glasgow seminar, april 20, 2012

ABC Methods for Bayesian Model Choice

Christian P. Robert

Universite Paris-Dauphine, IuF, & CRESThttp://www.ceremade.dauphine.fr/~xian

Glasgow University, April 20, 2012Joint work with J.-M. Cornuet, A. Grelaud,

J.-M. Marin, N. Pillai, & J. Rousseau

Outline

Approximate Bayesian computation

ABC for model choice

Model choice consistency

Approximate Bayesian computationABC basicsAlphabet soupSemi-automatic ABCsimulation-based methods inEconometrics

ABC basics

Bayesian computation issues

When faced with a non-standard posterior distribution

π(θ|y) ∝ π(θ)L(θ|y)

the standard solution is to use simulation (Monte Carlo) toproduce a sample

θ1, . . . , θT

from π(θ|y) (or approximately by Markov chain Monte Carlomethods)

[Robert & Casella, 2004]

ABC basics

Untractable likelihoods

Cases when the likelihood function f(y|θ) is unavailable and whenthe completion step

f(y|θ) =

∫Zf(y, z|θ) dz

is impossible or too costly because of the dimension of zc© MCMC cannot be implemented!

ABC basics

Illustrations

Example

Stochastic volatility model: fort = 1, . . . , T,

yt = exp(zt)εt , zt = a+bzt−1+σηt ,

T very large makes it difficult toinclude z within the simulatedparameters

0 200 400 600 800 1000

Highest weight trajectories

ABC basics

Illustrations

Example

Potts model: if y takes values on a grid Y of size kn and

f(y|θ) ∝ exp

{θ∑l∼i

Iyl=yi

where l∼i denotes a neighbourhood relation, n moderately largeprohibits the computation of the normalising constant

ABC basics

Illustrations

Example

Coalescence tree: in populationgenetics, reconstitution of a commonancestor from a sample of genes viaa phylogenetic tree that is close toimpossible to integrate out[100 processor days with 4parameters]

[Cornuet et al., 2009, Bioinformatics]

ABC basics

The ABC method

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) , z ∼ f(z|θ′) ,

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

[Tavare et al., 1997]

ABC basics

The ABC method

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

ABC algorithm

θ′ ∼ π(θ) , z ∼ f(z|θ′) ,

ABC basics

The ABC method

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

ABC algorithm

θ′ ∼ π(θ) , z ∼ f(z|θ′) ,

ABC basics

A as approximative

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

%(y, z) ≤ ε

where % is a distance

Output distributed from

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

ABC basics

A as approximative

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

%(y, z) ≤ ε

where % is a distanceOutput distributed from

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

ABC basics

ABC algorithm (with summary)

Algorithm 1 Likelihood-free rejection sampler

for i = 1 to N dorepeat

generate θ′ from the prior distribution π(·)generate z from the likelihood f(·|θ′)

until ρ{η(z), η(y)} ≤ εset θi = θ′

end for

where η(y) defines a (maybe in-sufficient) statistic

ABC basics

Output (with summary)

The likelihood-free algorithm samples from the marginal in z of:

πε(θ, z|y) =π(θ)f(z|θ)IAε,y(z)∫

Aε,y×Θ π(θ)f(z|θ)dzdθ,

where Aε,y = {z ∈ D|ρ(η(z), η(y)) < ε}.

The idea behind ABC is that the summary statistics coupled with asmall tolerance should provide a good approximation of theposterior distribution based on the summary statistic:

πε(θ|y) =

∫πε(θ, z|y)dz ≈ π(θ|η(y)) .

ABC basics

Output (with summary)

The likelihood-free algorithm samples from the marginal in z of:

πε(θ, z|y) =π(θ)f(z|θ)IAε,y(z)∫

Aε,y×Θ π(θ)f(z|θ)dzdθ,

where Aε,y = {z ∈ D|ρ(η(z), η(y)) < ε}.

The idea behind ABC is that the summary statistics coupled with asmall tolerance should provide a good approximation of theposterior distribution based on the summary statistic:

πε(θ|y) =

∫πε(θ, z|y)dz ≈ π(θ|η(y)) .

ABC basics

MA example

Consider the MA(q) model

xt = εt +

q∑i=1

ϑiεt−i

Simple prior: uniform prior over the identifiability zone, e.g.triangle for MA(2)

ABC basics

MA example (2)

ABC algorithm thus made of

1. picking a new value (ϑ1, ϑ2) in the triangle

2. generating an iid sequence (εt)−q<t≤T

3. producing a simulated series (x′t)1≤t≤T

Distance: basic distance between the series

ρ((x′t)1≤t≤T , (xt)1≤t≤T ) =T∑t=1

(xt − x′t)2

or between summary statistics like the first 2 autocorrelations

τj =T∑

xtxt−j

ABC basics

MA example (2)

ABC algorithm thus made of

1. picking a new value (ϑ1, ϑ2) in the triangle

2. generating an iid sequence (εt)−q<t≤T

3. producing a simulated series (x′t)1≤t≤T

Distance: basic distance between the series

ρ((x′t)1≤t≤T , (xt)1≤t≤T ) =T∑t=1

(xt − x′t)2

or between summary statistics like the first 2 autocorrelations

T∑t=j+1

xtxt−j

ABC basics

Comparison of distance impact

Impact of the tolerance on the ABC sample against both distances(ε = 100%, 10%, 1%, 0.1%) for an MA(2) model

ABC basics

0.0 0.2 0.4 0.6 0.8

−2.0 −1.0 0.0 0.5 1.0 1.5

0.00.5

1.01.5

Impact of the tolerance on the ABC sample against raw distance(ε = 100%, 10%, 1%, 0.1%) for an MA(2) model

ABC basics

0.0 0.2 0.4 0.6 0.8

−2.0 −1.0 0.0 0.5 1.0 1.5

0.00.5

1.01.5

Impact of the tolerance on the ABC sample againstautocorrelation distance (ε = 100%, 10%, 1%, 0.1%) for an MA(2)model

ABC basics

ABC advances

Simulating from the prior is often poor in efficiency

Either 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]

.....or even by including ε in the inferential framework [ABCµ][Ratmann et al., 2009]

ABC basics

ABC advances

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...

