séminaire de physique à besancon, nov. 22, 2012

MCMC and likelihood-free methods

Christian P. Robert

Universite Paris-Dauphine, IUF, & CREST

Universite de Besancon, November 22, 2012

Computational issues in Bayesian cosmology

Computational issues in Bayesiancosmology

The Metropolis-Hastings Algorithm

The Gibbs Sampler

Approximate Bayesian computation

Statistical problems in cosmology

I Potentially high dimensional parameter space [Not consideredhere]

I Immensely slow computation of likelihoods, e.g WMAP, CMB,because of numerically costly spectral transforms [Data is aFortran program]

I Nonlinear dependence and degeneracies between parametersintroduced by physical constraints or theoretical assumptions

Cosmological data

Posterior distribution of cosmological parameters for recentobservational data of CMB anisotropies (differences in temperaturefrom directions) [WMAP], SNIa, and cosmic shear.Combination of three likelihoods, some of which are available aspublic (Fortran) code, and of a uniform prior on a hypercube.

Cosmology parameters

Parameters for the cosmology likelihood(C=CMB, S=SNIa, L=lensing)

Symbol Description Minimum Maximum ExperimentΩb Baryon density 0.01 0.1 C LΩm Total matter density 0.01 1.2 C S Lw Dark-energy eq. of state -3.0 0.5 C S Lns Primordial spectral index 0.7 1.4 C L

∆2R Normalization (large scales) Cσ8 Normalization (small scales) C Lh Hubble constant C Lτ Optical depth CM Absolute SNIa magnitude Sα Colour response Sβ Stretch response Sa Lb galaxy z-distribution fit Lc L

For WMAP5, σ8 is a deduced quantity that depends on the other parameters

Adaptation of importance function

[Benabed et al., MNRAS, 2010]

Estimates

Parameter PMC MCMC

Ωb 0.0432+0.0027−0.0024 0.0432+0.0026

−0.0023

Ωm 0.254+0.018−0.017 0.253+0.018

−0.016

τ 0.088+0.018−0.016 0.088+0.019

−0.015

w −1.011± 0.060 −1.010+0.059−0.060

ns 0.963+0.015−0.014 0.963+0.015

−0.014

109∆2R 2.413+0.098

−0.093 2.414+0.098−0.092

h 0.720+0.022−0.021 0.720+0.023

−0.021

a 0.648+0.040−0.041 0.649+0.043

−0.042

b 9.3+1.4−0.9 9.3+1.7

−0.9

c 0.639+0.084−0.070 0.639+0.082

−0.070

−M 19.331± 0.030 19.332+0.029−0.031

α 1.61+0.15−0.14 1.62+0.16

−0.14

−β −1.82+0.17−0.16 −1.82± 0.16

σ8 0.795+0.028−0.030 0.795+0.030

−0.027

Means and 68% credible intervals using lensing, SNIa and CMB

Evidence/Marginal likelihood/Integrated Likelihood ...

Central quantity of interest in (Bayesian) model choice

∫π(x)dx =

∫π(x)

q(x)q(x)dx.

expressed as an expectation under any density q with large enoughsupport.

Importance sampling provides a sample x1, . . . xN ∼ q andapproximation of the above integral,

E ≈N∑n=1

where the wn =π(xn)q(xn)

are the (unnormalised) importance weights.

Evidence/Marginal likelihood/Integrated Likelihood ...

Central quantity of interest in (Bayesian) model choice

∫π(x)dx =

∫π(x)

q(x)q(x)dx.

expressed as an expectation under any density q with large enoughsupport.Importance sampling provides a sample x1, . . . xN ∼ q andapproximation of the above integral,

E ≈N∑n=1

where the wn =π(xn)q(xn)

are the (unnormalised) importance weights.

Back to cosmology questions

Standard cosmology successful in explaining recent observations,such as CMB, SNIa, galaxy clustering, cosmic shear, galaxy clustercounts, and Lyα forest clustering.

