ψ1→2 (θ1 , v1→2 )
α(θ1 , v1→2 ) = 1 ∧
π(M2 , θ2 ) 2→1 ∂(θ1 , v1→2 )
110.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Generic reversible jump acceptance probability
If several models are considered simultaneously, with probability
1→2 of choosing move to M2 while in M1 , as in
∞
XZ
K(x, B) = ρm (x, y)qm (x, dy) + ω(x)IB (x)
m=1
acceptance probability of θ2 = Ψ1→2 (θ1 , v1→2 ) is
π(M2 , θ2 ) 2→1 ∂Ψ1→2 (θ1 , v1→2 )
α(θ1 , v1→2 ) = 1 ∧
π(M1 , θ1 ) 1→2 ϕ1→2 (v1→2 ) ∂(θ1 , v1→2 )
while acceptance probability of θ1 with (θ1 , v1→2 ) = Ψ−1 (θ2 ) is
1→2
−1
π(M1 , θ1 ) 1→2 ϕ1→2 (v1→2 ) ∂Ψ1→2 (θ1 , v1→2 )
α(θ1 , v1→2 ) = 1 ∧
π(M2 , θ2 ) 2→1 ∂(θ1 , v1→2 )
111.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Generic reversible jump acceptance probability
If several models are considered simultaneously, with probability
1→2 of choosing move to M2 while in M1 , as in
∞
XZ
K(x, B) = ρm (x, y)qm (x, dy) + ω(x)IB (x)
m=1
acceptance probability of θ2 = Ψ1→2 (θ1 , v1→2 ) is
π(M2 , θ2 ) 2→1 ∂Ψ1→2 (θ1 , v1→2 )
α(θ1 , v1→2 ) = 1 ∧
π(M1 , θ1 ) 1→2 ϕ1→2 (v1→2 ) ∂(θ1 , v1→2 )
while acceptance probability of θ1 with (θ1 , v1→2 ) = Ψ−1 (θ2 ) is
1→2
−1
π(M1 , θ1 ) 1→2 ϕ1→2 (v1→2 ) ∂Ψ1→2 (θ1 , v1→2 )
α(θ1 , v1→2 ) = 1 ∧
π(M2 , θ2 ) 2→1 ∂(θ1 , v1→2 )
112.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Green’s sampler
Algorithm
Iteration t (t ≥ 1): if x(t) = (m, θ(m) ),
Select model Mn with probability πmn
1
Generate umn ∼ ϕmn (u) and set
2
(θ(n) , vnm ) = Ψm→n (θ(m) , umn )
Take x(t+1) = (n, θ(n) ) with probability
3
π(n, θ(n) ) πnm ϕnm (vnm ) ∂Ψm→n (θ(m) , umn )
min ,1
π(m, θ(m) ) πmn ϕmn (umn ) ∂(θ(m) , umn )
and take x(t+1) = x(t) otherwise.
113.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Mixture of normal distributions
k
2
pjk N (µjk , σjk )
Mk = (pjk , µjk , σjk );
j=1
Restrict moves from Mk to adjacent models, like Mk+1 and
Mk−1 , with probabilities πk(k+1) and πk(k−1) .
114.
On some computational methods for Bayesian model choice
Cross-model solutions
Reversible jump
Mixture of normal distributions
k
2
pjk N (µjk , σjk )
Mk = (pjk , µjk , σjk );
j=1
Restrict moves from Mk to adjacent models, like Mk+1 and
Mk−1 , with probabilities πk(k+1) and πk(k−1) .
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