ABC basics

ABC advances

ABC basics

ABC advances

Alphabet soup

ABC-NP

Better usage of [prior] simulations byadjustement: instead of throwing awayθ′ such that ρ(η(z), η(y)) > ε, replaceθ’s with locally regressed transforms

θ∗ = θ − {η(z)− η(y)}Tβ[Csillery et al., TEE, 2010]

where β is obtained by [NP] weighted least square regression on(η(z)− η(y)) with weights

Kδ {ρ(η(z), η(y))}

[Beaumont et al., 2002, Genetics]

Alphabet soup

ABC-NP (density estimation)

Use of the kernel weights

Kδ {ρ(η(z(θ)), η(y))}

leads to the NP estimate of the posterior expectation∑i θiKδ {ρ(η(z(θi)), η(y))}∑iKδ {ρ(η(z(θi)), η(y))}

[Blum, JASA, 2010]

Alphabet soup

ABC-NP (density estimation)

Use of the kernel weights

Kδ {ρ(η(z(θ)), η(y))}

leads to the NP estimate of the posterior conditional density∑i Kb(θi − θ)Kδ {ρ(η(z(θi)), η(y))}∑

iKδ {ρ(η(z(θi)), η(y))}

[Blum, JASA, 2010]

Alphabet soup

ABC-NP (density estimations)

Other versions incorporating regression adjustments∑i Kb(θ

∗i − θ)Kδ {ρ(η(z(θi)), η(y))}∑iKδ {ρ(η(z(θi)), η(y))}

In all cases, error

E[g(θ|y)]− g(θ|y) = cb2 + cδ2 + OP (b2 + δ2) + OP (1/nδd)

var(g(θ|y)) =c

nbδd(1 + oP (1))

Alphabet soup

In all cases, error

var(g(θ|y)) =c

nbδd(1 + oP (1))

[Blum, JASA, 2010]

Alphabet soup

In all cases, error

var(g(θ|y)) =c

nbδd(1 + oP (1))

[standard NP calculations]

Alphabet soup

Which summary statistics?

Fundamental difficulty of the choice of the summary statistic whenthere is no non-trivial sufficient statistic [except when done by theexperimenters in the field]

Alphabet soup

Which summary statistics?

Fundamental difficulty of the choice of the summary statistic whenthere is no non-trivial sufficient statistic [except when done by theexperimenters in the field]

Starting from a large collection of summary statistics is available,Joyce and Marjoram (2008) consider the sequential inclusion intothe ABC target, with a stopping rule based on a likelihood ratiotest.

Semi-automatic ABC

Fearnhead and Prangle (2012) study ABC and the selection of thesummary statistic in close proximity to Wilkinson’s (2008) proposalABC then considered from a purely inferential viewpoint andcalibrated for estimation purposes

Semi-automatic ABC

Fearnhead and Prangle (2012) study ABC and the selection of thesummary statistic in close proximity to Wilkinson’s (2008) proposalABC then considered from a purely inferential viewpoint andcalibrated for estimation purposesUse of a randomised (or ‘noisy’) version of the summary statistics

η(y) = η(y) + τε

Derivation of a well-calibrated version of ABC, i.e. an algorithmthat gives proper predictions for the distribution associated withthis randomised summary statistic [calibration constraint: ABCapproximation with same posterior mean as the true randomisedposterior]

Semi-automatic ABC

Fearnhead and Prangle (2012) study ABC and the selection of thesummary statistic in close proximity to Wilkinson’s (2008) proposalABC then considered from a purely inferential viewpoint andcalibrated for estimation purposes

I Optimality of the posterior expectation E[θ|y] of theparameter of interest as summary statistics η(y)!

I Use of the standard quadratic loss function

(θ − θ0)TA(θ − θ0) .

Semi-automatic ABC

Details on Fearnhead and Prangle (F&P) ABC

Use of a summary statistic S(·), an importance proposal g(·), akernel K(·) ≤ 1 and a bandwidth h > 0 such that

(θ,ysim) ∼ g(θ)f(ysim|θ)

is accepted with probability (hence the bound)

K[{S(ysim)− sobs}/h]

and the corresponding importance weight defined by

π(θ)/g(θ)

[Fearnhead & Prangle, 2012]

Semi-automatic ABC

Errors, errors, and errors

Three levels of approximation

I π(θ|yobs) by π(θ|sobs) loss of information[ignored]

I π(θ|sobs) by

πABC(θ|sobs) =

∫π(s)K[{s− sobs}/h]π(θ|s) ds∫

π(s)K[{s− sobs}/h] ds

noisy observations

I πABC(θ|sobs) by importance Monte Carlo based on Nsimulations, represented by var(a(θ)|sobs)/Nacc [expectednumber of acceptances]

Semi-automatic ABC

Average acceptance asymptotics

For the average acceptance probability/approximate likelihood

p(θ|sobs) =

∫f(ysim|θ)K[{S(ysim)− sobs}/h] dysim ,

overall acceptance probability

p(sobs) =

∫p(θ|sobs)π(θ) dθ = π(sobs)h

d + o(hd)

[F&P, Lemma 1]

Semi-automatic ABC

Calibration of h

“This result gives insight into how S(·) and h affect the MonteCarlo error. To minimize Monte Carlo error, we need hd to be nottoo small. Thus ideally we want S(·) to be a low dimensionalsummary of the data that is sufficiently informative about θ thatπ(θ|sobs) is close, in some sense, to π(θ|yobs)” (F&P, p.5)

I turns h into an absolute value while it should becontext-dependent and user-calibrated

I only addresses one term in the approximation error andacceptance probability (“curse of dimensionality”)

I h large prevents πABC(θ|sobs) to be close to π(θ|sobs)

I d small prevents π(θ|sobs) to be close to π(θ|yobs) (“curse of[dis]information”)

Semi-automatic ABC

Converging ABC

Theorem (F&P)

For noisy ABC, the expected noisy-ABC log-likelihood,

E {log[p(θ|sobs)]} =

∫ ∫log[p(θ|S(yobs) + ε)]π(yobs|θ0)K(ε)dyobsdε,

has its maximum at θ = θ0.

True for any choice of summary statistic? even ancilary statistics?![Imposes at least identifiability...]

Relevant in asymptotia and not for the data

Semi-automatic ABC

Converging ABC

Corollary (F&P)

For noisy ABC, the ABC posterior converges onto a point mass onthe true parameter value as m→∞.

For standard ABC, not always the case (unless h goes to zero).

Strength of regularity conditions (c1) and (c2) in Bernardo& Smith, 1994?

[out-of-reach constraints on likelihood and posterior]Again, there must be conditions imposed upon summarystatistics...

Semi-automatic ABC

Caveat

Since E(θ|yobs) is most usually unavailable, F&P suggest

(i) use a pilot run of ABC to determine a region of non-negligibleposterior mass;

(ii) simulate sets of parameter values and data;

(iii) use the simulated sets of parameter values and data toestimate the summary statistic; and

(iv) run ABC with this choice of summary statistic.

where is the assessment of the first stage error?

Semi-automatic ABC

Caveat

Since E(θ|yobs) is most usually unavailable, F&P suggest

(i) use a pilot run of ABC to determine a region of non-negligibleposterior mass;

(ii) simulate sets of parameter values and data;

(iii) use the simulated sets of parameter values and data toestimate the summary statistic; and

(iv) run ABC with this choice of summary statistic.

where is the assessment of the first stage error?

simulation-based methods in Econometrics

Econ’ections

Model choice

Similar exploration of simulation-based techniques in Econometrics

I Simulated method of moments

I Method of simulated moments

I Simulated pseudo-maximum-likelihood

I Indirect inference

[Gourieroux & Monfort, 1996]

Simulated method of moments

Given observations yo1:n from a model

yt = r(y1:(t−1), εt, θ) , εt ∼ g(·)

simulate ε?1:n, derive

y?t (θ) = r(y1:(t−1), ε?t , θ)

and estimate θ by

arg minθ

n∑t=1

(yot − y?t (θ))2

Simulated method of moments

Given observations yo1:n from a model

yt = r(y1:(t−1), εt, θ) , εt ∼ g(·)

simulate ε?1:n, derive

y?t (θ) = r(y1:(t−1), ε?t , θ)

and estimate θ by

arg minθ

{n∑t=1

yot −n∑t=1

y?t (θ)

Method of simulated moments

Given a statistic vector K(y) with

Eθ[K(Yt)|y1:(t−1)] = k(y1:(t−1); θ)

find an unbiased estimator of k(y1:(t−1); θ),

k(εt, y1:(t−1); θ)

Estimate θ by

arg minθ

∣∣∣∣∣∣∣∣∣∣n∑t=1

[K(yt)−

S∑s=1

k(εst , y1:(t−1); θ)/S

]∣∣∣∣∣∣∣∣∣∣

[Pakes & Pollard, 1989]

Indirect inference

Minimise (in θ) the distance between estimators β based onpseudo-models for genuine observations and for observationssimulated under the true model and the parameter θ.

[Gourieroux, Monfort, & Renault, 1993;Smith, 1993; Gallant & Tauchen, 1996]

Indirect inference (PML vs. PSE)

Example of the pseudo-maximum-likelihood (PML)

β(y) = arg maxβ

log f?(yt|β, y1:(t−1))

leading to

arg minθ||β(yo)− β(y1(θ), . . . ,yS(θ))||2

whenys(θ) ∼ f(y|θ) s = 1, . . . , S

Indirect inference (PML vs. PSE)

Example of the pseudo-score-estimator (PSE)

β(y) = arg minβ

∂ log f?

∂β(yt|β, y1:(t−1))

leading to

arg minθ||β(yo)− β(y1(θ), . . . ,yS(θ))||2

whenys(θ) ∼ f(y|θ) s = 1, . . . , S

ABC using indirect inference (1)

We present a novel approach for developing summary statisticsfor use in approximate Bayesian computation (ABC) algorithms byusing indirect inference(...) In the indirect inference approach toABC the parameters of an auxiliary model fitted to the data becomethe summary statistics. Although applicable to any ABC technique,we embed this approach within a sequential Monte Carlo algorithmthat is completely adaptive and requires very little tuning(...)

[Drovandi, Pettitt & Faddy, 2011]

c© Indirect inference provides summary statistics for ABC...

ABC using indirect inference (2)

...the above result shows that, in the limit as h→ 0, ABC willbe more accurate than an indirect inference method whose auxiliarystatistics are the same as the summary statistic that is used forABC(...) Initial analysis showed that which method is moreaccurate depends on the true value of θ.

[Fearnhead and Prangle, 2012]

c© Indirect inference provides estimates rather than global inference...

ABC for model choicePrincipleGibbs random fieldsGeneric ABC model choice

Principle

Bayesian model choice

Principle

Several modelsM1,M2, . . .

are considered simultaneously for dataset y and model index Mcentral to inference.Use of a prior π(M = m), plus a prior distribution on theparameter conditional on the value m of the model index, πm(θm)Goal is to derive the posterior distribution of M,

π(M = m|data)

a challenging computational target when models are complex.

Principle

Generic ABC for model choice

Algorithm 2 Likelihood-free model choice sampler (ABC-MC)

for t = 1 to T dorepeat

Generate m from the prior π(M = m)Generate θm from the prior πm(θm)Generate z from the model fm(z|θm)

until ρ{η(z), η(y)} < εSet m(t) = m and θ(t) = θm

end for

[Grelaud & al., 2009; Toni & al., 2009]

Principle

ABC estimates

Posterior probability π(M = m|y) approximated by the frequencyof acceptances from model m

T∑t=1

Im(t)=m .

Principle

ABC estimates

Posterior probability π(M = m|y) approximated by the frequencyof acceptances from model m

T∑t=1

Im(t)=m .

Extension to a weighted polychotomous logistic regressionestimate of π(M = m|y), with non-parametric kernel weights

[Cornuet et al., DIYABC, 2009]

Gibbs random fields

Potts model

Skip MRFs

Potts model

Distribution with an energy function of the form

θS(y) = θ∑l∼i

δyl=yi

where l∼i denotes a neighbourhood structure

In most realistic settings, summation

Zθ =∑x∈X

exp{θS(x)}

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

Gibbs random fields

Potts model

Skip MRFs

Potts model

Distribution with an energy function of the form

θS(y) = θ∑l∼i

δyl=yi

where l∼i denotes a neighbourhood structure

In most realistic settings, summation

Zθ =∑x∈X

exp{θS(x)}

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

Gibbs random fields

Neighbourhood relations

SetupChoice 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.

Gibbs random fields

Model index

Computational target:

P(M = m|x) ∝∫

fm(x|θm)πm(θm) dθm π(M = m)

If S(x) sufficient statistic for the joint parameters(M, θ0, . . . , θM−1),

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

Gibbs random fields

Model index

Computational target:

P(M = m|x) ∝∫

fm(x|θm)πm(θm) dθm π(M = m)

If S(x) sufficient statistic for the joint parameters(M, θ0, . . . , θM−1),

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

Gibbs random fields

Sufficient statistics in Gibbs random fields

Each model m has its own sufficient statistic Sm(·) andS(·) = (S0(·), . . . , SM−1(·)) is also (model-)sufficient.

Explanation: 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)}

c© S(x) is sufficient for the joint parameters

Gibbs random fields

Sufficient statistics in Gibbs random fields

Each model m has its own sufficient statistic Sm(·) andS(·) = (S0(·), . . . , SM−1(·)) is also (model-)sufficient.Explanation: 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)}

c© S(x) is sufficient for the joint parameters

Gibbs random fields

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) =1

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).

Gibbs random fields

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.

Generic ABC model choice

About sufficiency

If η1(x) sufficient statistic for model m = 1 and parameter θ1 andη2(x) sufficient statistic for model m = 2 and parameter θ2,(η1(x), η2(x)) is not always sufficient for (m, θm)

c© Potential loss of information at the testing level

About sufficiency

If η1(x) sufficient statistic for model m = 1 and parameter θ1 andη2(x) sufficient statistic for model m = 2 and parameter θ2,(η1(x), η2(x)) is not always sufficient for (m, θm)

c© Potential loss of information at the testing level

Poisson/geometric example

Samplex = (x1, . . . , xn)

from either a Poisson P(λ) or from a geometric G(p)Sum

n∑i=1

xi = η(x)

sufficient statistic for either model but not simultaneously

Limiting behaviour of B12 (T →∞)

ABC approximation

B12(y) =

∑Tt=1 Imt=1 Iρ{η(zt),η(y)}≤ε∑Tt=1 Imt=2 Iρ{η(zt),η(y)}≤ε

where the (mt, zt)’s are simulated from the (joint) prior

As T go to infinity, limit

Bε12(y) =

∫Iρ{η(z),η(y)}≤επ1(θ1)f1(z|θ1) dz dθ1∫Iρ{η(z),η(y)}≤επ2(θ2)f2(z|θ2) dz dθ2

∫Iρ{η,η(y)}≤επ1(θ1)fη1 (η|θ1) dη dθ1∫Iρ{η,η(y)}≤επ2(θ2)fη2 (η|θ2) dη dθ2

where fη1 (η|θ1) and fη2 (η|θ2) distributions of η(z)

Limiting behaviour of B12 (T →∞)

ABC approximation

B12(y) =

∑Tt=1 Imt=1 Iρ{η(zt),η(y)}≤ε∑Tt=1 Imt=2 Iρ{η(zt),η(y)}≤ε

where the (mt, zt)’s are simulated from the (joint) priorAs T go to infinity, limit

Bε12(y) =

∫Iρ{η(z),η(y)}≤επ1(θ1)f1(z|θ1) dz dθ1∫Iρ{η(z),η(y)}≤επ2(θ2)f2(z|θ2) dz dθ2

∫Iρ{η,η(y)}≤επ1(θ1)fη1 (η|θ1) dη dθ1∫Iρ{η,η(y)}≤επ2(θ2)fη2 (η|θ2) dη dθ2

where fη1 (η|θ1) and fη2 (η|θ2) distributions of η(z)

Limiting behaviour of B12 (ε→ 0)

When ε goes to zero,

Bη12(y) =

∫π1(θ1)fη1 (η(y)|θ1) dθ1∫π2(θ2)fη2 (η(y)|θ2) dθ2

c© Bayes factor based on the sole observation of η(y)

Limiting behaviour of B12 (ε→ 0)

When ε goes to zero,

Bη12(y) =

∫π1(θ1)fη1 (η(y)|θ1) dθ1∫π2(θ2)fη2 (η(y)|θ2) dθ2

c© Bayes factor based on the sole observation of η(y)

Limiting behaviour of B12 (under sufficiency)

If η(y) sufficient statistic in both models,

fi(y|θi) = gi(y)fηi (η(y)|θi)

B12(y) =

∫Θ1π(θ1)g1(y)fη1 (η(y)|θ1) dθ1∫

Θ2π(θ2)g2(y)fη2 (η(y)|θ2) dθ2

=g1(y)

∫π1(θ1)fη1 (η(y)|θ1) dθ1

g2(y)∫π2(θ2)fη2 (η(y)|θ2) dθ2

=g1(y)

g2(y)Bη

12(y) .

[Didelot, Everitt, Johansen & Lawson, 2011]

c© No discrepancy only when cross-model sufficiency

Limiting behaviour of B12 (under sufficiency)

If η(y) sufficient statistic in both models,

fi(y|θi) = gi(y)fηi (η(y)|θi)

B12(y) =

∫Θ1π(θ1)g1(y)fη1 (η(y)|θ1) dθ1∫

Θ2π(θ2)g2(y)fη2 (η(y)|θ2) dθ2

=g1(y)

∫π1(θ1)fη1 (η(y)|θ1) dθ1

g2(y)∫π2(θ2)fη2 (η(y)|θ2) dθ2

=g1(y)

g2(y)Bη

12(y) .

[Didelot, Everitt, Johansen & Lawson, 2011]

c© No discrepancy only when cross-model sufficiency

Poisson/geometric example (back)

Samplex = (x1, . . . , xn)

from either a Poisson P(λ) or from a geometric G(p)

Discrepancy ratio

g2(x)=S!n−S/

∏i xi!

n+S−1S

Poisson/geometric discrepancy

Range of B12(x) versus Bη12(x): The values produced have

nothing in common.

Formal recovery

Creating an encompassing exponential family

f(x|θ1, θ2, α1, α2) ∝ exp{θT1 η1(x) + θT

2 η2(x) +α1t1(x) +α2t2(x)}

leads to a sufficient statistic (η1(x), η2(x), t1(x), t2(x))[Didelot, Everitt, Johansen & Lawson, 2011]

Formal recovery

2 η2(x) +α1t1(x) +α2t2(x)}

In the Poisson/geometric case, if∏i xi! is added to S, no

discrepancy

Formal recovery

2 η2(x) +α1t1(x) +α2t2(x)}

Only applies in genuine sufficiency settings...

c© Inability to evaluate information loss due to summarystatistics

Meaning of the ABC-Bayes factor

The ABC approximation to the Bayes Factor is based solely on thesummary statistics....

In the Poisson/geometric case, if E[yi] = θ0 > 0,

limn→∞

Bη12(y) =

(θ0 + 1)2

θ0e−θ0

Meaning of the ABC-Bayes factor

The ABC approximation to the Bayes Factor is based solely on thesummary statistics....In the Poisson/geometric case, if E[yi] = θ0 > 0,

limn→∞

Bη12(y) =

(θ0 + 1)2

θ0e−θ0

MA example

Evolution [against ε] of ABC Bayes factor, in terms of frequencies ofvisits to models MA(1) (left) and MA(2) (right) when ε equal to10, 1, .1, .01% quantiles on insufficient autocovariance distances. Sampleof 50 points from a MA(2) with θ1 = 0.6, θ2 = 0.2. True Bayes factorequal to 17.71.

MA example

Evolution [against ε] of ABC Bayes factor, in terms of frequencies ofvisits to models MA(1) (left) and MA(2) (right) when ε equal to10, 1, .1, .01% quantiles on insufficient autocovariance distances. Sampleof 50 points from a MA(1) model with θ1 = 0.6. True Bayes factor B21

equal to .004.

A population genetics evaluation

Gene free

Population genetics example with

I 3 populations

I 2 scenari

I 15 individuals

I 5 loci

I single mutation parameter

I 24 summary statistics

I 2 million ABC proposal

I importance [tree] sampling alternative

A population genetics evaluation

Gene free

Population genetics example with

I 3 populations

I 2 scenari

I 15 individuals

I 5 loci

I single mutation parameter

I 24 summary statistics

I 2 million ABC proposal

I importance [tree] sampling alternative

Stability of importance sampling

Comparison with ABC

Use of 24 summary statistics and DIY-ABC logistic correction

●●

● ●

●●

0.0 0.2 0.4 0.6 0.8 1.0

importance sampling

●●

●● ●

●●

● ●

●●

● ●

Comparison with ABC

Use of 15 summary statistics

●●

● ●

●●

● ●

●●

● ●●

●●

−4 −2 0 2 4 6

importance sampling

Comparison with ABC

Use of 15 summary statistics and DIY-ABC logistic correction

●●

● ●

●●

● ●

●●

● ●●

●●

−4 −2 0 2 4 6

importance sampling

●●

● ●

●●

● ●

●●

The only safe cases???

Besides specific models like Gibbs random fields,

using distances over the data itself escapes the discrepancy...[Toni & Stumpf, 2010; Sousa & al., 2009]

...and so does the use of more informal model fitting measures[Ratmann & al., 2009]

The only safe cases???

Besides specific models like Gibbs random fields,

using distances over the data itself escapes the discrepancy...[Toni & Stumpf, 2010; Sousa & al., 2009]

...and so does the use of more informal model fitting measures[Ratmann & al., 2009]

ABC model choice consistency

Model choice consistencyFormalised frameworkConsistency resultsSummary statisticsConclusions

Formalised framework

The starting point

Central question to the validation of ABC for model choice:

When is a Bayes factor based on an insufficient statisticT (y) consistent?

Note: c© drawn on T (y) through BT12(y) necessarily differs from

c© drawn on y through B12(y)

The starting point

Central question to the validation of ABC for model choice:

When is a Bayes factor based on an insufficient statisticT (y) consistent?

Note: c© drawn on T (y) through BT12(y) necessarily differs from

c© drawn on y through B12(y)

A benchmark if toy example

Comparison suggested by referee of PNAS paper [thanks]:[X, Cornuet, Marin, & Pillai, Aug. 2011]

Model M1: y ∼ N (θ1, 1) opposedto model M2: y ∼ L(θ2, 1/

√2), Laplace distribution with mean θ2

and scale parameter 1/√

2 (variance one).

2 (variance one).Four possible statistics

1. sample mean y (sufficient for M1 if not M2);

2. sample median med(y) (insufficient);

3. sample variance var(y) (ancillary);

4. median absolute deviation mad(y) = med(|y −med(y)|);

2 (variance one).

●●

Gauss Laplace

2 (variance one).

●●

Gauss Laplace

●●

Gauss Laplace

Framework

Starting from sample

y = (y1, . . . , yn)

the observed sample, not necessarily iid with true distribution

y ∼ Pn

Summary statistics

T (y) = T n = (T1(y), T2(y), · · · , Td(y)) ∈ Rd

with true distribution T n ∼ Gn.

Framework

– under M1, y ∼ F1,n(·|θ1) where θ1 ∈ Θ1 ⊂ Rp1

– under M2, y ∼ F2,n(·|θ2) where θ2 ∈ Θ2 ⊂ Rp2

turned into

– under M1, T (y) ∼ G1,n(·|θ1), and θ1|T (y) ∼ π1(·|T n)

– under M2, T (y) ∼ G2,n(·|θ2), and θ2|T (y) ∼ π2(·|T n)

Consistency results

Assumptions

A collection of asymptotic “standard” assumptions:

[A1] is a standard central limit theorem under the true model[A2] controls the large deviations of the estimator T n from the estimandµ(θ)[A3] is the standard prior mass condition found in Bayesian asymptotics(di effective dimension of the parameter)[A4] restricts the behaviour of the model density against the true density

[Think CLT!]

Consistency results

Assumptions

[A1] There exist a sequence {vn} converging to +∞,a distribution Q,a symmetric, d× d positive definite matrix V0and a vector µ0 ∈ Rd such that

vnV−1/20 (T n − µ0)

n→∞ Q, under Gn

[Think CLT!]

Consistency results

Assumptions

[Think CLT!][A2] For i = 1, 2, there exist sets Fn,i ⊂ Θi,

functions µi(θi) and constants εi, τi, αi > 0 such that for all τ > 0,

supθi∈Fn,i

[|T n − µi(θi)| > τ |µi(θi)− µ0| ∧ εi |θi

](|τµi(θi)− µ0| ∧ εi)−αi

. v−αin

withπi(Fcn,i) = o(v−τin ).

Consistency results

Assumptions

[Think CLT!]

[A3] For u > 0

Sn,i(u) ={θi ∈ Fn,i; |µi(θi)− µ0| ≤ u v−1n

}if inf{|µi(θi)− µ0|; θi ∈ Θi} = 0, there exist constants di < τi ∧ αi − 1such that

πi(Sn,i(u)) ∼ udiv−din , ∀u . vn

Consistency results

Assumptions

[Think CLT!]

[A4] If inf{|µi(θi)− µ0|; θi ∈ Θi} = 0, for any ε > 0, there existU, δ > 0 and (En)n such that, if θi ∈ Sn,i(U)

En ⊂ {t; gi(t|θi) < δgn(t)} and Gn (Ecn) < ε.

Consistency results

Assumptions

[Think CLT!]

Again[A1] is a standard central limit theorem under the true model[A2] controls the large deviations of the estimator T n from the estimandµ(θ)[A3] is the standard prior mass condition found in Bayesian asymptotics(di effective dimension of the parameter)

[A4] restricts the behaviour of the model density against the true density

Consistency results

Effective dimension

Understanding di in [A3]: defined only whenµ0 ∈ {µi(θi), θi ∈ Θi},

πi(θi : |µi(θi)− µ0| < n−1/2) = O(n−di/2)

is the effective dimension of the model Θi around µ0

Consistency results

Effective dimension

Understanding di in [A3]:when inf{|µi(θi)− µ0|; θi ∈ Θi} = 0,

mi(Tn)

gn(T n)∼ v−din ,

thuslog(mi(T

n)/gn(T n)) ∼ −di log vn

and v−din penalization factor resulting from integrating θi out asthe effective number of parameters appears in the DIC

Consistency results

Effective dimension

Understanding di in [A3]:In regular models, di dimension of T (Θi), leading to BIC. Innon-regular models, di can be smaller.

Consistency results

Asymptotic marginals

Asymptotically, under [A1]–[A4]

mi(t) =

∫Θi

gi(t|θi)πi(θi) dθi

is such that(i) if inf{|µi(θi)− µ0|; θi ∈ Θi} = 0,

Clvd−din ≤ mi(T

n) ≤ Cuvd−din

and(ii) if inf{|µi(θi)− µ0|; θi ∈ Θi} > 0

mi(Tn) = oPn [vd−τin + vd−αin ].

Consistency results

Within-model consistency

Under same assumptions,if inf{|µi(θi)− µ0|; θi ∈ Θi} = 0,

the posterior distribution of µi(θi) given T n is consistent at rate1/vn provided αi ∧ τi > di.

Consistency results

Between-model consistency

Consequence of above is that asymptotic behaviour of the Bayesfactor is driven by the asymptotic mean value of T n under bothmodels. And only by this mean value!

Consistency results

Indeed, if

inf{|µ0 − µ2(θ2)|; θ2 ∈ Θ2} = inf{|µ0 − µ1(θ1)|; θ1 ∈ Θ1} = 0

Clv−(d1−d2)n ≤ m1(T n)

/m2(T n) ≤ Cuv−(d1−d2)

Consistency results

Else, if

inf{|µ0 − µ2(θ2)|; θ2 ∈ Θ2} > inf{|µ0 − µ1(θ1)|; θ1 ∈ Θ1} = 0

thenm1(T n)

m2(T n)≥ Cu min

(v−(d1−α2)n , v−(d1−τ2)

Consistency results

Consistency theorem

inf{|µ0 − µ2(θ2)|; θ2 ∈ Θ2} = inf{|µ0 − µ1(θ1)|; θ1 ∈ Θ1} = 0,

Bayes factorBT

12 = O(v−(d1−d2)n )

irrespective of the true model. It is inconsistent since it alwayspicks the model with the smallest dimension

Consistency results

Consistency theorem

If Pn belongs to one of the two models and if µ0 cannot beattained by the other one :

0 = min (inf{|µ0 − µi(θi)|; θi ∈ Θi}, i = 1, 2)

< max (inf{|µ0 − µi(θi)|; θi ∈ Θi}, i = 1, 2) ,

then the Bayes factor BT12 is consistent

Summary statistics

Consequences on summary statistics

Bayes factor driven by the means µi(θi) and the relative position ofµ0 wrt both sets {µi(θi); θi ∈ Θi}, i = 1, 2.

For ABC, this implies the most likely statistics T n are ancillarystatistics with different mean values under both models

Else, if T n asymptotically depends on some of the parameters ofthe models, it is quite likely that there exists θi ∈ Θi such thatµi(θi) = µ0 even though model M1 is misspecified

Summary statistics

Consequences on summary statistics

Bayes factor driven by the means µi(θi) and the relative position ofµ0 wrt both sets {µi(θi); θi ∈ Θi}, i = 1, 2.

For ABC, this implies the most likely statistics T n are ancillarystatistics with different mean values under both models

Else, if T n asymptotically depends on some of the parameters ofthe models, it is quite likely that there exists θi ∈ Θi such thatµi(θi) = µ0 even though model M1 is misspecified

Summary statistics

Toy example: Laplace versus Gauss [1]

T n = n−1n∑i=1

X4i , µ1(θ) = 3 + θ4 + 6θ2, µ2(θ) = 6 + · · ·

and the true distribution is Laplace with mean θ0 = 1, under theGaussian model the value θ∗ = 2

√3− 3 leads to µ0 = µ(θ∗)

[here d1 = d2 = d = 1]

Summary statistics

T n = n−1n∑i=1

X4i , µ1(θ) = 3 + θ4 + 6θ2, µ2(θ) = 6 + · · ·

and the true distribution is Laplace with mean θ0 = 1, under theGaussian model the value θ∗ = 2

√3− 3 leads to µ0 = µ(θ∗)

Summary statistics

T n = n−1n∑i=1

X4i , µ1(θ) = 3 + θ4 + 6θ2, µ2(θ) = 6 + · · ·

Summary statistics

T n = n−1n∑i=1

X4i , µ1(θ) = 3 + θ4 + 6θ2, µ2(θ) = 6 + · · ·

●●

n=1000

Fourth moment

Summary statistics

WhenT (y) =

{y(4)n , y(6)

}and the true distribution is Laplace with mean θ0 = 0, thenµ0 = 6, µ1(θ∗1) = 6 with θ∗1 = 2

√3− 3

[d1 = 1 and d2 = 1/2]thus

B12 ∼ n−1/4 → 0 : consistent

Under the Gaussian model µ0 = 3, µ2(θ2) ≥ 6 > 3 ∀θ2

B12 → +∞ : consistent

Summary statistics

WhenT (y) =

{y(4)n , y(6)

}and the true distribution is Laplace with mean θ0 = 0, thenµ0 = 6, µ1(θ∗1) = 6 with θ∗1 = 2

√3− 3

[d1 = 1 and d2 = 1/2]thus

B12 ∼ n−1/4 → 0 : consistent

Under the Gaussian model µ0 = 3, µ2(θ2) ≥ 6 > 3 ∀θ2

B12 → +∞ : consistent

Summary statistics

WhenT (y) =

{y(4)n , y(6)

0.0 0.2 0.4 0.6 0.8 1.0

posterior probabilities

Fourth and sixth moments

Summary statistics

Checking for adequate statistics

After running ABC, i.e. creating reference tables of (θi, xi) fromboth joints, straightforward derivation of ABC estimates θ1 and θ2.

Evaluation of E1θ1

[T (X)] and E2θ2

[T (X)] allows for detection of

different means under both models via Monte Carlo simulations

Summary statistics

Toy example: Laplace versus Gauss

●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●

Gauss Laplace Gauss Laplace

Normalised χ2 without and with mad

Summary statistics

A population genetic illustration

Two populations (1 and 2) having diverged at a fixed known timein the past and third population (3) which diverged from one ofthose two populations (models 1 and 2, respectively).

Observation of 50 diploid individuals/population genotyped at 5,50 or 100 independent microsatellite loci.

Model 2

Summary statistics

Two populations (1 and 2) having diverged at a fixed known timein the past and third population (3) which diverged from one ofthose two populations (models 1 and 2, respectively).

Observation of 50 diploid individuals/population genotyped at 5,50 or 100 independent microsatellite loci.

Stepwise mutation model: the number of repeats of the mutatedgene increases or decreases by one. Mutation rate µ common to allloci set to 0.005 (single parameter) with uniform prior distribution

µ ∼ U [0.0001, 0.01]

Summary statistics

Summary statistics associated to the (δµ)2 distance

xl,i,j repeated number of allele in locus l = 1, . . . , L for individuali = 1, . . . , 100 within the population j = 1, 2, 3. Then

(δµ)2j1,j2 =

L∑l=1

100∑i1=1

xl,i1,j1 −1

100∑i2=1

xl,i2,j2

Summary statistics

For two copies of locus l with allele sizes xl,i,j1 and xl,i′,j2 , mostrecent common ancestor at coalescence time τj1,j2 , gene genealogydistance of 2τj1,j2 , hence number of mutations Poisson withparameter 2µτj1,j2 . Therefore,

E{(xl,i,j1 − xl,i′,j2

)2 |τj1,j2} = 2µτj1,j2

andModel 1 Model 2

(δµ)21,2

}2µ1t

′ 2µ2t′

(δµ)21,3

}2µ1t 2µ2t

(δµ)22,3

}2µ1t

′ 2µ2t

Summary statistics

I Bayes factor based only on distance (δµ)21,2 not convergent: if

µ1 = µ2, same expectations.

I Bayes factor based only on distance (δµ)21,3 or (δµ)2

2,3 notconvergent: if µ1 = 2µ2 or 2µ1 = µ2 same expectation.

I if two of the three distances are used, Bayes factor converges:there is no (µ1, µ2) for which all expectations are equal.

Summary statistics

● ●

5 50 100

DM2(12)

●●

● ●

5 50 100

DM2(13)

●●

●●●●●

●●●●●●●

5 50 100

DM2(13) & DM2(23)

Posterior probabilities that the data is from model 1 for 5, 50and 100 loci

Summary statistics

Embedded models

When M1 submodel of M2, and if the true distribution belongs tothe smaller model M1, Bayes factor is of order

v−(d1−d2)n

Summary statistics

Embedded models

v−(d1−d2)n

If summary statistic only informative on a parameter that is thesame under both models, i.e if d1 = d2, then

Summary statistics

Embedded models

v−(d1−d2)n

Else, d1 < d2 and Bayes factor is going to ∞ under M1. If truedistribution not in M1, then

Summary statistics

Another toy example: Quantile distribution

Q(p;A,B, g, k) = A+B

1− exp{−gz(p)}1 + exp{−gz(p)}

] [1 + z(p)2

]kz(p)

A,B, g and k, location, scale, skewness and kurtosis parametersEmbedded models:

I M1 : g = 0 and k ∼ U [−1/2, 5]

I M2 : g ∼ U [0, 4] and k ∼ U [−1/2, 5].

Summary statistics

Consistency [or not]

●●

●●●

●●

Conclusions

• Model selection feasible with ABC• Choice of summary statistics is paramount• At best, ABC → π(. | T (y)) which concentrates around µ0

• For estimation : {θ;µ(θ) = µ0} = θ0

• For testing {µ1(θ1), θ1 ∈ Θ1} ∩ {µ2(θ2), θ2 ∈ Θ2} = ∅

Conclusions

• Model selection feasible with ABC• Choice of summary statistics is paramount• At best, ABC → π(. | T (y)) which concentrates around µ0

• For estimation : {θ;µ(θ) = µ0} = θ0

• For testing {µ1(θ1), θ1 ∈ Θ1} ∩ {µ2(θ2), θ2 ∈ Θ2} = ∅

glasgow seminar, april 20, 2012

bayesian model choicechristian

abc algorithmfor

observation y f y

approximativewhen y

zydz y

tfrom y

y f zdzdwhere

likelihood function

Education

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