Flat ΛCDM model with only six free parameters(Ωm,Ωb,h,ns, τ,σ8)

Extensions to ΛCDM may be based on independent evidence(massive neutrinos from oscillation experiments), predicted bycompelling hypotheses (primordial gravitational waves frominflation) or reflect ignorance about fundamental physics(dynamical dark energy).

Testing for dark energy, curvature, and inflationary models

Back to cosmology questions

Standard cosmology successful in explaining recent observations,such as CMB, SNIa, galaxy clustering, cosmic shear, galaxy clustercounts, and Lyα forest clustering.

Flat ΛCDM model with only six free parameters(Ωm,Ωb,h,ns, τ,σ8)

Extensions to ΛCDM may be based on independent evidence(massive neutrinos from oscillation experiments), predicted bycompelling hypotheses (primordial gravitational waves frominflation) or reflect ignorance about fundamental physics(dynamical dark energy).

Testing for dark energy, curvature, and inflationary models

Extended models

Focus on the dark energy equation-of-state parameter, modeled as

w = −1 ΛCDM

w = w0 wCDM

w = w0 +w1(1− a) w(z)CDM

In addition, curvature parameter ΩK for each of the above is eitherΩK = 0 (‘flat’) or ΩK 6= 0 (‘curved’).Choice of models represents simplest models beyond a“cosmological constant” model able to explain the observed,recent accelerated expansion of the Universe.

Cosmology priors

Prior ranges for dark energy and curvature models. In case ofw(a) models, the prior on w1 depends on w0

Parameter Description Min. Max.

Ωm Total matter density 0.15 0.45Ωb Baryon density 0.01 0.08h Hubble parameter 0.5 0.9

ΩK Curvature −1 1w0 Constant dark-energy par. −1 −1/3

w1 Linear dark-energy par. −1−w0−1/3−w01−aacc

Results

In most cases evidence in favour of the standard model. especiallywhen more datasets/experiments are combined.

Largest evidence is lnB12 = 1.8, for the w(z)CDM model andCMB alone. Case where a large part of the prior range is stillallowed by the data, and a region of comparable size is excluded.Hence weak evidence that both w0 and w1 are required, butexcluded when adding SNIa and BAO datasets.

Results on the curvature are compatible with current findings:non-flat Universe(s) strongly disfavoured for the three dark-energycases.

Evidence

Posterior outcome

Posterior on dark-energy parameters w0 and w1 as 68%- and 95% credible regions forWMAP (solid blue lines), WMAP+SNIa (dashed green) and WMAP+SNIa+BAO(dotted red curves). Allowed prior range as red straight lines.

The Gibbs Sampler

Monte Carlo basics

General purpose

A major computational issue in Bayesian statistics:

Given a density π known up to a normalizing constant, and anintegrable function h, compute

Π(h) =

∫h(x)π(x)µ(dx) =

∫h(x)π(x)µ(dx)∫π(x)µ(dx)

when∫h(x)π(x)µ(dx) is intractable.

Monte Carlo basics

Monte Carlo 101

Generate an iid sample x1, . . . , xN from π and estimate Π(h) by

ΠMCN (h) = N−1N∑i=1

h(xi).

LLN: ΠMCN (h)as−→ Π(h)

If Π(h2) =∫h2(x)π(x)µ(dx) <∞,

CLT:√N(ΠMCN (h) − Π(h)

(0,Π[h− Π(h)]2

Caveat conducting to MCMC

Often impossible or inefficient to simulate directly from Π

Monte Carlo basics

Monte Carlo 101

Generate an iid sample x1, . . . , xN from π and estimate Π(h) by

ΠMCN (h) = N−1N∑i=1

h(xi).

