Pittsburgh and Toronto "Halloween US trip" seminars

Relevant statistics for (approximate) Bayesianmodel choice

Christian P. RobertUniversite Paris-Dauphine, Paris & University of Warwick, Coventry

bayesianstatistics@gmail.com

a meeting of potential interest?!

MCMSki IV to be held in Chamonix Mt Blanc, France, fromMonday, Jan. 6 to Wed., Jan. 8 (and BNP workshop onJan. 9), 2014All aspects of MCMC++ theory and methodology and applicationsand more! All posters welcomed!

Website http://www.pages.drexel.edu/∼mwl25/mcmski/

Outline

Intractable likelihoods

ABC methods

ABC as an inference machine

ABC for model choice

Model choice consistency

Conclusion and perspectives

Intractable likelihood

Case of a well-defined statistical model where the likelihoodfunction

`(θ|y) = f (y1, . . . , yn|θ)

I is (really!) not available in closed form

I cannot (easily!) be either completed or demarginalised

I cannot be (at all!) estimated by an unbiased estimator

c© Prohibits direct implementation of a generic MCMC algorithmlike Metropolis–Hastings

Intractable likelihood

`(θ|y) = f (y1, . . . , yn|θ)

I is (really!) not available in closed form

I cannot (easily!) be either completed or demarginalised

I cannot be (at all!) estimated by an unbiased estimator

c© Prohibits direct implementation of a generic MCMC algorithmlike Metropolis–Hastings

Getting approximative

`(θ|y) = f (y1, . . . , yn|θ)

is out of reach

Empirical approximations to the original B problem

I Degrading the data precision down to a tolerance ε

I Replacing the likelihood with a non-parametric approximation

I Summarising/replacing the data with insufficient statistics

`(θ|y) = f (y1, . . . , yn|θ)

is out of reach

`(θ|y) = f (y1, . . . , yn|θ)

is out of reach

Different [levels of] worries about abc

Impact on B inference

I a mere computational issue (that will eventually end up beingsolved by more powerful computers, &tc, even if too costly inthe short term, as for regular Monte Carlo methods) Not!

I an inferential issue (opening opportunities for new inferencemachine on complex/Big Data models, with legitimitydiffering from classical B approach)

I a Bayesian conundrum (how closely related to the/a Bapproach? more than recycling B tools? true B with rawdata? a new type of empirical B inference?)

Approximate Bayesian computation

ABC methodsGenesis of ABCABC basicsAdvances and interpretationsABC as knn

Genetic background of ABC

skip genetics

ABC is a recent computational technique that only requires beingable to sample from the likelihood f (·|θ)

This technique stemmed from population genetics models, about15 years ago, and population geneticists still contributesignificantly to methodological developments of ABC.

[Griffith & al., 1997; Tavare & al., 1999]

Demo-genetic inference

Each model is characterized by a set of parameters θ that coverhistorical (time divergence, admixture time ...), demographics(population sizes, admixture rates, migration rates, ...) and genetic(mutation rate, ...) factors

The goal is to estimate these parameters from a dataset ofpolymorphism (DNA sample) y observed at the present time

Problem:

most of the time, we cannot calculate the likelihood of thepolymorphism data f (y|θ)...

Demo-genetic inference

Each model is characterized by a set of parameters θ that coverhistorical (time divergence, admixture time ...), demographics(population sizes, admixture rates, migration rates, ...) and genetic(mutation rate, ...) factors

The goal is to estimate these parameters from a dataset ofpolymorphism (DNA sample) y observed at the present time

Problem:

most of the time, we cannot calculate the likelihood of thepolymorphism data f (y|θ)...

Neutral model at a given microsatellite locus, in a closedpanmictic population at equilibrium

Sample of 8 genes

Mutations according tothe Simple stepwiseMutation Model(SMM)• date of the mutations ∼

Poisson process withintensity θ/2 over thebranches• MRCA = 100• independent mutations:±1 with pr. 1/2

Kingman’s genealogyWhen time axis isnormalized,T (k) ∼ Exp(k(k − 1)/2)

Observations: leafs of the treeθ = ?

Much more interesting models. . .

I several independent locusIndependent gene genealogies and mutations

I different populationslinked by an evolutionary scenario made of divergences,admixtures, migrations between populations, selectionpressure, etc.

I larger sample sizeusually between 50 and 100 genes

A typical evolutionary scenario:

POP 0 POP 1 POP 2

Instance of ecological questions

I How the Asian ladybirdbeetle arrived in Europe?

I Why does they swarm rightnow?

I What are the routes ofinvasion?

I How to get rid of them?

I Why did the chicken crossthe road?

[Lombaert & al., 2010, PLoS ONE]

Worldwide invasion routes of Harmonia axyridis

Worldwide routes of invasion of Harmonia axyridis

For each outbreak, the arrow indicates the most likely invasionpathway and the associated posterior probability, with 95% credibleintervals in brackets

[Estoup et al., 2012, Molecular Ecology Res.]

c© Intractable likelihood

Missing (too much missing!) data structure:

f (y|θ) =

f (y|G ,θ)f (G |θ)dG

cannot be computed in a manageable way...[Stephens & Donnelly, 2000]

The genealogies are considered as nuisance parameters

This modelling clearly differs from the phylogenetic perspective

where the tree is the parameter of interest.

c© Intractable likelihood

Missing (too much missing!) data structure:

f (y|θ) =

f (y|G ,θ)f (G |θ)dG

cannot be computed in a manageable way...[Stephens & Donnelly, 2000]

The genealogies are considered as nuisance parameters

This modelling clearly differs from the phylogenetic perspective

where the tree is the parameter of interest.

A?B?C?

I A stands for approximate[wrong likelihood / pic?]

I B stands for Bayesian

I C stands for computation[producing a parametersample]

A?B?C?

ESS=155.6

−0.5 0.0 0.5 1.0

ESS=75.93

−0.4 −0.2 0.0 0.2 0.4

ESS=76.87

−0.4 −0.2 0.0 0.2

ESS=91.54

−0.6 −0.4 −0.2 0.0 0.2

ESS=108.4

−0.4 0.0 0.2 0.4 0.6

ESS=85.13

−0.2 0.0 0.2 0.4 0.6

ESS=149.1

−0.5 0.0 0.5 1.0

ESS=96.31

−0.4 0.0 0.2 0.4 0.6

ESS=83.77

−0.6 −0.4 −0.2 0.0 0.2 0.4

ESS=155.7

−0.5 0.0 0.5

ESS=92.42

−0.4 0.0 0.2 0.4 0.6

3.0 ESS=95.01

−0.4 0.0 0.2 0.4 0.6

ESS=139.2

−0.6 −0.2 0.2 0.6

ESS=99.33

−0.4 −0.2 0.0 0.2 0.4

ESS=87.28

−0.2 0.0 0.2 0.4 0.6

How Bayesian is aBc?

Could we turn the resolution into a Bayesian answer?

I ideally so (not meaningfull: requires ∞-ly powerful computer

I approximation error unknown (w/o costly simulation)

I true B for wrong model (formal and artificial)

I true B for noisy model (much more convincing)

I true B for estimated likelihood (back to econometrics?)

I illuminating the tension between information and precision

ABC methodology

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

Foundation

For an observation y ∼ f (y|θ), under the prior π(θ), if one keepsjointly simulating

θ′ ∼ π(θ) , z ∼ f (z|θ′) ,

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

θ′ ∼ π(θ|y)

[Rubin, 1984; Diggle & Gratton, 1984; Tavare et al., 1997]

ABC methodology

Foundation

θ′ ∼ π(θ) , z ∼ f (z|θ′) ,

θ′ ∼ π(θ|y)

ABC methodology

Foundation

θ′ ∼ π(θ) , z ∼ f (z|θ′) ,

θ′ ∼ π(θ|y)

A as A...pproximative

When y is a continuous random variable, strict equality z = y isreplaced with a tolerance zone

ρ(y, z) 6 ε

where ρ is a distanceOutput distributed from

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

[Pritchard et al., 1999]

A as A...pproximative

When y is a continuous random variable, strict equality z = y isreplaced with a tolerance zone

ρ(y, z) 6 ε

where ρ is a distanceOutput distributed from

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

[Pritchard et al., 1999]

ABC algorithm

In most implementations, further degree of A...pproximation:

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)} 6 εset θi = θ

end for

where η(y) defines a (not necessarily sufficient) statistic

Output

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:

πε(θ|y) =

∫πε(θ, z|y)dz ≈ π(θ|y) .

...does it?!

Output

πε(θ|y) =

...does it?!

Output

πε(θ|y) =

...does it?!

Output

The idea behind ABC is that the summary statistics coupled with asmall tolerance should provide a good approximation of therestricted posterior distribution:

πε(θ|y) =

∫πε(θ, z|y)dz ≈ π(θ|η(y)) .

Not so good..!

Comments

I Role of distance paramount (because ε 6= 0)

I Scaling of components of η(y) is also determinant

I ε matters little if “small enough”

I representative of “curse of dimensionality”

I small is beautiful!

I the data as a whole may be paradoxically weakly informativefor ABC

curse of dimensionality in summaries

Consider that

I the simulated summary statistics η(y1), . . . ,η(yN)

I the observed summary statistics η(yobs)

are iid with uniform distribution on [0, 1]d

Take δ∞(d ,N) = E[

mini=1,...,N

∥∥η(yobs) − η(yi )∥∥∞]

N = 100 N = 1, 000 N = 10, 000 N = 100, 000δ∞(1,N) 0.0025 0.00025 0.000025 0.0000025δ∞(2,N) 0.033 0.01 0.0033 0.001δ∞(10,N) 0.28 0.22 0.18 0.14δ∞(200,N) 0.48 0.48 0.47 0.46

ABC (simul’) advances

how approximative is ABC? ABC as knn

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]

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

ABC as knn

[Biau et al., 2013, Annales de l’IHP]

Practice of ABC: determine tolerance ε as a quantile on observeddistances, say 10% or 1% quantile,

ε = εN = qα(d1, . . . , dN)

I Interpretation of ε as nonparametric bandwidth onlyapproximation of the actual practice

[Blum & Francois, 2010]

I ABC is a k-nearest neighbour (knn) method with kN = NεN[Loftsgaarden & Quesenberry, 1965]

ABC as knn

ε = εN = qα(d1, . . . , dN)

ABC as knn

ε = εN = qα(d1, . . . , dN)

ABC consistency

Provided

kN/ log logN −→∞ and kN/N −→ 0

as N →∞, for almost all s0 (with respect to the distribution ofS), with probability 1,

kN∑j=1

ϕ(θj) −→ E[ϕ(θj)|S = s0]

[Devroye, 1982]

Biau et al. (2013) also recall pointwise and integrated mean square errorconsistency results on the corresponding kernel estimate of theconditional posterior distribution, under constraints

kN →∞, kN/N → 0, hN → 0 and hpNkN →∞,

ABC consistency

Provided

kN/ log logN −→∞ and kN/N −→ 0

as N →∞, for almost all s0 (with respect to the distribution ofS), with probability 1,

kN∑j=1

ϕ(θj) −→ E[ϕ(θj)|S = s0]

[Devroye, 1982]

Biau et al. (2013) also recall pointwise and integrated mean square errorconsistency results on the corresponding kernel estimate of theconditional posterior distribution, under constraints

kN →∞, kN/N → 0, hN → 0 and hpNkN →∞,

Rates of convergence

Further assumptions (on target and kernel) allow for precise(integrated mean square) convergence rates (as a power of thesample size N), derived from classical k-nearest neighbourregression, like

I when m = 1, 2, 3, kN ≈ N(p+4)/(p+8) and rate N− 4p+8

I when m = 4, kN ≈ N(p+4)/(p+8) and rate N− 4p+8 logN

I when m > 4, kN ≈ N(p+4)/(m+p+4) and rate N− 4m+p+4

[Biau et al., 2013]

Drag: Only applies to sufficient summary statistics

Rates of convergence

Further assumptions (on target and kernel) allow for precise(integrated mean square) convergence rates (as a power of thesample size N), derived from classical k-nearest neighbourregression, like

I when m = 1, 2, 3, kN ≈ N(p+4)/(p+8) and rate N− 4p+8

I when m = 4, kN ≈ N(p+4)/(p+8) and rate N− 4p+8 logN

I when m > 4, kN ≈ N(p+4)/(m+p+4) and rate N− 4m+p+4

[Biau et al., 2013]

Drag: Only applies to sufficient summary statistics

ABC inference machine

ABC methods

ABC as an inference machineError inc.Exact BC and approximatetargetssummary statistic

How Bayesian is aBc..?

I may be a convergent method of inference (meaningful?sufficient? foreign?)

I approximation error unknown (w/o massive simulation)

I pragmatic/empirical B (there is no other solution!)

I many calibration issues (tolerance, distance, statistics)

I the NP side should be incorporated into the whole B picture

I the approximation error should also be part of the B inference

Idea Infer about the error as well as about the parameter:Use of a joint density

f (θ, ε|y) ∝ ξ(ε|y, θ)× πθ(θ)× πε(ε)

where y is the data, and ξ(ε|y, θ) is the prior predictive density ofρ(η(z),η(y)) given θ and y when z ∼ f (z|θ)Warning! Replacement of ξ(ε|y, θ) with a non-parametric kernelapproximation.

[Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS]

f (θ, ε|y) ∝ ξ(ε|y, θ)× πθ(θ)× πε(ε)

Wilkinson’s exact BC (not exactly!)

ABC approximation error (i.e. non-zero tolerance) replaced withexact simulation from a controlled approximation to the target,convolution of true posterior with kernel function

πε(θ, z|y) =π(θ)f (z|θ)Kε(y− z)∫π(θ)f (z|θ)Kε(y− z)dzdθ

with Kε kernel parameterised by bandwidth ε.[Wilkinson, 2013]

Theorem

The ABC algorithm based on the production of a randomisedobservation y = y+ ξ, ξ ∼ Kε, and an acceptance probability of

Kε(y− z)/M

gives draws from the posterior distribution π(θ|y).

Wilkinson’s exact BC (not exactly!)

ABC approximation error (i.e. non-zero tolerance) replaced withexact simulation from a controlled approximation to the target,convolution of true posterior with kernel function

πε(θ, z|y) =π(θ)f (z|θ)Kε(y− z)∫π(θ)f (z|θ)Kε(y− z)dzdθ

with Kε kernel parameterised by bandwidth ε.[Wilkinson, 2013]

Theorem

The ABC algorithm based on the production of a randomisedobservation y = y+ ξ, ξ ∼ Kε, and an acceptance probability of

Kε(y− z)/M

gives draws from the posterior distribution π(θ|y).

How exact a BC?

I Pseudo-data from true model vs. observed data from noisymodel

I Outcome is completely controlled

I Link with ABCµ when assuming y observed withmeasurement error with density Kε

I Relates to the theory of model approximation[Kennedy & O’Hagan, 2001]

I leads to “noisy ABC”: justdo it!, i.e. perturbates the data down to precision ε and proceed

[Fearnhead & Prangle, 2012]

I Requires Kε to be bounded by M

I True approximation error never assessed

How exact a BC?

I Pseudo-data from true model vs. observed data from noisymodel

I Outcome is completely controlled

I Link with ABCµ when assuming y observed withmeasurement error with density Kε

I Relates to the theory of model approximation[Kennedy & O’Hagan, 2001]

I leads to “noisy ABC”: justdo it!, i.e. perturbates the data down to precision ε and proceed

[Fearnhead & Prangle, 2012]

I Requires Kε to be bounded by M

I True approximation error never assessed

Which summary?

Fundamental difficulty of the choice of the summary statistic whenthere is no non-trivial sufficient statistics

I Loss of statistical information balanced against gain in dataroughening

I Approximation error and information loss remain unknown

I Choice of statistics induces choice of distance functiontowards standardisation

I may be imposed for external/practical reasons (e.g., DIYABC)

I may gather several non-B point estimates

I can learn about efficient combination

Which summary?

Semi-automatic ABC

Fearnhead and Prangle (2012) study ABC and the selection of thesummary statistic in close proximity to Wilkinson’s proposal

I ABC considered as inferential method and calibrated as such

I randomised (or ‘noisy’) version of the summary statistics

η(y) = η(y) + τε

I derivation of a well-calibrated version of ABC, i.e. analgorithm that gives proper predictions for the distributionassociated with this randomised summary statistic

[weak property]

Summary [of F&P/statistics)

I optimality of the posterior expectation

E[θ|y]

of the parameter of interest as summary statistics η(y)![requires iterative process]

I use of the standard quadratic loss function

(θ− θ0)TA(θ− θ0)

I recent extension to model choice, optimality of Bayes factor

B12(y)

[F&P, ISBA 2012, Kyoto]

Summary [of F&P/statistics)

I optimality of the posterior expectation

E[θ|y]

of the parameter of interest as summary statistics η(y)![requires iterative process]

I use of the standard quadratic loss function

(θ− θ0)TA(θ− θ0)

I recent extension to model choice, optimality of Bayes factor

B12(y)

[F&P, ISBA 2012, Kyoto]

LDA summaries for model choice

In parallel to F& P semi-automatic ABC, selection of mostdiscriminant subvector out of a collection of summary statistics,based on Linear Discriminant Analysis (LDA)

[Estoup & al., 2012, Mol. Ecol. Res.]

Solution now implemented in DIYABC.2[Cornuet & al., 2008, Bioinf.; Estoup & al., 2013]

Implementation

Step 1: Rake a subset of the α% (e.g., 1%) best simulationsfrom an ABC reference table usually including 106–109

simulations for each of the M compared scenarios/models

Selection based on normalized Euclidian distance computedbetween observed and simulated raw summaries

Step 2: run LDA on this subset to transform summaries into(M − 1) discriminant variables

When computing LDA functions, weight simulated data with anEpanechnikov kernel

Step 3: Estimation of the posterior probabilities of eachcompeting scenario/model by polychotomous local logisticregression against the M − 1 most discriminant variables

[Cornuet & al., 2008, Bioinformatics]

LDA advantages

I much faster computation of scenario probabilities viapolychotomous regression

I a (much) lower number of explanatory variables improves theaccuracy of the ABC approximation, reduces the tolerance εand avoids extra costs in constructing the reference table

I allows for a large collection of initial summaries

I ability to evaluate Type I and Type II errors on more complexmodels

I LDA reduces correlation among explanatory variables

When available, using both simulated and real data sets, posteriorprobabilities of scenarios computed from LDA-transformed and rawsummaries are strongly correlated

LDA advantages

I much faster computation of scenario probabilities viapolychotomous regression

I a (much) lower number of explanatory variables improves theaccuracy of the ABC approximation, reduces the tolerance εand avoids extra costs in constructing the reference table

I allows for a large collection of initial summaries

I ability to evaluate Type I and Type II errors on more complexmodels

I LDA reduces correlation among explanatory variables

When available, using both simulated and real data sets, posteriorprobabilities of scenarios computed from LDA-transformed and rawsummaries are strongly correlated

ABC methods

ABC for model choiceBMC PrincipleGibbs random fields (counterexample)Generic ABC model choice

Bayesian model choice

BMC Principle

Several modelsM1,M2, . . .

are considered simultaneously for dataset y and model index M

central to inference.Use of

I prior π(M = m), plus

I prior distribution on the parameter conditional on the value mof the model index, πm(θm)

Bayesian model choice

BMC Principle

Several modelsM1,M2, . . .

are considered simultaneously for dataset y and model index M

central to inference.Goal is to derive the posterior distribution of M,

π(M = m|data)

a challenging computational target when models are complex.

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]

ABC estimates

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

T∑t=1

Im(t)=m .

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]

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

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

Neighbourhood relations

SetupChoice to be made between M neighbourhood relations

im∼ i ′ (0 6 m 6 M − 1)

withSm(x) =

∑im∼i ′

I{xi=xi ′ }

driven by the posterior probabilities of the models.

Model index

Computational target:

P(M = m|x) ∝∫Θm

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)) .

Model index

Computational target:

P(M = m|x) ∝∫Θm

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)) .

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) = f 1m(x|S(x))f2m(S(x)|θm)

n(S(x))f 2m(S(x)|θm)

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

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

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) = f 1m(x|S(x))f2m(S(x)|θm)

n(S(x))f 2m(S(x)|θm)

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

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

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)}

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).

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)}6ε∑Tt=1 Imt=2 Iρ{η(zt),η(y)}6ε

where the (mt , z t)’s are simulated from the (joint) priorAs T goes to infinity, limit

Bε12(y) =

∫Iρ{η(z),η(y)}6επ1(θ1)f1(z|θ1) dz dθ1∫Iρ{η(z),η(y)}6επ2(θ2)f2(z|θ2) dz dθ2

∫Iρ{η,η(y)}6επ1(θ1)f

η1 (η|θ1) dη dθ1∫

Iρ{η,η(y)}6επ2(θ2)fη2 (η|θ2) dη dθ2

where f η1 (η|θ1) and f η2 (η|θ2) distributions of η(z)

Limiting behaviour of B12 (T →∞)

ABC approximation

B12(y) =

∑Tt=1 Imt=1 Iρ{η(zt),η(y)}6ε∑Tt=1 Imt=2 Iρ{η(zt),η(y)}6ε

where the (mt , z t)’s are simulated from the (joint) priorAs T goes to infinity, limit

Bε12(y) =

∫Iρ{η(z),η(y)}6επ1(θ1)f1(z|θ1) dz dθ1∫Iρ{η(z),η(y)}6επ2(θ2)f2(z|θ2) dz dθ2

∫Iρ{η,η(y)}6επ1(θ1)f

η1 (η|θ1) dη dθ1∫

Iρ{η,η(y)}6επ2(θ2)fη2 (η|θ2) dη dθ2

where f η1 (η|θ1) and f η2 (η|θ2) distributions of η(z)

Limiting behaviour of B12 (ε→ 0)

When ε goes to zero,

Bη12(y) =

∫π1(θ1)f

η1 (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)f

η1 (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)f

η1 (η(y)|θ1) dθ1∫

Θ2π(θ2)g2(y)f

η2 (η(y)|θ2) dθ2

=g1(y)

∫π1(θ1)f

η1 (η(y)|θ1) dθ1

g2(y)∫π2(θ2)f

η2 (η(y)|θ2) dθ2

=g1(y)

g2(y)Bη12(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)f

η1 (η(y)|θ1) dθ1∫

Θ2π(θ2)g2(y)f

η2 (η(y)|θ2) dθ2

=g1(y)

∫π1(θ1)f

η1 (η(y)|θ1) dθ1

g2(y)∫π2(θ2)f

η2 (η(y)|θ2) dθ2

=g1(y)

g2(y)Bη12(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 nothingin common.

Formal recovery

Creating an encompassing exponential family

f (x|θ1, θ2,α1,α2) ∝ exp{θT1 η1(x) + θT2 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

In the Poisson/geometric case, if∏

i xi ! is added to S , nodiscrepancy

Formal recovery

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

ABC model choice consistency

ABC methods

Model choice consistencyFormalised frameworkConsistency resultsSummary statistics

The starting point

Central question to the validation of ABC for model choice:

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

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

from c© drawn on y through B12(y)[Marin, Pillai, X, & Rousseau, JRSS B, 2013]

The starting point

Central question to the validation of ABC for model choice:

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

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

from c© drawn on y through B12(y)[Marin, Pillai, X, & Rousseau, JRSS B, 2013]

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) = Tn = (T1(y),T2(y), · · · ,Td(y)) ∈ Rd

with true distribution Tn ∼ 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(·|Tn)

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

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 Tn from theestimand µ(θ)[A3] is the standard prior mass condition found in Bayesianasymptotics (di effective dimension of the parameter)[A4] restricts the behaviour of the model density against the truedensity

[Think CLT!]

Assumptions

[Think CLT!]

[A1] There exist

I a sequence {vn} converging to +∞,

I a distribution Q,

I a symmetric, d × d positive definite matrix V0 and

I a vector µ0 ∈ Rd

such thatvnV

−1/20 (Tn − µ0)

n→∞ Q, under Gn

Assumptions

[Think CLT!]

[A2] For i = 1, 2, there exist sets Fn,i ⊂ Θi , functions µi (θi ) andconstants εi , τi ,αi > 0 such that for all τ > 0,

supθi∈Fn,i

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

](|τµi (θi ) − µ0| ∧ εi )

−αi. v−αi

withπi (F

cn,i ) = o(v−τin ).

Assumptions

[Think CLT!]

[A3] If inf{|µi (θi ) − µ0|; θi ∈ Θi } = 0, defining (u > 0)

Sn,i (u) ={θi ∈ Fn,i ; |µi (θi ) − µ0| 6 u v−1

there exist constants di < τi ∧ αi − 1 such that

πi (Sn,i (u)) ∼ udi v−din , ∀u . vn

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.

Assumptions

[Think CLT!]

Again (sumup)[A1] is a standard central limit theorem under the true model[A2] controls the large deviations of the estimator Tn from theestimand µ(θ)[A3] is the standard prior mass condition found in Bayesianasymptotics (di effective dimension of the parameter)[A4] restricts the behaviour of the model density against the truedensity

Effective dimension

Understanding di in [A3]:defined only when µ0 ∈ {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

Effective dimension

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

mi (Tn)

gn(Tn)

∼ v−din ,

thuslog(mi (T

n)/gn(Tn)) ∼ −di log vn

and v−din penalization factor resulting from integrating θi out (see

effective number of parameters in DIC)

Effective dimension

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

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 6 mi (T

n) 6 Cuvd−din

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

mi (Tn) = oPn [vd−τin + vd−αi

Within-model consistency

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

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

Between-model consistency

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

Indeed, if

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

Clv−(d1−d2)n 6 m1(T

n)/m2(T

n) 6 Cuv−(d1−d2)n ,

Else, if

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

thenm1(T

m2(Tn)> Cu min

(v−(d1−α2)n , v

−(d1−τ2)n

Consistency theorem

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

Bayes factor

BT12 = O(v

−(d1−d2)n )

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

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

Consequences on summary statistics

Bayes factor driven by the means µi (θi ) and the relative positionof µ0 wrt both sets {µi (θi ); θi ∈ Θi }, i = 1, 2.

For ABC, this implies the simplest summary statistics Tn areancillary statistics with different mean values under both models

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

Consequences on summary statistics

Bayes factor driven by the means µi (θi ) and the relative positionof µ0 wrt both sets {µi (θi ); θi ∈ Θi }, i = 1, 2.

For ABC, this implies the simplest summary statistics Tn areancillary statistics with different mean values under both models

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

Toy example: Laplace versus Gauss [1]

Tn = n−1n∑

X 4i , 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 also leads to µ0 = µ(θ

∗)[here d1 = d2 = d = 1]

Tn = n−1n∑

X 4i , 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 also leads to µ0 = µ(θ

∗)[here d1 = d2 = d = 1]

Tn = n−1n∑

X 4i , µ1(θ) = 3 + θ4 + 6θ2, µ2(θ) = 6 + · · ·

Tn = n−1n∑

X 4i , µ1(θ) = 3 + θ4 + 6θ2, µ2(θ) = 6 + · · ·

●●

n=1000

Fourth moment

WhenT(y) =

{y(4)n , y

}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

WhenT(y) =

{y(4)n , y

}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

WhenT(y) =

{y(4)n , y

●●

Gauss Laplace

●●●●●

●●

●●●

Gauss Laplace

n=1000

●●

Gauss Laplace

n=10000

Fourth and sixth moments

Checking for adequate statistics

Run a practical check of the relevance (or non-relevance) of Tn

null hypothesis that both models are compatible with the statisticTn

H0 : inf{|µ2(θ2) − µ0|; θ2 ∈ Θ2} = 0

againstH1 : inf{|µ2(θ2) − µ0|; θ2 ∈ Θ2} > 0

testing procedure provides estimates of mean of Tn under eachmodel and checks for equality

Checking in practice

I Under each model Mi , generate ABC sample θi ,l , l = 1, · · · , L

I For each θi ,l , generate yi ,l ∼ Fi ,n(·|ψi ,l), derive Tn(yi ,l) andcompute

µi =1

L∑l=1

Tn(yi ,l), i = 1, 2 .

I Conditionally on Tn(y),

√L(µi − Eπ [µi (θi )|Tn(y)]) N(0,Vi ),

I Test for a common mean

H0 : µ1 ∼ N(µ0,V1) , µ2 ∼ N(µ0,V2)

against the alternative of different means

H1 : µi ∼ N(µi ,Vi ), with µ1 6= µ2 .

Toy example: Laplace versus Gauss

●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●

Gauss Laplace Gauss Laplace

Normalised χ2 without and with mad

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

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 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

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,

xl ,i ,j1 − xl ,i ′,j2)2

|τj1,j2

}= 2µτj1,j2

andModel 1 Model 2

E{(δµ)

}2µ1t

′ 2µ2t′

E{(δµ)

}2µ1t 2µ2t

E{(δµ)

}2µ1t

′ 2µ2t

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

µ1 = µ2, same expectation

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

22,3 not

convergent: 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

● ●

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

●●●●●

DM2(12) DM2(13) & DM2(23)

●●●●●●●●●

DM2(12) DM2(13) & DM2(23)

p−values

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

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

Embedded models

v−(d1−d2)n

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

Summary

I Model selection feasible with ABC

I Choice of summary statistics is paramount

I At best, ABC → π(. | T(y)) which concentrates around µ0

I For consistent estimation : {θ;µ(θ) = µ0} = θ0

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

Summary

I Model selection feasible with ABC

I Choice of summary statistics is paramount

I At best, ABC → π(. | T(y)) which concentrates around µ0

I For consistent estimation : {θ;µ(θ) = µ0} = θ0

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

Conclusion/perspectives

I abc part of a wider pictureto handle complex/Big Datamodels, able to start fromrudimentary machinelearning summaries

I many formats of empirical[likelihood] Bayes methodsavailable

I lack of comparative toolsand of an assessment forinformation loss

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