Computational methods for Bayesian model choice

On some computational methods for Bayesian model choice

On some computational methods for Bayesianmodel choice

Christian P. Robert

CREST-INSEE and Universite Paris Dauphinehttp://www.ceremade.dauphine.fr/~xian

c©Cours OFPR, CREST, Malakoff 2-12 mars 2009

Outline

1 Introduction

2 Importance sampling solutions

3 Cross-model solutions

4 Nested sampling

5 ABC model choice

Introduction

Bayes tests

Construction of Bayes tests

Definition (Test)

Given an hypothesis H0 : θ ∈ Θ0 on the parameter θ ∈ Θ0 of astatistical model, a test is a statistical procedure that takes itsvalues in {0, 1}.

Example (Normal mean)

For x ∼ N (θ, 1), decide whether or not θ ≤ 0.

Introduction

Bayes tests

Construction of Bayes tests

Definition (Test)

Given an hypothesis H0 : θ ∈ Θ0 on the parameter θ ∈ Θ0 of astatistical model, a test is a statistical procedure that takes itsvalues in {0, 1}.

Example (Normal mean)

For x ∼ N (θ, 1), decide whether or not θ ≤ 0.

Introduction

Bayes tests

The 0− 1 loss

Neyman–Pearson loss for testing hypotheses

Test of H0 : θ ∈ Θ0 versus H1 : θ 6∈ Θ0.Then

D = {0, 1}

The 0− 1 loss

L(θ, d) =

{1− d if θ ∈ Θ0

d otherwise,

Introduction

Bayes tests

The 0− 1 loss

Neyman–Pearson loss for testing hypotheses

Test of H0 : θ ∈ Θ0 versus H1 : θ 6∈ Θ0.Then

D = {0, 1}

The 0− 1 loss

L(θ, d) =

{1− d if θ ∈ Θ0

d otherwise,

Introduction

Bayes tests

Type–one and type–two errors

Associated with the risk

R(θ, δ) = Eθ[L(θ, δ(x))]

{Pθ(δ(x) = 0) if θ ∈ Θ0,

Pθ(δ(x) = 1) otherwise,

Theorem (Bayes test)

The Bayes estimator associated with π and with the 0− 1 loss is

δπ(x) =

{1 if π(θ ∈ Θ0|x) > π(θ 6∈ Θ0|x),0 otherwise,

Introduction

Bayes tests

Type–one and type–two errors

Associated with the risk

R(θ, δ) = Eθ[L(θ, δ(x))]

{Pθ(δ(x) = 0) if θ ∈ Θ0,

Pθ(δ(x) = 1) otherwise,

Theorem (Bayes test)

The Bayes estimator associated with π and with the 0− 1 loss is

δπ(x) =

{1 if π(θ ∈ Θ0|x) > π(θ 6∈ Θ0|x),0 otherwise,

Introduction

Bayes factor

Definition (Bayes factors)

For testing hypotheses H0 : θ ∈ Θ0 vs. Ha : θ 6∈ Θ0, under prior

π(Θ0)π0(θ) + π(Θc0)π1(θ) ,

central quantity

B01 =π(Θ0|x)π(Θc

/π(Θ0)π(Θc

∫Θ0

f(x|θ)π0(θ)dθ∫Θc0

f(x|θ)π1(θ)dθ

[Jeffreys, 1939]

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:

if log10(Bπ10) between 0 and 0.5, evidence against H0 weak,if log10(Bπ10) 0.5 and 1, evidence substantial,if log10(Bπ10) 1 and 2, evidence strong andif log10(Bπ10) above 2, evidence decisive

Requires the computation of the marginal/evidence underboth hypotheses/models

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 uniformprior on E[pi|ξ, ω] = ξ and on p (ω is fixed)

B10 =∫ 1

G∏i=1

0pyii (1− pi)ni−yipα−1

i (1− pi)β−1d pi

×Γ(1/ω)/[Γ(ξ/ω)Γ((1− ξ)/ω)]dξ∫ 10 p

Pi yi(1− p)

Pi(ni−yi)d p

For instance, log10(B10) = −0.79 for ω = 0.005 and G = 138slightly favours H0.

[Kass & Raftery, 1995]

Introduction

Bayes factor

Hot hand

B10 =∫ 1

G∏i=1

×Γ(1/ω)/[Γ(ξ/ω)Γ((1− ξ)/ω)]dξ∫ 10 p

Pi yi(1− p)

Pi(ni−yi)d p

Introduction

Bayes factor

Hot hand

B10 =∫ 1

G∏i=1

×Γ(1/ω)/[Γ(ξ/ω)Γ((1− ξ)/ω)]dξ∫ 10 p

Pi yi(1− p)

Pi(ni−yi)d p

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 andBayes factors

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 andBayes factors

Introduction

Model choice

Bayesian model choiceProbabilise the entire model/parameter space

allocate probabilities pi to all models Mi

define priors πi(θi) for each parameter space Θi

compute

π(Mi|x) =pi

∫Θi

fi(x|θi)πi(θi)dθi∑j

∫Θj

fj(x|θj)πj(θj)dθj

take largest π(Mi|x) to determine “best” model,or use averaged predictive∑

π(Mj |x)∫

fj(x′|θj)πj(θj |x)dθj

Introduction

Model choice

compute

π(Mi|x) =pi

∫Θi

∫Θj

π(Mj |x)∫

Introduction

Model choice

compute

π(Mi|x) =pi

∫Θi

∫Θj

π(Mj |x)∫

Introduction

Model choice

compute

π(Mi|x) =pi

∫Θi

∫Θj

π(Mj |x)∫

Introduction

Evidence

All these problems end up with a similar quantity, the evidence

Zk =∫

πk(θk)Lk(θk) dθk,

aka the marginal likelihood.

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

Ef [h(X)] =∫X

f(x)g(x)

]g(x) dx .

which allows us to use other distributions than f

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

Ef [h(X)] =∫X

f(x)g(x)

]g(x) dx .

which allows us to use other distributions than f

Regular importance

Importance sampling algorithm

Evaluation of

Ef [h(X)] =∫Xh(x) f(x) dx

1 Generate a sample X1, . . . , Xn from a distribution g

2 Use the approximation

m∑j=1

f(Xj)g(Xj)

Regular importance

Bayes factor approximation

When approximating the Bayes factor

∫Θ0

f0(x|θ0)π0(θ0)dθ0∫Θ1

f1(x|θ1)π1(θ1)dθ1

use of importance functions $0 and $1 and

B01 =n−1

∑n0i=1 f0(x|θi0)π0(θi0)/$0(θi0)

n−11

∑n1i=1 f1(x|θi1)π1(θi1)/$1(θi1)

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

B12 ≈1n

n∑i=1

π1(θi|x)π2(θi|x)

θi ∼ π2(θ|x)

[Gelman & Meng, 1998; Chen, Shao & Ibrahim, 2000]

Regular importance

Bridge sampling variance

The bridge sampling estimator does poorly if

var(B12)B2

[(π1(θ)− π2(θ)

π2(θ)

is large, i.e. if π1 and π2 have little overlap...

Regular importance

Bridge sampling variance

The bridge sampling estimator does poorly if

var(B12)B2

[(π1(θ)− π2(θ)

π2(θ)

is large, i.e. if π1 and π2 have little overlap...

Regular importance

(Further) bridge sampling

In addition

∫π2(θ|x)α(θ)π1(θ|x)dθ∫π1(θ|x)α(θ)π2(θ|x)dθ

∀ α(·)

n1∑i=1

π2(θ1i|x)α(θ1i)

n2∑i=1

π1(θ2i|x)α(θ2i)θji ∼ πj(θ|x)

Regular importance

An infamous example

α(θ) =1

π1(θ)π2(θ)

harmonic mean approximation to B12

n1∑i=1

(θ1i|x)

n2∑i=1

1/π2(θ2i|x)

θji ∼ πj(θ|x)

[Newton & Raftery, 1994]Infamous: Most often leads to an infinite variance!!!

[Radford Neal’s blog, 2008]

Regular importance

An infamous example

α(θ) =1

π1(θ)π2(θ)

harmonic mean approximation to B12

n1∑i=1

(θ1i|x)

n2∑i=1

1/π2(θ2i|x)

θji ∼ πj(θ|x)

[Newton & Raftery, 1994]Infamous: Most often leads to an infinite variance!!!

[Radford Neal’s blog, 2008]

Regular importance

“The Worst Monte Carlo Method Ever”

“The good news is that the Law of Large Numbers guarantees thatthis estimator is consistent ie, it will very likely be very close to thecorrect answer if you use a sufficiently large number of points fromthe posterior distribution.The bad news is that the number of points required for thisestimator to get close to the right answer will often be greaterthan the number of atoms in the observable universe. The evenworse news is that itws easy for people to not realize this, and tonaively accept estimates that are nowhere close to the correctvalue of the marginal likelihood.”

[Radford Neal’s blog, Aug. 23, 2008]

Regular importance

“The Worst Monte Carlo Method Ever”

“The good news is that the Law of Large Numbers guarantees thatthis estimator is consistent ie, it will very likely be very close to thecorrect answer if you use a sufficiently large number of points fromthe posterior distribution.The bad news is that the number of points required for thisestimator to get close to the right answer will often be greaterthan the number of atoms in the observable universe. The evenworse news is that itws easy for people to not realize this, and tonaively accept estimates that are nowhere close to the correctvalue of the marginal likelihood.”

[Radford Neal’s blog, Aug. 23, 2008]

Regular importance

Optimal bridge sampling

The optimal choice of auxiliary function is

α? =n1 + n2

n1π1(θ|x) + n2π2(θ|x)

leading to

B12 ≈

n1∑i=1

π2(θ1i|x)n1π1(θ1i|x) + n2π2(θ1i|x)

n2∑i=1

π1(θ2i|x)n1π1(θ2i|x) + n2π2(θ2i|x)

Back later!

Regular importance

Optimal bridge sampling (2)

Reason:

Var(B12)B2

≈ 1n1n2

{∫π1(θ)π2(θ)[n1π1(θ) + n2π2(θ)]α(θ)2 dθ(∫

π1(θ)π2(θ)α(θ) dθ)2 − 1

(by the δ method)Dependence on the unknown normalising constants solvediteratively

Regular importance

Optimal bridge sampling (2)

Reason:

Var(B12)B2

≈ 1n1n2

{∫π1(θ)π2(θ)[n1π1(θ) + n2π2(θ)]α(θ)2 dθ(∫

π1(θ)π2(θ)α(θ) dθ)2 − 1

(by the δ method)Dependence on the unknown normalising constants solvediteratively

Regular importance

Ratio importance sampling

Another identity:

B12 =Eϕ [π1(θ)/ϕ(θ)]Eϕ [π2(θ)/ϕ(θ)]

for any density ϕ with sufficiently large support[Torrie & Valleau, 1977]

Use of a single sample θ1, . . . , θn from ϕ

B12 =∑

i=1 π1(θi)/ϕ(θi)∑i=1 π2(θi)/ϕ(θi)

Regular importance

Ratio importance sampling

Another identity:

B12 =Eϕ [π1(θ)/ϕ(θ)]Eϕ [π2(θ)/ϕ(θ)]

for any density ϕ with sufficiently large support[Torrie & Valleau, 1977]

Use of a single sample θ1, . . . , θn from ϕ

B12 =∑

i=1 π1(θi)/ϕ(θi)∑i=1 π2(θi)/ϕ(θi)

Regular importance

Ratio importance sampling (2)

Approximate variance:

var(B12)B2

[((π1(θ)− π2(θ))2

ϕ(θ)2

Optimal choice:

ϕ∗(θ) =| π1(θ)− π2(θ) |∫| π1(η)− π2(η) | dη

[Chen, Shao & Ibrahim, 2000]

Regular importance

Ratio importance sampling (2)

Approximate variance:

var(B12)B2

[((π1(θ)− π2(θ))2

ϕ(θ)2

Optimal choice:

ϕ∗(θ) =| π1(θ)− π2(θ) |∫| π1(η)− π2(η) | dη

[Chen, Shao & Ibrahim, 2000]

Regular importance

Improving upon bridge sampler

Theorem 5.5.3: The asymptotic variance of the optimal ratioimportance sampling estimator is smaller than the asymptoticvariance 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) | .

Regular importance

Improving upon bridge sampler

Theorem 5.5.3: The asymptotic variance of the optimal ratioimportance sampling estimator is smaller than the asymptoticvariance 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) | .

Varying dimensions

Generalisation to point null situations

∫Θ1

π1(θ1)dθ1∫Θ2

π2(θ2)dθ2

and Θ2 = Θ1 ×Ψ, we get θ2 = (θ1, ψ) and

B12 = Eπ2

[π1(θ1)ω(ψ|θ1)π2(θ1, ψ)

]holds for any conditional density ω(ψ|θ1).

Varying dimensions

X-dimen’al bridge sampling

Generalisation of the previous identity:For any α,

B12 =Eπ2 [π1(θ1)ω(ψ|θ1)α(θ1, ψ)]Eπ1×ω [π2(θ1, ψ)α(θ1, ψ)]

and, for any density ϕ,

B12 =Eϕ [π1(θ1)ω(ψ|θ1)/ϕ(θ1, ψ)]

Eϕ [π2(θ1, ψ)/ϕ(θ1, ψ)]

[Chen, Shao, & Ibrahim, 2000]Optimal choice: ω(ψ|θ1) = π2(ψ|θ1)

[Theorem 5.8.2]

Varying dimensions

X-dimen’al bridge sampling

Generalisation of the previous identity:For any α,

B12 =Eπ2 [π1(θ1)ω(ψ|θ1)α(θ1, ψ)]Eπ1×ω [π2(θ1, ψ)α(θ1, ψ)]

and, for any density ϕ,

B12 =Eϕ [π1(θ1)ω(ψ|θ1)/ϕ(θ1, ψ)]

Eϕ [π2(θ1, ψ)/ϕ(θ1, ψ)]

[Chen, Shao, & Ibrahim, 2000]Optimal choice: ω(ψ|θ1) = π2(ψ|θ1)

[Theorem 5.8.2]

Harmonic means

Approximating Zk from a posterior sample

Use of the [harmonic mean] identity

ϕ(θk)πk(θk)Lk(θk)

∣∣∣∣x] =∫

πk(θk)Lk(θk)Zk

dθk =1Zk

no matter what the proposal ϕ(·) is.[Gelfand & Dey, 1994; Bartolucci et al., 2006]

Direct exploitation of the MCMC outputRB-RJ

Harmonic means

Approximating Zk from a posterior sample

Use of the [harmonic mean] identity

∣∣∣∣x] =∫

πk(θk)Lk(θk)Zk

dθk =1Zk

no matter what the proposal ϕ(·) is.[Gelfand & Dey, 1994; Bartolucci et al., 2006]

Direct exploitation of the MCMC outputRB-RJ

Harmonic means

Comparison with regular importance sampling

Harmonic mean: Constraint opposed to usual importance samplingconstraints: ϕ(θ) must have lighter (rather than fatter) tails thanπk(θk)Lk(θk) for the approximation

Z1k = 1

T∑t=1

ϕ(θ(t)k )

πk(θ(t)k )Lk(θ

(t)k )

to have a finite variance.E.g., use finite support kernels (like Epanechnikov’s kernel) for ϕ

Harmonic means

Comparison with regular importance sampling

Harmonic mean: Constraint opposed to usual importance samplingconstraints: ϕ(θ) must have lighter (rather than fatter) tails thanπk(θk)Lk(θk) for the approximation

Z1k = 1

T∑t=1

ϕ(θ(t)k )

πk(θ(t)k )Lk(θ

(t)k )

to have a finite variance.E.g., use finite support kernels (like Epanechnikov’s kernel) for ϕ

Harmonic means

Comparison with regular importance sampling (cont’d)

Compare Z1k with a standard importance sampling approximation

Z2k =1T

T∑t=1

πk(θ(t)k )Lk(θ

(t)k )

ϕ(θ(t)k )

where the θ(t)k ’s are generated from the density ϕ(·) (with fatter

tails like t’s)

Harmonic means

Approximating Zk using a mixture representation

Bridge sampling redux

Design a specific mixture for simulation [importance sampling]purposes, with density

ϕk(θk) ∝ ω1πk(θk)Lk(θk) + ϕ(θk) ,

where ϕ(·) is arbitrary (but normalised)Note: ω1 is not a probability weight

Harmonic means

Approximating Zk using a mixture representation

Bridge sampling redux

Design a specific mixture for simulation [importance sampling]purposes, with density

ϕk(θk) ∝ ω1πk(θk)Lk(θk) + ϕ(θk) ,

where ϕ(·) is arbitrary (but normalised)Note: ω1 is not a probability weight

Harmonic means

Approximating Z using a mixture representation (cont’d)

Corresponding MCMC (=Gibbs) sampler

At iteration t

1 Take δ(t) = 1 with probability

ω1πk(θ(t−1)k )Lk(θ

(t−1)k )

/(ω1πk(θ

(t−1)k )Lk(θ

(t−1)k ) + ϕ(θ(t−1)

and δ(t) = 2 otherwise;

2 If δ(t) = 1, generate θ(t)k ∼ MCMC(θ(t−1)

k , θk) whereMCMC(θk, θ′k) denotes an arbitrary MCMC kernel associatedwith the posterior πk(θk|x) ∝ πk(θk)Lk(θk);

3 If δ(t) = 2, generate θ(t)k ∼ ϕ(θk) independently

Harmonic means

At iteration t

(t−1)k )

/(ω1πk(θ

(t−1)k )Lk(θ

(t−1)k ) + ϕ(θ(t−1)

Harmonic means

At iteration t

(t−1)k )

/(ω1πk(θ

(t−1)k )Lk(θ

(t−1)k ) + ϕ(θ(t−1)