LLN: ΠMCN (h)as−→ Π(h)

If Π(h2) =∫h2(x)π(x)µ(dx) <∞,

CLT:√N(ΠMCN (h) − Π(h)

(0,Π[h− Π(h)]2

Caveat conducting to MCMC

Often impossible or inefficient to simulate directly from Π

Monte Carlo Methods based on Markov Chains

Running Monte Carlo via Markov Chains (MCMC)

It is not necessary to use a sample from the distribution f toapproximate the integral

∫h(x)f(x)dx ,

[notation warnin: π turned to f!]

∫h(x)f(x)dx ,

We can obtain X1, . . . ,Xn ∼ f (approx)without directly simulating from f,using an ergodic Markov chain withstationary distribution f

∫h(x)f(x)dx ,

We can obtain X1, . . . ,Xn ∼ f (approx)without directly simulating from f,using an ergodic Markov chain withstationary distribution f

Andreı Markov

Running Monte Carlo via Markov Chains (2)

For an arbitrary starting value x(0), an ergodic chain (X(t)) isgenerated using a transition kernel with stationary distribution f

I irreducible Markov chain with stationary distribution f isergodic with limiting distribution f under weak conditions

I hence convergence in distribution of (X(t)) to a randomvariable from f.

I for T0 “large enough” T0, X(T0) distributed from f

I Markov sequence is dependent sample X(T0),X(T0+1), . . .generated from f

I Birkoff’s ergodic theorem extends LLN, sufficient for mostapproximation purposes

Problem: How can one build a Markov chain with a givenstationary distribution?

The Metropolis–Hastings algorithm

Arguments: The algorithm uses theobjective (target) density

and a conditional density

q(y|x)

called the instrumental (or proposal)distribution

Nicholas Metropolis

The MH algorithm

Algorithm (Metropolis–Hastings)

Given x(t),

1. Generate Yt ∼ q(y|x(t)).

2. Take

X(t+1) =

Yt with prob. ρ(x(t), Yt),

x(t) with prob. 1− ρ(x(t), Yt),

ρ(x,y) = min

q(x|y)

q(y|x), 1

Features

I Independent of normalizing constants for both f and q(·|x)(ie, those constants independent of x)

I Never move to values with f(y) = 0

I The chain (x(t))t may take the same value several times in arow, even though f is a density wrt Lebesgue measure

I The sequence (yt)t is usually not a Markov chain

Convergence properties

1. The M-H Markov chain is reversible, withinvariant/stationary density f since it satisfies the detailedbalance condition

f(y)K(y, x) = f(x)K(x,y)

2. As f is a probability measure, the chain is positive recurrent

[f(Yt) q(X

(t)|Yt)

f(X(t)) q(Yt|X(t))> 1

]< 1. (1)

that is, the event X(t+1) = X(t) is possible, then the chain isaperiodic

[f(Yt) q(X

(t)|Yt)

f(X(t)) q(Yt|X(t))> 1

]< 1. (1)

[f(Yt) q(X

(t)|Yt)

f(X(t)) q(Yt|X(t))> 1

]< 1. (1)

Random-walk Metropolis-Hastings algorithms

Random walk Metropolis–Hastings

Use of a local perturbation as proposal

Yt = X(t) + εt,

where εt ∼ g, independent of X(t).The instrumental density is of the form g(y− x) and the Markovchain is a random walk if we take g to be symmetric g(x) = g(−x)

Random-walk Metropolis-Hastings algorithms

Random walk Metropolis–Hastings [code]

Algorithm (Random walk Metropolis)

Given x(t)

1. Generate Yt ∼ g(y− x(t))

2. Take

X(t+1) =

Yt with prob. min

1,f(Yt)

f(x(t))

x(t) otherwise.

Extensions

Langevin Algorithms

Proposal based on the Langevin diffusion Lt is defined by thestochastic differential equation

dLt = dBt +1

2∇ log f(Lt)dt,

where Bt is the standard Brownian motion

Theorem

The Langevin diffusion is the only non-explosive diffusion which isreversible with respect to f.

Extensions

Discretization

Instead, consider the sequence

x(t+1) = x(t) +σ2

2∇ log f(x(t)) + σεt, εt ