These are my abridged slides for the Italian annual stat conference in Padova/Padua.

1. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling methods for Bayesian
discrimination between embedded models
Christian P. Robert
Universit´ Paris Dauphine & CREST-INSEE
e
http://www.ceremade.dauphine.fr/~xian
45th Scientific Meeting of the Italian Statistical Society
Padova, 16 giugno 2010
Joint work with J.-M. Marin

2. Importance sampling methods for Bayesian discrimination between embedded models
Outline
1 Bayesian model choice
2 Importance sampling model comparison solutions compared

3. Importance sampling methods for Bayesian discrimination between embedded models
Bayesian model choice
Model choice
Model choice as model comparison
Choice between models
Several models available for the same observation
Mi : x ∼ fi (x|θi ), i∈I
Subsitute hypotheses with models

4. Importance sampling methods for Bayesian discrimination between embedded models
Bayesian model choice
Model choice
Model choice as model comparison
Choice between models
Several models available for the same observation
Mi : x ∼ fi (x|θi ), i∈I
Subsitute hypotheses with models

5. Importance sampling methods for Bayesian discrimination between embedded models
Bayesian model choice
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
j Θj
take largest π(Mi |x) to determine “best” model,

6. Importance sampling methods for Bayesian discrimination between embedded models
Bayesian model choice
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
j Θj
take largest π(Mi |x) to determine “best” model,

7. Importance sampling methods for Bayesian discrimination between embedded models
Bayesian model choice
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
j Θj
take largest π(Mi |x) to determine “best” model,

8. Importance sampling methods for Bayesian discrimination between embedded models
Bayesian model choice
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
j Θj
take largest π(Mi |x) to determine “best” model,

9. Importance sampling methods for Bayesian discrimination between embedded models
Bayesian model choice
Bayes factor
Bayes factor
Definition (Bayes factors)
For comparing model M0 with θ ∈ Θ0 vs. M1 with θ ∈ Θ1 , under
priors π0 (θ) and π1 (θ), central quantity
f0 (x|θ0 )π0 (θ0 )dθ0
π(Θ0 |x) π(Θ0 ) Θ0
B01 = =
π(Θ1 |x) π(Θ1 )
f1 (x|θ)π1 (θ1 )dθ1
Θ1
[Jeffreys, 1939]

10. Importance sampling methods for Bayesian discrimination between embedded models
Bayesian model choice
Evidence
Evidence
Problems using a similar quantity, the evidence
Zk = πk (θk )Lk (θk ) dθk ,
Θk
aka the marginal likelihood.
[Jeffreys, 1939]

11. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
A comparison of importance sampling solutions
1 Bayesian model choice
2 Importance sampling model comparison solutions compared
Regular importance
Bridge sampling
Mixtures to bridge
Harmonic means
Chib’s solution
The Savage–Dickey ratio
[Marin & Robert, 2010]

12. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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 0 and 1 and
n−1
0
n0 i i
i=1 f0 (x|θ0 )π0 (θ0 )/
i
0 (θ0 )
B01 =
n−1
1
n1 i i
i=1 f1 (x|θ1 )π1 (θ1 )/
i
1 (θ1 )

13. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Regular importance
Diabetes in Pima Indian women
Example (R benchmark)
“A population of women who were at least 21 years old, of Pima
Indian heritage and living near Phoenix (AZ), was tested for
diabetes according to WHO criteria. The data were collected by
the US National Institute of Diabetes and Digestive and Kidney
Diseases.
200 Pima Indian women with observed variables
plasma glucose concentration in oral glucose tolerance test
diastolic blood pressure
diabetes pedigree function
presence/absence of diabetes

14. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Regular importance
Probit modelling on Pima Indian women
Probability of diabetes function of above variables
P(y = 1|x) = Φ(x1 β1 + x2 β2 + x3 β3 ) ,
Test of H0 : β3 = 0 for 200 observations of Pima.tr based on a
g-prior modelling:
β ∼ N3 (0, n XT X)−1

15. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Regular importance
Probit modelling on Pima Indian women
Probability of diabetes function of above variables
P(y = 1|x) = Φ(x1 β1 + x2 β2 + x3 β3 ) ,
Test of H0 : β3 = 0 for 200 observations of Pima.tr based on a
g-prior modelling:
β ∼ N3 (0, n XT X)−1

16. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Regular importance
Importance sampling for the Pima Indian dataset
Use of the importance function inspired from the MLE estimate
distribution
ˆ ˆ
β ∼ N (β, Σ)
R Importance sampling code
model1=summary(glm(y~-1+X1,family=binomial(link=probit)))
is1=rmvnorm(Niter,mean=model1$coeff[,1],sigma=2*model1$cov.unscaled)
is2=rmvnorm(Niter,mean=model2$coeff[,1],sigma=2*model2$cov.unscaled)
bfis=mean(exp(probitlpost(is1,y,X1)-dmvlnorm(is1,mean=model1$coeff[,1],
sigma=2*model1$cov.unscaled))) / mean(exp(probitlpost(is2,y,X2)-
dmvlnorm(is2,mean=model2$coeff[,1],sigma=2*model2$cov.unscaled)))

17. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Regular importance
Importance sampling for the Pima Indian dataset
Use of the importance function inspired from the MLE estimate
distribution
ˆ ˆ
β ∼ N (β, Σ)
R Importance sampling code
model1=summary(glm(y~-1+X1,family=binomial(link=probit)))
is1=rmvnorm(Niter,mean=model1$coeff[,1],sigma=2*model1$cov.unscaled)
is2=rmvnorm(Niter,mean=model2$coeff[,1],sigma=2*model2$cov.unscaled)
bfis=mean(exp(probitlpost(is1,y,X1)-dmvlnorm(is1,mean=model1$coeff[,1],
sigma=2*model1$cov.unscaled))) / mean(exp(probitlpost(is2,y,X2)-
dmvlnorm(is2,mean=model2$coeff[,1],sigma=2*model2$cov.unscaled)))

18. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Regular importance
Diabetes in Pima Indian women
Comparison of the variation of the Bayes factor approximations
based on 100 replicas for 20, 000 simulations from the prior and
the above MLE importance sampler
5
4
q
3
2
Monte Carlo Importance sampling

19. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Bridge sampling
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
1 π1 (θi |x)
˜
B12 ≈ θi ∼ π2 (θ|x)
n π2 (θi |x)
˜
i=1
[Gelman & Meng, 1998; Chen, Shao & Ibrahim, 2000]

20. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Bridge sampling
(Further) bridge sampling
General identity:
π2 (θ|x)α(θ)π1 (θ|x)dθ
˜
B12 = ∀ α(·)
π1 (θ|x)α(θ)π2 (θ|x)dθ
˜
n1
1
π2 (θ1i |x)α(θ1i )
˜
n1
i=1
≈ n2 θji ∼ πj (θ|x)
1
π1 (θ2i |x)α(θ2i )
˜
n2
i=1

21. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Bridge sampling
Optimal bridge sampling
The optimal choice of auxiliary function is
n1 + n2
α =
n1 π1 (θ|x) + n2 π2 (θ|x)
leading to
n1
1 π2 (θ1i |x)
˜
n1 n1 π1 (θ1i |x) + n2 π2 (θ1i |x)
i=1
B12 ≈ n2
1 π1 (θ2i |x)
˜
n2 n1 π1 (θ2i |x) + n2 π2 (θ2i |x)
i=1
Back later!
Drawback: Dependence on the unknown normalising constants
solved iteratively

22. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Bridge sampling
Optimal bridge sampling
The optimal choice of auxiliary function is
n1 + n2
α =
n1 π1 (θ|x) + n2 π2 (θ|x)
leading to
n1
1 π2 (θ1i |x)
˜
n1 n1 π1 (θ1i |x) + n2 π2 (θ1i |x)
i=1
B12 ≈ n2
1 π1 (θ2i |x)
˜
n2 n1 π1 (θ2i |x) + n2 π2 (θ2i |x)
i=1
Back later!
Drawback: Dependence on the unknown normalising constants
solved iteratively

23. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Bridge sampling
Extension to varying dimensions
When dim(Θ1 ) = dim(Θ2 ), e.g. θ2 = (θ1 , ψ), introduction of a
pseudo-posterior density, ω(ψ|θ1 , x), augmenting π1 (θ1 |x) into
joint distribution
π1 (θ1 |x) × ω(ψ|θ1 , x)
on Θ2 so that
π1 (θ1 |x)α(θ1 , ψ)π2 (θ1 , ψ|x)dθ1 ω(ψ|θ1 , x) dψ
˜
B12 =
π2 (θ1 , ψ|x)α(θ1 , ψ)π1 (θ1 |x)dθ1 ω(ψ|θ1 , x) dψ
˜
π1 (θ1 )ω(ψ|θ1 )
˜ Eϕ [˜1 (θ1 )ω(ψ|θ1 )/ϕ(θ1 , ψ)]
π
= Eπ2 =
π2 (θ1 , ψ)
˜ Eϕ [˜2 (θ1 , ψ)/ϕ(θ1 , ψ)]
π
for any conditional density ω(ψ|θ1 ) and any joint density ϕ.

24. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Bridge sampling
Extension to varying dimensions
When dim(Θ1 ) = dim(Θ2 ), e.g. θ2 = (θ1 , ψ), introduction of a
pseudo-posterior density, ω(ψ|θ1 , x), augmenting π1 (θ1 |x) into
joint distribution
π1 (θ1 |x) × ω(ψ|θ1 , x)
on Θ2 so that
π1 (θ1 |x)α(θ1 , ψ)π2 (θ1 , ψ|x)dθ1 ω(ψ|θ1 , x) dψ
˜
B12 =
π2 (θ1 , ψ|x)α(θ1 , ψ)π1 (θ1 |x)dθ1 ω(ψ|θ1 , x) dψ
˜
π1 (θ1 )ω(ψ|θ1 )
˜ Eϕ [˜1 (θ1 )ω(ψ|θ1 )/ϕ(θ1 , ψ)]
π
= Eπ2 =
π2 (θ1 , ψ)
˜ Eϕ [˜2 (θ1 , ψ)/ϕ(θ1 , ψ)]
π
for any conditional density ω(ψ|θ1 ) and any joint density ϕ.

25. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Bridge sampling
Illustration for the Pima Indian dataset
Use of the MLE induced conditional of β3 given (β1 , β2 ) as a
pseudo-posterior and mixture of both MLE approximations on β3
in bridge sampling estimate
R bridge sampling code
cova=model2$cov.unscaled
expecta=model2$coeff[,1]
covw=cova[3,3]-t(cova[1:2,3])%*%ginv(cova[1:2,1:2])%*%cova[1:2,3]
probit1=hmprobit(Niter,y,X1)
probit2=hmprobit(Niter,y,X2)
pseudo=rnorm(Niter,meanw(probit1),sqrt(covw))
probit1p=cbind(probit1,pseudo)
bfbs=mean(exp(probitlpost(probit2[,1:2],y,X1)+dnorm(probit2[,3],meanw(probit2[,1:2]),
sqrt(covw),log=T))/ (dmvnorm(probit2,expecta,cova)+dnorm(probit2[,3],expecta[3],
cova[3,3])))/ mean(exp(probitlpost(probit1p,y,X2))/(dmvnorm(probit1p,expecta,cova)+
dnorm(pseudo,expecta[3],cova[3,3])))

26. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Bridge sampling
Illustration for the Pima Indian dataset
Use of the MLE induced conditional of β3 given (β1 , β2 ) as a
pseudo-posterior and mixture of both MLE approximations on β3
in bridge sampling estimate
R bridge sampling code
cova=model2$cov.unscaled
expecta=model2$coeff[,1]
covw=cova[3,3]-t(cova[1:2,3])%*%ginv(cova[1:2,1:2])%*%cova[1:2,3]
probit1=hmprobit(Niter,y,X1)
probit2=hmprobit(Niter,y,X2)
pseudo=rnorm(Niter,meanw(probit1),sqrt(covw))
probit1p=cbind(probit1,pseudo)
bfbs=mean(exp(probitlpost(probit2[,1:2],y,X1)+dnorm(probit2[,3],meanw(probit2[,1:2]),
sqrt(covw),log=T))/ (dmvnorm(probit2,expecta,cova)+dnorm(probit2[,3],expecta[3],
cova[3,3])))/ mean(exp(probitlpost(probit1p,y,X2))/(dmvnorm(probit1p,expecta,cova)+
dnorm(pseudo,expecta[3],cova[3,3])))

27. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Bridge sampling
Diabetes in Pima Indian women (cont’d)
Comparison of the variation of the Bayes factor approximations
based on 100 × 20, 000 simulations from the prior (MC), the above
bridge sampler and the above importance sampler
5
4
q
q
3
q
2
MC Bridge IS

28. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Mixtures to bridge
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
[Chopin & Robert, 2010]

29. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Mixtures to bridge
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
[Chopin & Robert, 2010]

30. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Mixtures to bridge
Approximating Z using a mixture representation (cont’d)
Corresponding MCMC (=Gibbs) sampler
At iteration t
1 Take δ (t) = 1 with probability
(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)
2 If δ (t) = 1, generate θk ∼ MCMC(θk , θk ) where
MCMC(θk , θk ) denotes an arbitrary MCMC kernel associated
with the posterior πk (θk |x) ∝ πk (θk )Lk (θk );
(t)
3 If δ (t) = 2, generate θk ∼ ϕ(θk ) independently

31. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Mixtures to bridge
Approximating Z using a mixture representation (cont’d)
Corresponding MCMC (=Gibbs) sampler
At iteration t
1 Take δ (t) = 1 with probability
(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)
2 If δ (t) = 1, generate θk ∼ MCMC(θk , θk ) where
MCMC(θk , θk ) denotes an arbitrary MCMC kernel associated
with the posterior πk (θk |x) ∝ πk (θk )Lk (θk );
(t)
3 If δ (t) = 2, generate θk ∼ ϕ(θk ) independently

32. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Mixtures to bridge
Approximating Z using a mixture representation (cont’d)
Corresponding MCMC (=Gibbs) sampler
At iteration t
1 Take δ (t) = 1 with probability
(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)
2 If δ (t) = 1, generate θk ∼ MCMC(θk , θk ) where
MCMC(θk , θk ) denotes an arbitrary MCMC kernel associated
with the posterior πk (θk |x) ∝ πk (θk )Lk (θk );
(t)
3 If δ (t) = 2, generate θk ∼ ϕ(θk ) independently

33. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Mixtures to bridge
Evidence approximation by mixtures
Rao-Blackwellised estimate
T
ˆ 1
ξ=
(t)
ω1 πk (θk )Lk (θk )
(t) (t) (t)
ω1 πk (θk )Lk (θk ) + ϕ(θk ) ,
(t)
T
t=1
converges to ω1 Zk /{ω1 Zk + 1}
3k
ˆ ˆ ˆ
Deduce Zˆ from ω1 Z3k /{ω1 Z3k + 1} = ξ ie
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]

34. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Mixtures to bridge
Evidence approximation by mixtures
Rao-Blackwellised estimate
T
ˆ 1
ξ=
(t)
ω1 πk (θk )Lk (θk )
(t) (t) (t)
ω1 πk (θk )Lk (θk ) + ϕ(θk ) ,
(t)
T
t=1
converges to ω1 Zk /{ω1 Zk + 1}
3k
ˆ ˆ ˆ
Deduce Zˆ from ω1 Z3k /{ω1 Z3k + 1} = ξ ie
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]

35. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Harmonic means
The original harmonic mean estimator
When θki ∼ πk (θ|x),
T
1 1
T L(θkt |x)
t=1
is an unbiased estimator of 1/mk (x)
[Newton & Raftery, 1994]
Highly dangerous: Most often leads to an infinite variance!!!

36. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Harmonic means
The original harmonic mean estimator
When θki ∼ πk (θ|x),
T
1 1
T L(θkt |x)
t=1
is an unbiased estimator of 1/mk (x)
[Newton & Raftery, 1994]
Highly dangerous: Most often leads to an infinite variance!!!

37. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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

38. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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

39. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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)
1 ϕ(θk )
Z1k = 1 (t) (t)
T πk (θk )Lk (θk )
t=1
to have a finite variance.
E.g., use finite support kernels for ϕ

40. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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)
1 ϕ(θk )
Z1k = 1 (t) (t)
T πk (θk )Lk (θk )
t=1
to have a finite variance.
E.g., use finite support kernels for ϕ

41. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Harmonic means
Comparison with regular importance sampling (cont’d)
Compare Z1k with a standard importance sampling approximation
T (t) (t)
1 πk (θk )Lk (θk )
Z2k = (t)
T ϕ(θk )
t=1
(t)
where the θk ’s are generated from the density ϕ(·) (with fatter
tails like t’s)

42. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Harmonic means
HPD indicator as ϕ
Use the convex hull of MCMC simulations corresponding to the
10% HPD region (easily derived!) and ϕ as indicator:
10
ϕ(θ) = Id(θ,θ(t) )≤
T
t∈HPD

43. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Harmonic means
Diabetes in Pima Indian women (cont’d)
Comparison of the variation of the Bayes factor approximations
based on 100 replicas for 20, 000 simulations for a simulation from
the above harmonic mean sampler and importance samplers
3.102 3.104 3.106 3.108 3.110 3.112 3.114 3.116
q
q
Harmonic mean Importance sampling

44. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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)

45. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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)

46. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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
Skip difficulties...

47. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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

48. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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

49. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Chib’s solution
Connected difficulties
1 Number of modes of the likelihood of order O(k!):
c Maximization and even [MCMC] exploration of the
posterior surface harder
2 Under exchangeable priors on (θ, p) [prior invariant under
permutation of the indices], all posterior marginals are
identical:
c Posterior expectation of θ1 equal to posterior expectation
of θ2

50. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Chib’s solution
Connected difficulties
1 Number of modes of the likelihood of order O(k!):
c Maximization and even [MCMC] exploration of the
posterior surface harder
2 Under exchangeable priors on (θ, p) [prior invariant under
permutation of the indices], all posterior marginals are
identical:
c Posterior expectation of θ1 equal to posterior expectation
of θ2

51. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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
0 100 200
n
300 400 500
pi −1 0
µ
1
i
2 3
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
n σi

52. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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!!!

53. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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!!!

54. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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!!!

55. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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]

56. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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]

57. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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

58. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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]

59. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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]

60. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
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]

61. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Chib’s solution
Case of the probit model
For the completion by z,
1
π (θ|x) =
ˆ π(θ|x, z (t) )
T t
is a simple average of normal densities
R Bridge sampling code
gibbs1=gibbsprobit(Niter,y,X1)
gibbs2=gibbsprobit(Niter,y,X2)
bfchi=mean(exp(dmvlnorm(t(t(gibbs2$mu)-model2$coeff[,1]),mean=rep(0,3),
sigma=gibbs2$Sigma2)-probitlpost(model2$coeff[,1],y,X2)))/
mean(exp(dmvlnorm(t(t(gibbs1$mu)-model1$coeff[,1]),mean=rep(0,2),
sigma=gibbs1$Sigma2)-probitlpost(model1$coeff[,1],y,X1)))

62. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Chib’s solution
Case of the probit model
For the completion by z,
1
π (θ|x) =
ˆ π(θ|x, z (t) )
T t
is a simple average of normal densities
R Bridge sampling code
gibbs1=gibbsprobit(Niter,y,X1)
gibbs2=gibbsprobit(Niter,y,X2)
bfchi=mean(exp(dmvlnorm(t(t(gibbs2$mu)-model2$coeff[,1]),mean=rep(0,3),
sigma=gibbs2$Sigma2)-probitlpost(model2$coeff[,1],y,X2)))/
mean(exp(dmvlnorm(t(t(gibbs1$mu)-model1$coeff[,1]),mean=rep(0,2),
sigma=gibbs1$Sigma2)-probitlpost(model1$coeff[,1],y,X1)))

63. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
Chib’s solution
Diabetes in Pima Indian women (cont’d)
Comparison of the variation of the Bayes factor approximations
based on 100 replicas for 20, 000 simulations for a simulation from
the above Chib’s and importance samplers
q
0.0255
q
q q
q
0.0250
0.0245
0.0240
q
q
Chib's method importance sampling

64. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
The Savage–Dickey ratio
Special representation of the Bayes factor used for simulation
Original version (Dickey, AoMS, 1971)

65. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Savage’s density ratio theorem
Given a test H0 : θ = θ0 in a model f (x|θ, ψ) with a nuisance
parameter ψ, under priors π0 (ψ) and π1 (θ, ψ) such that
π1 (ψ|θ0 ) = π0 (ψ)
then
π1 (θ0 |x)
B01 = ,
π1 (θ0 )
with the obvious notations
π1 (θ) = π1 (θ, ψ)dψ , π1 (θ|x) = π1 (θ, ψ|x)dψ ,
[Dickey, 1971; Verdinelli & Wasserman, 1995]

66. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Measure-theoretic difficulty
Representation depends on the choice of versions of conditional
densities:
π0 (ψ)f (x|θ0 , ψ) dψ
B01 = [by definition]
π1 (θ, ψ)f (x|θ, ψ) dψdθ
π1 (ψ|θ0 )f (x|θ0 , ψ) dψ π1 (θ0 )
= [specific version of π1 (ψ|θ0 )
π1 (θ, ψ)f (x|θ, ψ) dψdθ π1 (θ0 )
and arbitrary version of π1 (θ0 )]
π1 (θ0 , ψ)f (x|θ0 , ψ) dψ
= [specific version of π1 (θ0 , ψ)]
m1 (x)π1 (θ0 )
π1 (θ0 |x)
= [version dependent]
π1 (θ0 )

67. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Measure-theoretic difficulty
Representation depends on the choice of versions of conditional
densities:
π0 (ψ)f (x|θ0 , ψ) dψ
B01 = [by definition]
π1 (θ, ψ)f (x|θ, ψ) dψdθ
π1 (ψ|θ0 )f (x|θ0 , ψ) dψ π1 (θ0 )
= [specific version of π1 (ψ|θ0 )
π1 (θ, ψ)f (x|θ, ψ) dψdθ π1 (θ0 )
and arbitrary version of π1 (θ0 )]
π1 (θ0 , ψ)f (x|θ0 , ψ) dψ
= [specific version of π1 (θ0 , ψ)]
m1 (x)π1 (θ0 )
π1 (θ0 |x)
= [version dependent]
π1 (θ0 )

68. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Choice of density version
c Dickey’s (1971) condition is not a condition:
If
π1 (θ0 |x) π0 (ψ)f (x|θ0 , ψ) dψ
=
π1 (θ0 ) m1 (x)
is chosen as a version, then Savage–Dickey’s representation holds

69. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Choice of density version
c Dickey’s (1971) condition is not a condition:
If
π1 (θ0 |x) π0 (ψ)f (x|θ0 , ψ) dψ
=
π1 (θ0 ) m1 (x)
is chosen as a version, then Savage–Dickey’s representation holds

70. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Savage–Dickey paradox
Verdinelli-Wasserman extension:
π1 (θ0 |x) π1 (ψ|x,θ0 ,x) π0 (ψ)
B01 = E
π1 (θ0 ) π1 (ψ|θ0 )
similarly depends on choices of versions...
...but Monte Carlo implementation relies on specific versions of all
densities without making mention of it
[Chen, Shao & Ibrahim, 2000]

71. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Savage–Dickey paradox
Verdinelli-Wasserman extension:
π1 (θ0 |x) π1 (ψ|x,θ0 ,x) π0 (ψ)
B01 = E
π1 (θ0 ) π1 (ψ|θ0 )
similarly depends on choices of versions...
...but Monte Carlo implementation relies on specific versions of all
densities without making mention of it
[Chen, Shao & Ibrahim, 2000]

72. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Computational implementation
Starting from the (new) prior
π1 (θ, ψ) = π1 (θ)π0 (ψ)
˜
define the associated posterior
π1 (θ, ψ|x) = π0 (ψ)π1 (θ)f (x|θ, ψ) m1 (x)
˜ ˜
and impose
π1 (θ0 |x)
˜ π0 (ψ)f (x|θ0 , ψ) dψ
=
π0 (θ0 ) m1 (x)
˜
to hold.
Then
π1 (θ0 |x) m1 (x)
˜ ˜
B01 =
π1 (θ0 ) m1 (x)

73. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Computational implementation
Starting from the (new) prior
π1 (θ, ψ) = π1 (θ)π0 (ψ)
˜
define the associated posterior
π1 (θ, ψ|x) = π0 (ψ)π1 (θ)f (x|θ, ψ) m1 (x)
˜ ˜
and impose
π1 (θ0 |x)
˜ π0 (ψ)f (x|θ0 , ψ) dψ
=
π0 (θ0 ) m1 (x)
˜
to hold.
Then
π1 (θ0 |x) m1 (x)
˜ ˜
B01 =
π1 (θ0 ) m1 (x)

74. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
First ratio
If (θ(1) , ψ (1) ), . . . , (θ(T ) , ψ (T ) ) ∼ π (θ, ψ|x), then
˜
1
π1 (θ0 |x, ψ (t) )
˜
T t
converges to π1 (θ0 |x) (if the right version is used in θ0 ).
˜
π1 (θ0 )f (x|θ0 , ψ)
π1 (θ0 |x, ψ) =
˜
π1 (θ)f (x|θ, ψ) dθ

75. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Rao–Blackwellisation with latent variables
When π1 (θ0 |x, ψ) unavailable, replace with
˜
T
1
π1 (θ0 |x, z (t) , ψ (t) )
˜
T
t=1
via data completion by latent variable z such that
f (x|θ, ψ) = ˜
f (x, z|θ, ψ) dz
˜
and that π1 (θ, ψ, z|x) ∝ π0 (ψ)π1 (θ)f (x, z|θ, ψ) available in closed
˜
form, including the normalising constant, based on version
π1 (θ0 |x, z, ψ)
˜ ˜
f (x, z|θ0 , ψ)
= .
π1 (θ0 ) ˜
f (x, z|θ, ψ)π1 (θ) dθ

76. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Rao–Blackwellisation with latent variables
When π1 (θ0 |x, ψ) unavailable, replace with
˜
T
1
π1 (θ0 |x, z (t) , ψ (t) )
˜
T
t=1
via data completion by latent variable z such that
f (x|θ, ψ) = ˜
f (x, z|θ, ψ) dz
˜
and that π1 (θ, ψ, z|x) ∝ π0 (ψ)π1 (θ)f (x, z|θ, ψ) available in closed
˜
form, including the normalising constant, based on version
π1 (θ0 |x, z, ψ)
˜ ˜
f (x, z|θ0 , ψ)
= .
π1 (θ0 ) ˜
f (x, z|θ, ψ)π1 (θ) dθ

77. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Bridge revival (1)
Since m1 (x)/m1 (x) is unknown, apparent failure!
˜
Use of the bridge identity
π1 (θ, ψ)f (x|θ, ψ) π1 (ψ|θ) m1 (x)
Eπ1 (θ,ψ|x)
˜
= Eπ1 (θ,ψ|x)
˜
=
π0 (ψ)π1 (θ)f (x|θ, ψ) π0 (ψ) m1 (x)
˜
to (biasedly) estimate m1 (x)/m1 (x) by
˜
T
π1 (ψ (t) |θ(t) )
T
t=1
π0 (ψ (t) )
based on the same sample from π1 .
˜

78. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Bridge revival (1)
Since m1 (x)/m1 (x) is unknown, apparent failure!
˜
Use of the bridge identity
π1 (θ, ψ)f (x|θ, ψ) π1 (ψ|θ) m1 (x)
Eπ1 (θ,ψ|x)
˜
= Eπ1 (θ,ψ|x)
˜
=
π0 (ψ)π1 (θ)f (x|θ, ψ) π0 (ψ) m1 (x)
˜
to (biasedly) estimate m1 (x)/m1 (x) by
˜
T
π1 (ψ (t) |θ(t) )
T
t=1
π0 (ψ (t) )
based on the same sample from π1 .
˜

79. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Bridge revival (2)
Alternative identity
π0 (ψ)π1 (θ)f (x|θ, ψ) π0 (ψ) m1 (x)
˜
Eπ1 (θ,ψ|x) = Eπ1 (θ,ψ|x) =
π1 (θ, ψ)f (x|θ, ψ) π1 (ψ|θ) m1 (x)
¯ ¯
suggests using a second sample (θ(1) , ψ (1) , z (1) ), . . . ,
¯(T ) , ψ (T ) , z (T ) ) ∼ π1 (θ, ψ|x) and the ratio estimate
(θ ¯
T
1 ¯ ¯ ¯
π0 (ψ (t) ) π1 (ψ (t) |θ(t) )
T
t=1
Resulting unbiased estimate:
(t) , ψ (t) ) T ¯
1 t π1 (θ0 |x, z
˜ 1 π0 (ψ (t) )
B01 =
T π1 (θ0 ) T
t=1
π1 (ψ ¯
¯ (t) |θ (t) )

80. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Bridge revival (2)
Alternative identity
π0 (ψ)π1 (θ)f (x|θ, ψ) π0 (ψ) m1 (x)
˜
Eπ1 (θ,ψ|x) = Eπ1 (θ,ψ|x) =
π1 (θ, ψ)f (x|θ, ψ) π1 (ψ|θ) m1 (x)
¯ ¯
suggests using a second sample (θ(1) , ψ (1) , z (1) ), . . . ,
¯(T ) , ψ (T ) , z (T ) ) ∼ π1 (θ, ψ|x) and the ratio estimate
(θ ¯
T
1 ¯ ¯ ¯
π0 (ψ (t) ) π1 (ψ (t) |θ(t) )
T
t=1
Resulting unbiased estimate:
(t) , ψ (t) ) T ¯
1 t π1 (θ0 |x, z
˜ 1 π0 (ψ (t) )
B01 =
T π1 (θ0 ) T
t=1
π1 (ψ ¯
¯ (t) |θ (t) )

81. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Difference with Verdinelli–Wasserman representation
The above leads to the representation
π1 (θ0 |x) π1 (θ,ψ|x) π0 (ψ)
˜
B01 = E
π1 (θ0 ) π1 (ψ|θ)
which shows how our approach differs from Verdinelli and
Wasserman’s
π1 (θ0 |x) π1 (ψ|x,θ0 ,x) π0 (ψ)
B01 = E
π1 (θ0 ) π1 (ψ|θ0 )

82. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Difference with Verdinelli–Wasserman approximation
In terms of implementation,
(t) , ψ (t) ) T ¯
MR 1 t π1 (θ0 |x, z
˜ 1 π0 (ψ (t) )
B01 (x) = ¯ ¯
T π1 (θ0 ) T
t=1
π1 (ψ (t) |θ(t) )
formaly resembles
T T ˜
VW 1 π1 (θ0 |x, z (t) , ψ (t) ) 1 π0 (ψ (t) )
B01 (x) = .
T π1 (θ0 ) T ˜
π1 (ψ (t) |θ0 )
t=1 t=1
But the simulated sequences differ: first average involves
simulations from π1 (θ, ψ, z|x) and from π1 (θ, ψ, z|x), while second
˜
average relies on simulations from π1 (θ, ψ, z|x) and from
π1 (ψ, z|x, θ0 ),

83. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Difference with Verdinelli–Wasserman approximation
In terms of implementation,
(t) , ψ (t) ) T ¯
MR 1 t π1 (θ0 |x, z
˜ 1 π0 (ψ (t) )
B01 (x) = ¯ ¯
T π1 (θ0 ) T
t=1
π1 (ψ (t) |θ(t) )
formaly resembles
T T ˜
VW 1 π1 (θ0 |x, z (t) , ψ (t) ) 1 π0 (ψ (t) )
B01 (x) = .
T π1 (θ0 ) T ˜
π1 (ψ (t) |θ0 )
t=1 t=1
But the simulated sequences differ: first average involves
simulations from π1 (θ, ψ, z|x) and from π1 (θ, ψ, z|x), while second
˜
average relies on simulations from π1 (θ, ψ, z|x) and from
π1 (ψ, z|x, θ0 ),

84. Importance sampling methods for Bayesian discrimination between embedded models
Importance sampling model comparison solutions compared
The Savage–Dickey ratio
Diabetes in Pima Indian women (cont’d)
Comparison of the variation of the Bayes factor approximations
based on 100 replicas for 20, 000 simulations for a simulation from
the above importance, Chib’s, Savage–Dickey’s and bridge
samplers
q
3.4
3.2
3.0
2.8
q
IS Chib Savage−Dickey Bridge