Harmonic means

Evidence approximation by mixtures

Rao-Blackwellised estimate

ξ =1T

T∑t=1

ω1πk(θ(t)k )Lk(θ

(t)k )/ω1πk(θ

(t)k )Lk(θ

(t)k ) + ϕ(θ(t)

converges to ω1Zk/{ω1Zk + 1}Deduce Z3k from ω1Z3k/{ω1Z3k + 1} = ξ ie

∑Tt=1 ω1πk(θ

(t)k )Lk(θ

(t)k )/ω1π(θ(t)

k )Lk(θ(t)k ) + ϕ(θ(t)

∑Tt=1 ϕ(θ(t)

k )/ω1πk(θ

(t)k )Lk(θ

(t)k ) + ϕ(θ(t)

[Bridge sampler]

Harmonic means

Evidence approximation by mixtures

Rao-Blackwellised estimate

ξ =1T

T∑t=1

ω1πk(θ(t)k )Lk(θ

(t)k )/ω1πk(θ

(t)k )Lk(θ

(t)k ) + ϕ(θ(t)

converges to ω1Zk/{ω1Zk + 1}Deduce Z3k from ω1Z3k/{ω1Z3k + 1} = ξ ie

∑Tt=1 ω1πk(θ

(t)k )Lk(θ

(t)k )/ω1π(θ(t)

k )Lk(θ(t)k ) + ϕ(θ(t)

∑Tt=1 ϕ(θ(t)

k )/ω1πk(θ

(t)k )Lk(θ

(t)k ) + ϕ(θ(t)

[Bridge sampler]

Chib’s solution

Chib’s representation

Direct application of Bayes’ theorem: given x ∼ fk(x|θk) andθk ∼ πk(θk),

Zk = mk(x) =fk(x|θk)πk(θk)

πk(θk|x)

Use of an approximation to the posterior

Zk = mk(x) =fk(x|θ∗k)πk(θ∗k)

πk(θ∗k|x).

Chib’s solution

Chib’s representation

Direct application of Bayes’ theorem: given x ∼ fk(x|θk) andθk ∼ πk(θk),

Zk = mk(x) =fk(x|θk)πk(θk)

πk(θk|x)

Use of an approximation to the posterior

Zk = mk(x) =fk(x|θ∗k)πk(θ∗k)

πk(θ∗k|x).

Chib’s solution

Case of latent variables

For missing variable z as in mixture models, natural Rao-Blackwellestimate

πk(θ∗k|x) =1T

T∑t=1

πk(θ∗k|x, z(t)k ) ,

where the z(t)k ’s are Gibbs sampled latent variables

Chib’s solution

Label switching

A mixture model [special case of missing variable model] isinvariant under permutations of the indices of the components.E.g., mixtures

0.3N (0, 1) + 0.7N (2.3, 1)

and0.7N (2.3, 1) + 0.3N (0, 1)

are exactly the same!c© The component parameters θi are not identifiablemarginally since they are exchangeable

Chib’s solution

Label switching

A mixture model [special case of missing variable model] isinvariant under permutations of the indices of the components.E.g., mixtures

0.3N (0, 1) + 0.7N (2.3, 1)

and0.7N (2.3, 1) + 0.3N (0, 1)

are exactly the same!c© The component parameters θi are not identifiablemarginally since they are exchangeable

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 underpermutation of the indices], all posterior marginals areidentical:c© Posterior expectation of θ1 equal to posterior expectation

of θ2

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 underpermutation of the indices], all posterior marginals areidentical:c© Posterior expectation of θ1 equal to posterior expectation

of θ2

Chib’s solution

License

Since Gibbs output does not produce exchangeability, the Gibbssampler has not explored the whole parameter space: it lacksenergy to switch simultaneously enough component allocations atonce

0 100 200 300 400 500

−1 0 1 2 3

0.20.3

0.40.5

0 100 200 300 400 500

0.20.3

0.40.5

0.2 0.3 0.4 0.5

0.40.6

0.81.0

0 100 200 300 400 500

0.40.6

0.81.0

0.4 0.6 0.8 1.0

Chib’s solution

Label switching paradox

We should observe the exchangeability of the components [labelswitching] to conclude about convergence of the Gibbs sampler.If we observe it, then we do not know how to estimate theparameters.If we do not, then we are uncertain about the convergence!!!

Chib’s solution

Compensation for label switching

For mixture models, z(t)k usually fails to visit all configurations in a

balanced way, despite the symmetry predicted by the theory

πk(θk|x) = πk(σ(θk)|x) =1k!

∑σ∈S

πk(σ(θk)|x)

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

πk(θ∗k|x) =1T k!

∑σ∈Sk

T∑t=1

πk(σ(θ∗k)|x, z(t)k ) .

[Berkhof, Mechelen, & Gelman, 2003]

Chib’s solution

Compensation for label switching

For mixture models, z(t)k usually fails to visit all configurations in a

balanced way, despite the symmetry predicted by the theory

πk(θk|x) = πk(σ(θk)|x) =1k!

∑σ∈S

πk(σ(θk)|x)

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

πk(θ∗k|x) =1T k!

∑σ∈Sk

T∑t=1

πk(σ(θ∗k)|x, z(t)k ) .

[Berkhof, Mechelen, & Gelman, 2003]

Chib’s solution

Galaxy dataset

n = 82 galaxies as a mixture of k normal distributions with bothmean and variance unknown.

[Roeder, 1992]Average density

−2 −1 0 1 2 3

Chib’s solution

Galaxy dataset (k)Using only the original estimate, with θ∗k as the MAP estimator,

log(mk(x)) = −105.1396

for k = 3 (based on 103 simulations), while introducing thepermutations leads to

log(mk(x)) = −103.3479

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’sapproximation (based on 105 Gibbs iterations and, for k > 5, 100permutations selected at random in Sk).

[Lee, Marin, Mengersen & Robert, 2008]

Chib’s solution

log(mk(x)) = −105.1396

log(mk(x)) = −103.3479

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

Chib’s solution

log(mk(x)) = −105.1396

log(mk(x)) = −103.3479

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

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 ofexplanatory variables for the regression of y [intercept included inevery corresponding model]

Computational issue

2k models in competition...

Variable selection

Computational issue

Variable selection

Computational issue

Variable selection

Computational issue

Variable selection

Model notations

X =[1n x1 · · · xk

]is the matrix containing 1n and all the k potential predictorvariables

2 Each model Mγ associated with binary indicator vectorγ ∈ Γ = {0, 1}k where γi = 1 means that the variable xi isincluded in the model Mγ

3 qγ = 1Tnγ number of variables included in the model Mγ

4 t1(γ) and t0(γ) indices of variables included in the model andindices of variables not included in the model

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 thecolumns in t1(γ) have been left.

Variable selection

Models in competition

The model Mγ is thus defined as

y|γ, βγ , σ2, X ∼ Nn(Xγβγ , σ

where βγ ∈ Rqγ+1 and σ2 ∈ R∗+ are the unknown parameters.

Warning

σ2 is common to all models and thus uses the same prior for allmodels

Variable selection

Models in competition

The model Mγ is thus defined as

y|γ, βγ , σ2, X ∼ Nn(Xγβγ , σ

where βγ ∈ Rqγ+1 and σ2 ∈ R∗+ are the unknown parameters.

Warning

σ2 is common to all models and thus uses the same prior for allmodels

Variable selection

Informative G-prior

Many (2k) models in competition: we cannot expect a practitionerto specify a prior on every Mγ in a completely subjective andautonomous manner.

Shortcut: We derive all priors from a single global prior associatedwith the so-called full model that corresponds to γ = (1, . . . , 1).

Variable selection

Prior definitions

(i) For the full model, Zellner’s G-prior:

β|σ2, X ∼ Nk+1(β, cσ2(XTX)−1) and σ2 ∼ π(σ2|X) = σ−2

(ii) For each model Mγ , the prior distribution of βγ conditionalon σ2 is fixed as

βγ |γ, σ2 ∼ Nqγ+1

(βγ , cσ

2(XTγ Xγ

)−1),

where βγ =(XTγ Xγ

)−1XTγ β and same prior on σ2.

Variable selection

Prior completion

The joint prior for model Mγ is the improper prior

π(βγ , σ2|γ) ∝(σ2)−(qγ+1)/2−1 exp

[− 1

2(cσ2)

(βγ − βγ

(XTγ Xγ)

(βγ − βγ

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 isproportional 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, σ2, X)π(β|γ, σ2, X) dβ

)π(σ2|X) dσ2 .

Variable selection

f(y|γ, σ2, X) =∫f(y|γ, β, σ2)π(β|γ, σ2) dβ

= (c+ 1)−(qγ+1)/2(2π)−n/2(σ2)−n/2

exp(− 1

2σ2yTy

2σ2(c+ 1)

{cyTXγ

(XTγXγ

)−1XTγ y − βT

γXTγXγ βγ

this posterior density satisfies

π(γ|y,X) ∝ (c+ 1)−(qγ+1)/2

[yTy − c

c+ 1yTXγ

(XTγ Xγ

)−1XTγ y

− 1c+ 1

βTγX

Tγ Xγ βγ

]−n/2.

Variable selection

Pine processionary caterpillars

t1(γ) π(γ|y, X)

0,1,2,4,5 0.23160,1,2,4,5,9 0.03740,1,9 0.03440,1,2,4,5,10 0.03280,1,4,5 0.03060,1,2,9 0.02500,1,2,4,5,7 0.02410,1,2,4,5,8 0.02380,1,2,4,5,6 0.02370,1,2,3,4,5 0.02320,1,6,9 0.01460,1,2,3,9 0.01450,9 0.01430,1,2,6,9 0.01350,1,4,5,9 0.01280,1,3,9 0.01170,1,2,8 0.0115

Variable selection

Pine processionary caterpillars (cont’d)

Interpretation

Model Mγ with the highest posterior probability ist1(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 regressionoutput

Variable selection

Pine processionary caterpillars (cont’d)

Interpretation

Model Mγ with the highest posterior probability ist1(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 regressionoutput

Variable selection

Noninformative extension

For Zellner noninformative prior with π(c) = 1/c, we have

π(γ|y,X) ∝∞∑c=1

c−1(c+ 1)−(qγ+1)/2[yTy−

c+ 1yTXγ

(XTγ Xγ

)−1XTγ y

]−n/2.

Variable selection

t1(γ) π(γ|y, X)

0,1,2,4,5 0.09290,1,2,4,5,9 0.03250,1,2,4,5,10 0.02950,1,2,4,5,7 0.02310,1,2,4,5,8 0.02280,1,2,4,5,6 0.02280,1,2,3,4,5 0.02240,1,2,3,4,5,9 0.01670,1,2,4,5,6,9 0.01670,1,2,4,5,8,9 0.01370,1,4,5 0.01100,1,2,4,5,9,10 0.01000,1,2,3,9 0.00970,1,2,9 0.00930,1,2,4,5,7,9 0.00920,1,2,6,9 0.0092

Variable selection

Stochastic search for the most likely model

When k gets large, impossible to compute the posteriorprobabilities of the 2k models.

Need of a tailored algorithm that samples from π(γ|y,X) andselects the most likely models.

Can be done by Gibbs sampling, given the availability of the fullconditional 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)

Variable selection

Stochastic search for the most likely model

When k gets large, impossible to compute the posteriorprobabilities of the 2k models.

Need of a tailored algorithm that samples from π(γ|y,X) andselects the most likely models.

Can be done by Gibbs sampling, given the availability of the fullconditional 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)

Variable selection

Gibbs sampling for variable selection

Initialization: Draw γ0 from the uniformdistribution on Γ

Iteration t: Given (γ(t−1)1 , . . . , γ

(t−1)k ), generate

1. γ(t)1 according to π(γ1|y, γ(t−1)

2 , . . . , γ(t−1)k , X)

2. γ(t)2 according to

π(γ2|y, γ(t)1 , γ

(t−1)3 , . . . , γ

(t−1)k , X)

p. γ(t)k according to π(γk|y, γ(t)

1 , . . . , γ(t)k−1, X)

Variable selection

MCMC interpretation

After T � 1 MCMC iterations, output used to approximate theposterior probabilities π(γ|y,X) by empirical averages

π(γ|y,X) =(

1T − T0 + 1

) T∑t=T0

Iγ(t)=γ .

where the T0 first values are eliminated as burnin.

And approximation of the probability to include i-th variable,

P π(γi = 1|y,X) =(

1T − T0 + 1

) T∑t=T0

Iγ(t)i =1

Variable selection

MCMC interpretation

After T � 1 MCMC iterations, output used to approximate theposterior probabilities π(γ|y,X) by empirical averages

π(γ|y,X) =(

1T − T0 + 1

) T∑t=T0

Iγ(t)=γ .

where the T0 first values are eliminated as burnin.

And approximation of the probability to include i-th variable,

P π(γi = 1|y,X) =(

1T − T0 + 1

) T∑t=T0

Iγ(t)i =1

Variable selection

γi Pπ(γi = 1|y, X) Pπ(γi = 1|y, X)γ1 0.8624 0.8844γ2 0.7060 0.7716γ3 0.1482 0.2978γ4 0.6671 0.7261γ5 0.6515 0.7006γ6 0.1678 0.3115γ7 0.1371 0.2880γ8 0.1555 0.2876γ9 0.4039 0.5168γ10 0.1151 0.2609

Probabilities of inclusion with both informative (β = 011, c = 100)and noninformative Zellner’s priors

Reversible jump

Idea: Set up a proper measure–theoretic framework for designingmoves between models Mk

[Green, 1995]Create a reversible kernel K on H =

⋃k{k} ×Θk such that∫

K(x, dy)π(x)dx =∫B

K(y, dx)π(y)dy

for the invariant density π [x is of the form (k, θ(k))]

Reversible jump

Idea: Set up a proper measure–theoretic framework for designingmoves between models Mk

[Green, 1995]Create a reversible kernel K on H =

⋃k{k} ×Θk such that∫

K(x, dy)π(x)dx =∫B

K(y, dx)π(y)dy

for the invariant density π [x is of the form (k, θ(k))]

Reversible jump

Local movesFor a move between two models, M1 and M2, the Markov chainbeing 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 dimensiondim(M2)− dim(M1), generated as

v1→2 ∼ ϕ1→2(v1→2) .

Reversible jump

Local movesFor a move between two models, M1 and M2, the Markov chainbeing 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 dimensiondim(M2)− dim(M1), generated as

v1→2 ∼ ϕ1→2(v1→2) .

Reversible jump

Local moves (2)

In this case, q1→2(θ1, dθ2) has density

ϕ1→2(v1→2)∣∣∣∣∂Ψ1→2(θ1, v1→2)

∂(θ1, v1→2)

∣∣∣∣−1

by the Jacobian rule.Reverse importance link

If probability $1→2 of choosing move to M2 while in M1,acceptance probability reduces to

α(θ1, v1→2) = 1∧ π(M2, θ2)$2→1

π(M1, θ1)$1→2 ϕ1→2(v1→2)

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

∣∣∣∣ .c©Difficult calibration

Reversible jump

Local moves (2)

ϕ1→2(v1→2)∣∣∣∣∂Ψ1→2(θ1, v1→2)

∂(θ1, v1→2)

∣∣∣∣−1

α(θ1, v1→2) = 1∧ π(M2, θ2)$2→1

π(M1, θ1)$1→2 ϕ1→2(v1→2)

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

Reversible jump

Local moves (2)

ϕ1→2(v1→2)∣∣∣∣∂Ψ1→2(θ1, v1→2)

∂(θ1, v1→2)

∣∣∣∣−1

α(θ1, v1→2) = 1∧ π(M2, θ2)$2→1

π(M1, θ1)$1→2 ϕ1→2(v1→2)

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

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−11→2(θ2)

appears like a regular Metropolis–Hastings move from thecouple (θ1, v1→2) to θ2 when stationary distributions areπ(M1, θ1)× ϕ1→2(v1→2) and π(M2, θ2), and when proposaldistribution is deterministic (??)

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−11→2(θ2)

appears like a regular Metropolis–Hastings move from thecouple (θ1, v1→2) to θ2 when stationary distributions areπ(M1, θ1)× ϕ1→2(v1→2) and π(M2, θ2), and when proposaldistribution is deterministic (??)

Reversible jump

Pseudo-deterministic reasoning

Consider the proposals

θ2 ∼ N (Ψ1→2(θ1, v1→2), ε) and Ψ1→2(θ1, v1→2) ∼ N (θ2, ε)

Reciprocal proposal has density

exp{−(θ2 −Ψ1→2(θ1, v1→2))2/2ε

2πε×∣∣∣∣∂Ψ1→2(θ1, v1→2)

∂(θ1, v1→2)

∣∣∣∣by the Jacobian rule.Thus Metropolis–Hastings acceptance probability is

1 ∧ π(M2, θ2)π(M1, θ1)ϕ1→2(v1→2)

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

∣∣∣∣Does not depend on ε: Let ε go to 0

Reversible jump

Pseudo-deterministic reasoning

Consider the proposals

θ2 ∼ N (Ψ1→2(θ1, v1→2), ε) and Ψ1→2(θ1, v1→2) ∼ N (θ2, ε)

Reciprocal proposal has density

exp{−(θ2 −Ψ1→2(θ1, v1→2))2/2ε

2πε×∣∣∣∣∂Ψ1→2(θ1, v1→2)

∂(θ1, v1→2)

∣∣∣∣by the Jacobian rule.Thus Metropolis–Hastings acceptance probability is

1 ∧ π(M2, θ2)π(M1, θ1)ϕ1→2(v1→2)

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

∣∣∣∣Does not depend on ε: Let ε go to 0

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

K(x,B) =∞Xm=1

Zρm(x, y)qm(x, dy) + ω(x)IB(x)

acceptance probability of θ2 = Ψ1→2(θ1, v1→2) is

α(θ1, v1→2) = 1 ∧ π(M2, θ2)$2→1

π(M1, θ1)$1→2 ϕ1→2(v1→2)

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

∣∣∣∣while acceptance probability of θ1 with (θ1, v1→2) = Ψ−1

1→2(θ2) is

α(θ1, v1→2) = 1 ∧ π(M1, θ1)$1→2 ϕ1→2(v1→2)π(M2, θ2)$2→1

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

∣∣∣∣−1

Reversible jump

K(x,B) =∞Xm=1

α(θ1, v1→2) = 1 ∧ π(M2, θ2)$2→1

π(M1, θ1)$1→2 ϕ1→2(v1→2)

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

1→2(θ2) is

α(θ1, v1→2) = 1 ∧ π(M1, θ1)$1→2 ϕ1→2(v1→2)π(M2, θ2)$2→1

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

∣∣∣∣−1

Reversible jump

K(x,B) =∞Xm=1

α(θ1, v1→2) = 1 ∧ π(M2, θ2)$2→1

π(M1, θ1)$1→2 ϕ1→2(v1→2)

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

1→2(θ2) is

α(θ1, v1→2) = 1 ∧ π(M1, θ1)$1→2 ϕ1→2(v1→2)π(M2, θ2)$2→1

∣∣∣∣∂Ψ1→2(θ1, v1→2)∂(θ1, v1→2)

∣∣∣∣−1

Reversible jump

Green’s sampler

Algorithm

Iteration t (t ≥ 1): if x(t) = (m, θ(m)),

1 Select model Mn with probability πmn2 Generate umn ∼ ϕmn(u) and set

(θ(n), vnm) = Ψm→n(θ(m), umn)3 Take x(t+1) = (n, θ(n)) with probability

min(π(n, θ(n))π(m, θ(m))

πnmϕnm(vnm)πmnϕmn(umn)

∣∣∣∣∂Ψm→n(θ(m), umn)∂(θ(m), umn)

∣∣∣∣ , 1)and take x(t+1) = x(t) otherwise.

Reversible jump

Mixture of normal distributions

(pjk, µjk, σjk);k∑j=1

pjkN (µjk, σ2jk)

Restrict moves from Mk to adjacent models, like Mk+1 andMk−1, with probabilities πk(k+1) and πk(k−1).

Reversible jump

Mixture of normal distributions

(pjk, µjk, σjk);k∑j=1

pjkN (µjk, σ2jk)

Restrict moves from Mk to adjacent models, like Mk+1 andMk−1, with probabilities πk(k+1) and πk(k−1).

Reversible jump

Mixture birth

Take Ψk→k+1 as a birth step: i.e. add a new normal component inthe mixture, by generating the parameters of the new componentfrom the prior distribution

(µk+1, σk+1) ∼ π(µ, σ) and pk+1 ∼ Be(a1, a2 + . . .+ ak)

if (p1, . . . , pk) ∼Mk(a1, . . . , ak)Jacobian is (1− pk+1)k−1

Death step then derived from the reversibility constraint byremoving one of the k components at random.

Reversible jump

Mixture birth

Take Ψk→k+1 as a birth step: i.e. add a new normal component inthe mixture, by generating the parameters of the new componentfrom the prior distribution

(µk+1, σk+1) ∼ π(µ, σ) and pk+1 ∼ Be(a1, a2 + . . .+ ak)

if (p1, . . . , pk) ∼Mk(a1, . . . , ak)Jacobian is (1− pk+1)k−1

Death step then derived from the reversibility constraint byremoving one of the k components at random.

Reversible jump

Mixture acceptance probability

Birth acceptance probability

min(π(k+1)k

πk(k+1)

(k + 1)!(k + 1)k!

π(k + 1, θk+1)π(k, θk) (k + 1)ϕk(k+1)(uk(k+1))

= min(π(k+1)k

πk(k+1)

%(k + 1)%(k)

`k+1(θk+1) (1− pk+1)k−1

`k(θk), 1),

where `k likelihood of the k component mixture model Mk and%(k) prior probability of model Mk.Combinatorial terms: there are (k + 1)! ways of defining a (k + 1)component mixture by adding one component, while, given a (k + 1)component mixture, there are (k+ 1) choices for a component to die and

then k! associated mixtures for the remaining components.

Reversible jump

Mixture acceptance probability

Birth acceptance probability

min(π(k+1)k

πk(k+1)

(k + 1)!(k + 1)k!

π(k + 1, θk+1)π(k, θk) (k + 1)ϕk(k+1)(uk(k+1))

= min(π(k+1)k

πk(k+1)

%(k + 1)%(k)

`k+1(θk+1) (1− pk+1)k−1

`k(θk), 1),

where `k likelihood of the k component mixture model Mk and%(k) prior probability of model Mk.Combinatorial terms: there are (k + 1)! ways of defining a (k + 1)component mixture by adding one component, while, given a (k + 1)component mixture, there are (k+ 1) choices for a component to die and

then k! associated mixtures for the remaining components.

Saturation schemes

AlternativeSaturation of the parameter space H =

⋃k{k} ×Θk by creating

θ = (θ1, . . . , θD)a model index Mpseudo-priors πj(θj |M = k) for j 6= k

[Carlin & Chib, 1995]Validation by

P(M = k|x) =∫P (M = k|x, θ)π(θ|x)dθ = Zk

where the (marginal) posterior is [not πk!]

π(θ|x) =D∑k=1

P(θ,M = k|x)

=D∑k=1

pk Zk πk(θk|x)∏j 6=k

πj(θj |M = k) .

Saturation schemes

AlternativeSaturation of the parameter space H =

⋃k{k} ×Θk by creating

θ = (θ1, . . . , θD)a model index Mpseudo-priors πj(θj |M = k) for j 6= k

[Carlin & Chib, 1995]Validation by

P(M = k|x) =∫P (M = k|x, θ)π(θ|x)dθ = Zk

where the (marginal) posterior is [not πk!]

π(θ|x) =D∑k=1

P(θ,M = k|x)

=D∑k=1

pk Zk πk(θk|x)∏j 6=k

πj(θj |M = k) .

Saturation schemes

MCMC implementation

Run a Markov chain (M (t), θ(t)1 , . . . , θ

(t)D ) with stationary

distribution π(θ,M |x) by

1 Pick M (t) = k with probability π(θ(t−1), k|x)

2 Generate θ(t−1)k from the posterior πk(θk|x) [or MCMC step]

3 Generate θ(t−1)j (j 6= k) from the pseudo-prior πj(θj |M = k)

Approximate P(M = k|x) = Zk by

pk(x) ∝ pkT∑t=1

fk(x|θ(t)k )πk(θ

(t)k )∏j 6=k

πj(θ(t)j |M = k)

/ D∑`=1

p` f`(x|θ(t)` )π`(θ

(t)` )∏j 6=`

πj(θ(t)j |M = `)

Saturation schemes

MCMC implementation

pk(x) ∝ pkT∑t=1

fk(x|θ(t)k )πk(θ

(t)k )∏j 6=k

πj(θ(t)j |M = k)

/ D∑`=1

p` f`(x|θ(t)` )π`(θ

(t)` )∏j 6=`

πj(θ(t)j |M = `)

Saturation schemes

MCMC implementation

pk(x) ∝ pkT∑t=1

fk(x|θ(t)k )πk(θ

(t)k )∏j 6=k

πj(θ(t)j |M = k)

/ D∑`=1

p` f`(x|θ(t)` )π`(θ

(t)` )∏j 6=`

πj(θ(t)j |M = `)

Implementation error

Scott’s (2002) proposal

Suggest estimating P(M = k|x) by

Zk ∝ pkT∑t=1

fk(x|θ(t)k )/ D∑

pj fj(x|θ(t)j )

based on D simultaneous and independent MCMC chains

(θ(t)k )t , 1 ≤ k ≤ D ,

with stationary distributions πk(θk|x) [instead of above joint!!]

Scott’s (2002) proposal

Suggest estimating P(M = k|x) by

Zk ∝ pkT∑t=1

fk(x|θ(t)k )/ D∑

pj fj(x|θ(t)j )

based on D simultaneous and independent MCMC chains

(θ(t)k )t , 1 ≤ k ≤ D ,

with stationary distributions πk(θk|x) [instead of above joint!!]

Congdon’s (2006) extension

Selecting flat [prohibited!] pseudo-priors, uses instead

Zk ∝ pkT∑t=1

fk(x|θ(t)k )πk(θ

(t)k )/ D∑

pj fj(x|θ(t)j )πj(θ

(t)j )

where again the θ(t)k ’s are MCMC chains with stationary

distributions πk(θk|x)

Examples

Example (Model choice)

Model M1 : x|θ ∼ U(0, θ) with prior θ ∼ Exp(1) is versus modelM2 : x|θ ∼ Exp(θ) with prior θ ∼ Exp(1). Equal prior weights onboth models: %1 = %2 = 0.5.

Approximations of P(M = 1|x):

Scott’s (2002) (blue), and

Congdon’s (2006) (red)

[N = 106 simulations].

1 2 3 4 5

Examples

Example (Model choice)

Model M1 : x|θ ∼ U(0, θ) with prior θ ∼ Exp(1) is versus modelM2 : x|θ ∼ Exp(θ) with prior θ ∼ Exp(1). Equal prior weights onboth models: %1 = %2 = 0.5.

Approximations of P(M = 1|x):

Scott’s (2002) (blue), and

Congdon’s (2006) (red)

[N = 106 simulations].

1 2 3 4 5

Examples (2)

Example (Model choice (2))

Normal model M1 : x ∼ N (θ, 1) with θ ∼ N (0, 1) vs. normalmodel M2 : x ∼ N (θ, 1) with θ ∼ N (5, 1)

Comparison of both

approximations with

P(M = 1|x): Scott’s (2002)

(green and mixed dashes) and

Congdon’s (2006) (brown and

long dashes) [N = 104

simulations].

−1 0 1 2 3 4 5 6

Examples (3)

Example (Model choice (3))

Model M1 : x ∼ N (0, 1/ω) with ω ∼ Exp(a) vs.M2 : exp(x) ∼ Exp(λ) with λ ∼ Exp(b).

Comparison of Congdon’s (2006)

(brown and dashed lines) with

P(M = 1|x) when (a, b) is equal

to (.24, 8.9), (.56, .7), (4.1, .46)and (.98, .081), resp. [N = 104

simulations].

−10 −5 0 5 10

Nested sampling

Purpose

Nested sampling: Goal

Skilling’s (2007) technique using the one-dimensionalrepresentation:

Z = Eπ[L(θ)] =∫ 1

0ϕ(x) dx

withϕ−1(l) = P π(L(θ) > l).

Note; ϕ(·) is intractable in most cases.

Nested sampling

Implementation

Nested sampling: First approximation

Approximate Z by a Riemann sum:

Z =j∑i=1

(xi−1 − xi)ϕ(xi)

where the xi’s are either:

deterministic: xi = e−i/N

or random:

x0 = 0, xi+1 = tixi, ti ∼ Be(N, 1)

so that E[log xi] = −i/N .

Nested sampling

Implementation

Extraneous white noise

Z =∫e−θ dθ =

∫1δe−(1−δ)θ e−δθ = Eδ

[1δe−(1−δ)θ

N∑i=1

δ−1 e−(1−δ)θi(xi−1 − xi) , θi ∼ E(δ) I(θi ≤ θi−1)

N deterministic random50 4.64 10.5

4.65 10.5100 2.47 4.9

2.48 5.02500 .549 1.01

.550 1.14

Comparison of variances and MSEs

Nested sampling

Implementation

Z =∫e−θ dθ =

[1δe−(1−δ)θ

N∑i=1

4.65 10.5100 2.47 4.9

2.48 5.02500 .549 1.01

.550 1.14

Nested sampling

Implementation

Z =∫e−θ dθ =

[1δe−(1−δ)θ

N∑i=1

4.65 10.5100 2.47 4.9

2.48 5.02500 .549 1.01

.550 1.14

Nested sampling

Implementation

Nested sampling: Second approximation

Replace (intractable) ϕ(xi) by ϕi, obtained by

Nested sampling

Start with N values θ1, . . . , θN sampled from πAt iteration i,

1 Take ϕi = L(θk), where θk is the point with smallestlikelihood in the pool of θi’s

2 Replace θk with a sample from the prior constrained toL(θ) > ϕi: the current N points are sampled from priorconstrained to L(θ) > ϕi.

Nested sampling

Implementation

Nested sampling

Implementation

Nested sampling

Implementation

Nested sampling: Third approximation

Iterate the above steps until a given stopping iteration j isreached: e.g.,

observe very small changes in the approximation Z;

reach the maximal value of L(θ) when the likelihood isbounded and its maximum is known;

truncate the integral Z at level ε, i.e. replace∫ 1

0ϕ(x) dx with

εϕ(x) dx

Nested sampling

Error rates

Approximation error

Error = Z− Z

=j∑i=1

(xi−1 − xi)ϕi −∫ 1

0ϕ(x) dx = −

∫ ε

0ϕ(x) dx (Truncation Error)

[j∑i=1

(xi−1 − xi)ϕ(xi)−∫ 1

εϕ(x) dx

](Quadrature Error)

[j∑i=1

(xi−1 − xi) {ϕi − ϕ(xi)}

](Stochastic Error)

[Dominated by Monte Carlo!]

Nested sampling

Error rates

A CLT for the Stochastic Error

The (dominating) stochastic error is OP (N−1/2):

N1/2 {Stochastic Error} D→ N (0, V )

V = −∫s,t∈[ε,1]

sϕ′(s)tϕ′(t) log(s ∨ t) ds dt.

[Proof based on Donsker’s theorem]

The number of simulated points equals the number of iterations j,and is a multiple of N : if one stops at first iteration j such thate−j/N < ε, then: j = Nd− log εe.

Nested sampling

Error rates

A CLT for the Stochastic Error

The (dominating) stochastic error is OP (N−1/2):

N1/2 {Stochastic Error} D→ N (0, V )

V = −∫s,t∈[ε,1]

sϕ′(s)tϕ′(t) log(s ∨ t) ds dt.

[Proof based on Donsker’s theorem]

The number of simulated points equals the number of iterations j,and is a multiple of N : if one stops at first iteration j such thate−j/N < ε, then: j = Nd− log εe.

Nested sampling

Impact of dimension

Curse of dimension

For a simple Gaussian-Gaussian model of dimension dim(θ) = d,the following 3 quantities are O(d):

1 asymptotic variance of the NS estimator;

2 number of iterations (necessary to reach a given truncationerror);

3 cost of one simulated sample.

Therefore, CPU time necessary for achieving error level e is

O(d3/e2)

Nested sampling

Impact of dimension

Curse of dimension

O(d3/e2)

Nested sampling

Impact of dimension

Curse of dimension

O(d3/e2)

Nested sampling

Impact of dimension

Curse of dimension

O(d3/e2)

Nested sampling

Constraints

Sampling from constr’d priors

Exact simulation from the constrained prior is intractable in mostcases!

Skilling (2007) proposes to use MCMC, but:

this introduces a bias (stopping rule).

if MCMC stationary distribution is unconst’d prior, more andmore difficult to sample points such that L(θ) > l as lincreases.

If implementable, then slice sampler can be devised at the samecost!

Nested sampling

Constraints

Nested sampling

Constraints

Nested sampling

Constraints

Illustration of MCMC bias

10 20 30 40 50 60 70 80 90100-50

10N=100, M=1

10 20 30 40 50 60 70 80 90100

N=100, M=3

10 20 30 40 50 60 70 80 90100

N=100, M=5

0 20 40 60 80 1000

80000N=100, M=5

10 20 30 40 50 60 70 80 90100-10

10N=500, M=1

Log-relative error against d (left), avg. number of iterations (right)vs dimension d, for a Gaussian-Gaussian model with d parameters,when using T = 10 iterations of the Gibbs sampler.

Nested sampling

Importance variant

A IS variant of nested sampling

Consider instrumental prior π and likelihood L, weight function

w(θ) =π(θ)L(θ)

π(θ)L(θ)

and weighted NS estimator

Z =j∑i=1

(xi−1 − xi)ϕiw(θi).

Then choose (π, L) so that sampling from π constrained toL(θ) > l is easy; e.g. N (c, Id) constrained to ‖c− θ‖ < r.

Nested sampling

Importance variant

A IS variant of nested sampling

Consider instrumental prior π and likelihood L, weight function

w(θ) =π(θ)L(θ)

π(θ)L(θ)

and weighted NS estimator

Z =j∑i=1

(xi−1 − xi)ϕiw(θi).

Then choose (π, L) so that sampling from π constrained toL(θ) > l is easy; e.g. N (c, Id) constrained to ‖c− θ‖ < r.

Nested sampling

A mixture comparison

Benchmark: Target distribution

Posterior distribution on (µ, σ) associated with the mixture

pN (0, 1) + (1− p)N (µ, σ) ,

when p is known

Nested sampling

Experiment

n observations withµ = 2 and σ = 3/2,

Use of a uniform priorboth on (−2, 6) for µand on (.001, 16) forlog σ2.

occurrences of posteriorbursts for µ = xi

computation of thevarious estimates of Z

Nested sampling

Experiment (cont’d)

MCMC sample for n = 16observations from the mixture.

Nested sampling sequencewith M = 1000 starting points.

Nested sampling

Experiment (cont’d)

MCMC sample for n = 50observations from the mixture.

Nested sampling sequencewith M = 1000 starting points.

Nested sampling

Comparison

Monte Carlo and MCMC (=Gibbs) outputs based on T = 104

simulations and numerical integration based on a 850× 950 grid inthe (µ, σ) parameter space.Nested sampling approximation based on a starting sample ofM = 1000 points followed by at least 103 further simulations fromthe constr’d prior and a stopping rule at 95% of the observedmaximum likelihood.Constr’d prior simulation based on 50 values simulated by randomwalk accepting only steps leading to a lik’hood higher than thebound

Nested sampling

Comparison (cont’d)

●●

V1 V2 V3 V4

Graph based on a sample of 10 observations for µ = 2 andσ = 3/2 (150 replicas).

Nested sampling

●●

V1 V2 V3 V4

Nested sampling

●●

●●●

●●

V1 V2 V3 V4

Nested sampling

Nested sampling gets less reliable as sample size increasesMost reliable approach is mixture Z3 although harmonic solutionZ1 close to Chib’s solution [taken as golden standard]Monte Carlo method Z2 also producing poor approximations to Z

(Kernel φ used in Z2 is a t non-parametric kernel estimate withstandard bandwidth estimation.)

ABC model choice

ABC method

Approximate Bayesian Computation

Bayesian setting: target is π(θ)f(x|θ)When likelihood f(x|θ) not in closed form, likelihood-free rejectiontechnique:

ABC algorithm

For an observation y ∼ f(y|θ), under the prior π(θ), keep jointlysimulating

θ′ ∼ π(θ) , x ∼ f(x|θ′) ,

until the auxiliary variable x is equal to the observed value, x = y.

[Pritchard et al., 1999]

ABC model choice

ABC method

ABC algorithm

θ′ ∼ π(θ) , x ∼ f(x|θ′) ,

ABC model choice

ABC method

ABC algorithm

θ′ ∼ π(θ) , x ∼ f(x|θ′) ,

ABC model choice

ABC method

Population genetics example

Tree of ancestors in a sample of genes

ABC model choice

ABC method

A as approximative

When y is a continuous random variable, equality x = y is replacedwith a tolerance condition,

%(x, y) ≤ ε

where % is a distance between summary statisticsOutput distributed from

π(θ)Pθ{%(x, y) < ε} ∝ π(θ|%(x, y) < ε)

ABC model choice

ABC method

A as approximative

When y is a continuous random variable, equality x = y is replacedwith a tolerance condition,

%(x, y) ≤ ε

where % is a distance between summary statisticsOutput distributed from

π(θ)Pθ{%(x, y) < ε} ∝ π(θ|%(x, y) < ε)

ABC model choice

ABC method

ABC improvements

Simulating from the prior is often poor in efficiencyEither modify the proposal distribution on θ to increase the densityof x’s within the vicinity of y...

[Marjoram et al, 2003; Bortot et al., 2007, Sisson et al., 2007]

...or by viewing the problem as a conditional density estimationand by developing techniques to allow for larger ε

[Beaumont et al., 2002]

ABC model choice

ABC method

ABC improvements

ABC model choice

ABC method

ABC improvements

ABC model choice

ABC method

ABC-MCMC

Markov chain (θ(t)) created via the transition function

θ(t+1) =

θ′ ∼ K(θ′|θ(t)) if x ∼ f(x|θ′) is such that x = y

and u ∼ U(0, 1) ≤ π(θ′)K(θ(t)|θ′)π(θ(t))K(θ′|θ(t)) ,

θ(t) otherwise,

has the posterior π(θ|y) as stationary distribution[Marjoram et al, 2003]

ABC model choice

ABC method

ABC-MCMC

Markov chain (θ(t)) created via the transition function

θ(t+1) =

θ′ ∼ K(θ′|θ(t)) if x ∼ f(x|θ′) is such that x = y

and u ∼ U(0, 1) ≤ π(θ′)K(θ(t)|θ′)π(θ(t))K(θ′|θ(t)) ,

θ(t) otherwise,

has the posterior π(θ|y) as stationary distribution[Marjoram et al, 2003]

ABC model choice

ABC method

ABC-PRC

Another sequential version producing a sequence of Markov

transition kernels Kt and of samples (θ(t)1 , . . . , θ

(t)N ) (1 ≤ t ≤ T )

ABC-PRC Algorithm

1 Pick a θ? is selected at random among the previous θ(t−1)i ’s

with probabilities ω(t−1)i (1 ≤ i ≤ N).

2 Generateθ

(t)i ∼ Kt(θ|θ?) , x ∼ f(x|θ(t)

3 Check that %(x, y) < ε, otherwise start again.

[Sisson et al., 2007]

ABC model choice

ABC method

ABC-PRC

Another sequential version producing a sequence of Markov

transition kernels Kt and of samples (θ(t)1 , . . . , θ

(t)N ) (1 ≤ t ≤ T )

ABC-PRC Algorithm

1 Pick a θ? is selected at random among the previous θ(t−1)i ’s

with probabilities ω(t−1)i (1 ≤ i ≤ N).

2 Generateθ

(t)i ∼ Kt(θ|θ?) , x ∼ f(x|θ(t)

3 Check that %(x, y) < ε, otherwise start again.

[Sisson et al., 2007]

ABC model choice

ABC method

ABC-PRC weight

Probability ω(t)i computed as

ω(t)i ∝ π(θ(t)

i )Lt−1(θ?|θ(t)i ){π(θ?)Kt(θ

(t)i |θ

?)}−1 ,

where Lt−1 is an arbitrary transition kernel.In case

Lt−1(θ′|θ) = Kt(θ|θ′) ,

all weights are equal under a uniform prior.Inspired from Del Moral et al. (2006), who use backward kernelsLt−1 in SMC to achieve unbiasedness

ABC model choice

ABC method

ABC-PRC weight

ω(t)i ∝ π(θ(t)

(t)i |θ

?)}−1 ,

Lt−1(θ′|θ) = Kt(θ|θ′) ,

ABC model choice

ABC method

ABC-PRC weight

ω(t)i ∝ π(θ(t)

(t)i |θ

?)}−1 ,

Lt−1(θ′|θ) = Kt(θ|θ′) ,

ABC model choice

ABC method

ABC-PRC bias

Lack of unbiasedness of the methodJoint density of the accepted pair (θ(t−1), θ(t)) proportional to

π(θ(t−1)|y)Kt(θ(t)|θ(t−1))f(y|θ(t)) ,

For an arbitrary function h(θ), E[ωth(θ(t))] proportional to

ZZh(θ

(t))π(θ(t))Lt−1(θ(t−1)|θ(t))π(θ(t−1))Kt(θ(t)|θ(t−1))

π(θ(t−1)|y)Kt(θ(t)|θ(t−1)

)f(y|θ(t))dθ(t−1)dθ(t)

h(θ(t)

)π(θ(t))Lt−1(θ(t−1)|θ(t))π(θ(t−1))Kt(θ(t)|θ(t−1))

π(θ(t−1)

)f(y|θ(t−1))

×Kt(θ(t)|θ(t−1))f(y|θ(t))dθ(t−1)dθ(t)

∝Zh(θ

(t))π(θ

(t)|y)Z

Lt−1(θ(t−1)|θ(t))f(y|θ(t−1)

)dθ(t−1)ff

dθ(t) .

ABC model choice

ABC method

ABC-PRC bias

Lack of unbiasedness of the methodJoint density of the accepted pair (θ(t−1), θ(t)) proportional to

π(θ(t−1)|y)Kt(θ(t)|θ(t−1))f(y|θ(t)) ,

For an arbitrary function h(θ), E[ωth(θ(t))] proportional to

ZZh(θ

(t))π(θ(t))Lt−1(θ(t−1)|θ(t))π(θ(t−1))Kt(θ(t)|θ(t−1))

π(θ(t−1)|y)Kt(θ(t)|θ(t−1)

)f(y|θ(t))dθ(t−1)dθ(t)

h(θ(t)

)π(θ(t))Lt−1(θ(t−1)|θ(t))π(θ(t−1))Kt(θ(t)|θ(t−1))

π(θ(t−1)

)f(y|θ(t−1))

×Kt(θ(t)|θ(t−1))f(y|θ(t))dθ(t−1)dθ(t)

∝Zh(θ

(t))π(θ

(t)|y)Z

Lt−1(θ(t−1)|θ(t))f(y|θ(t−1)

)dθ(t−1)ff

dθ(t) .

ABC model choice

ABC method

A mixture example

0.00.20.40.60.81.0

Comparison of τ = 0.15 and τ = 1/0.15 in Kt

ABC model choice

ABC-PMC

A PMC version

Use of the same kernel idea as ABC-PRC but with IS correctionGenerate a sample at iteration t by

πt(θ(t)) ∝N∑j=1

ω(t−1)j Kt(θ(t)|θ(t−1)

modulo acceptance of the associated xt, and use an importance

weight associated with an accepted simulation θ(t)i

ω(t)i ∝ π(θ(t)

i )/πt(θ

(t)i ) .

ABC model choice

ABC-PMC

The ABC-PMC algorithmGiven a decreasing sequence of approximation levels ε1 ≥ . . . ≥ εT ,

1. At iteration t = 1,

For i = 1, ..., NSimulate θ

(1)i ∼ π(θ) and x ∼ f(x|θ(1)i ) until %(x, y) < ε1

Set ω(1)i = 1/N

Take τ2 as twice the empirical variance of the θ(1)i ’s

2. At iteration 2 ≤ t ≤ T ,

For i = 1, ..., N , repeat

Pick θ?i from the θ(t−1)j ’s with probabilities ω

(t−1)j

generate θ(t)i |θ

?i ∼ N (θ?i , σ

2t ) and x ∼ f(x|θ(t)i )

until %(x, y) < εt

Set ω(t)i ∝ π(θ(t)i )/

∑Nj=1 ω

(t−1)j ϕ

(σ−1t

{θ(t)i − θ

(t−1)j )

})Take τ2

t+1 as twice the weighted empirical variance of the θ(t)i ’s

ABC model choice

ABC-PMC

A mixture example (0)

Toy model of Sisson et al. (2007): if

θ ∼ U(−10, 10) , x|θ ∼ 0.5N (θ, 1) + 0.5N (θ, 1/100) ,

then the posterior distribution associated with y = 0 is the normalmixture

θ|y = 0 ∼ 0.5N (0, 1) + 0.5N (0, 1/100)

restricted to [−10, 10].Furthermore, true target available as

π(θ||x| < ε) ∝ Φ(ε−θ)−Φ(−ε−θ)+Φ(10(ε−θ))−Φ(−10(ε+θ)) .

ABC model choice

ABC-PMC

A mixture example (2)

Recovery of the target, whether using a fixed standard deviation ofτ = 0.15 or τ = 1/0.15, or a sequence of adaptive τt’s.

ABC model choice

ABC for model choice in GRFs

Gibbs random fields

Gibbs distribution

The rv y = (y1, . . . , yn) is a Gibbs random field associated withthe graph G if

f(y) =1Z

{−∑c∈C

Vc(yc)

where Z is the normalising constant, C is the set of cliques of G

and Vc is any function also called potentialU(y) =

∑c∈C Vc(yc) is the energy function

ABC model choice

Gibbs random fields

Gibbs distribution

The rv y = (y1, . . . , yn) is a Gibbs random field associated withthe graph G if

f(y) =1Z

{−∑c∈C

Vc(yc)

where Z is the normalising constant, C is the set of cliques of G

and Vc is any function also called potentialU(y) =

∑c∈C Vc(yc) is the energy function

ABC model choice

Potts model

Vc(y) is of the form

Vc(y) = θS(y) = θ∑l∼i

δyl=yi

where l∼i denotes a neighbourhood structure

In most realistic settings, summation

Zθ =∑x∈X

exp{θTS(x)}

involves too many terms to be manageable and numericalapproximations cannot always be trusted

[Cucala, Marin, CPR & Titterington, 2009]

ABC model choice

Potts model

Vc(y) is of the form

Vc(y) = θS(y) = θ∑l∼i

δyl=yi

where l∼i denotes a neighbourhood structure

In most realistic settings, summation

Zθ =∑x∈X

exp{θTS(x)}

involves too many terms to be manageable and numericalapproximations cannot always be trusted

[Cucala, Marin, CPR & Titterington, 2009]

ABC model choice

Bayesian Model Choice

Comparing a model with potential S0 taking values in Rp0 versus amodel with potential S1 taking values in Rp1 can be done throughthe Bayes factor corresponding to the priors π0 and π1 on eachparameter space

Bm0/m1(x) =

∫exp{θT

0 S0(x)}/Zθ0,0π0(dθ0)∫exp{θT

1 S1(x)}/Zθ1,1π1(dθ1)

Use of Jeffreys’ scale to select most appropriate model

ABC model choice

Bayesian Model Choice

Comparing a model with potential S0 taking values in Rp0 versus amodel with potential S1 taking values in Rp1 can be done throughthe Bayes factor corresponding to the priors π0 and π1 on eachparameter space

Bm0/m1(x) =

∫exp{θT

0 S0(x)}/Zθ0,0π0(dθ0)∫exp{θT

1 S1(x)}/Zθ1,1π1(dθ1)

Use of Jeffreys’ scale to select most appropriate model

ABC model choice

Neighbourhood relations

Choice to be made between M neighbourhood relations

im∼ i′ (0 ≤ m ≤M − 1)

withSm(x) =

∑im∼i′

I{xi=xi′}

driven by the posterior probabilities of the models.

ABC model choice

Model index

Formalisation via a model index M that appears as a newparameter with prior distribution π(M = m) andπ(θ|M = m) = πm(θm)Computational target:

P(M = m|x) ∝∫

fm(x|θm)πm(θm) dθm π(M = m) ,

ABC model choice

Model index

Formalisation via a model index M that appears as a newparameter with prior distribution π(M = m) andπ(θ|M = m) = πm(θm)Computational target:

P(M = m|x) ∝∫

fm(x|θm)πm(θm) dθm π(M = m) ,

ABC model choice

Sufficient statisticsBy definition, if S(x) sufficient statistic for the joint parameters(M, θ0, . . . , θM−1),

P(M = m|x) = P(M = m|S(x)) .

For each model m, own sufficient statistic Sm(·) andS(·) = (S0(·), . . . , SM−1(·)) also sufficient.For Gibbs random fields,

x|M = m ∼ fm(x|θm) = f1m(x|S(x))f2

m(S(x)|θm)

n(S(x))f2m(S(x)|θm)

wheren(S(x)) = ] {x ∈ X : S(x) = S(x)}

ABC model choice

P(M = m|x) = P(M = m|S(x)) .

m(S(x)|θm)

wheren(S(x)) = ] {x ∈ X : S(x) = S(x)}

ABC model choice

P(M = m|x) = P(M = m|S(x)) .

m(S(x)|θm)

wheren(S(x)) = ] {x ∈ X : S(x) = S(x)}

ABC model choice

ABC model choice Algorithm

ABC-MC

Generate m∗ from the prior π(M = m).

Generate θ∗m∗ from the prior πm∗(·).

Generate x∗ from the model fm∗(·|θ∗m∗).

Compute the distance ρ(S(x0), S(x∗)).

Accept (θ∗m∗ ,m∗) if ρ(S(x0), S(x∗)) < ε.

[Cornuet, Grelaud, Marin & Robert, 2008]

Note When ε = 0 the algorithm is exact

ABC model choice

ABC approximation to the Bayes factor

Frequency ratio:

BFm0/m1(x0) =

P(M = m0|x0)P(M = m1|x0)

× π(M = m1)π(M = m0)

=]{mi∗ = m0}]{mi∗ = m1}

× π(M = m1)π(M = m0)

replaced with

BFm0/m1(x0) =

1 + ]{mi∗ = m0}1 + ]{mi∗ = m1}

× π(M = m1)π(M = m0)

to avoid indeterminacy (also Bayes estimate).

ABC model choice

ABC approximation to the Bayes factor

Frequency ratio:

BFm0/m1(x0) =

P(M = m0|x0)P(M = m1|x0)

× π(M = m1)π(M = m0)

=]{mi∗ = m0}]{mi∗ = m1}

× π(M = m1)π(M = m0)

replaced with

BFm0/m1(x0) =

1 + ]{mi∗ = m0}1 + ]{mi∗ = m1}

× π(M = m1)π(M = m0)

to avoid indeterminacy (also Bayes estimate).

ABC model choice

Illustrations

Toy example

iid Bernoulli model versus two-state first-order Markov chain, i.e.

f0(x|θ0) = exp

n∑i=1

I{xi=1}

)/{1 + exp(θ0)}n ,

versus

f1(x|θ1) =12

n∑i=2

I{xi=xi−1}

)/{1 + exp(θ1)}n−1 ,

with priors θ0 ∼ U(−5, 5) and θ1 ∼ U(0, 6) (inspired by “phasetransition” boundaries).

ABC model choice

Illustrations

Toy example (2)

−40 −20 0 10

−40 −20 0 10−10

(left) Comparison of the true BFm0/m1(x0) with BFm0/m1

(x0)(in logs) over 2, 000 simulations and 4.106 proposals from theprior. (right) Same when using tolerance ε corresponding to the1% quantile on the distances.

ABC model choice

Illustrations

Protein folding

Superposition of the native structure (grey) with the ST1structure (red.), the ST2 structure (orange), the ST3 structure(green), and the DT structure (blue).

ABC model choice

Illustrations

Protein folding (2)

% seq . Id. TM-score FROST score

1i5nA (ST1) 32 0.86 75.31ls1A1 (ST2) 5 0.42 8.91jr8A (ST3) 4 0.24 8.91s7oA (DT) 10 0.08 7.8

Characteristics of dataset. % seq. Id.: percentage of identity withthe query sequence. TM-score.: similarity between predicted andnative structure (uncertainty between 0.17 and 0.4) FROST score:quality of alignment of the query onto the candidate structure(uncertainty between 7 and 9).

ABC model choice

Illustrations

Protein folding (3)

NS/ST1 NS/ST2 NS/ST3 NS/DT

BF 1.34 1.22 2.42 2.76P(M = NS|x0) 0.573 0.551 0.708 0.734

Estimates of the Bayes factors between model NS and modelsST1, ST2, ST3, and DT, and corresponding posteriorprobabilities of model NS based on an ABC-MC algorithm using1.2 106 simulations and a tolerance ε equal to the 1% quantile ofthe distances.

Computational methods for Bayesian model choice

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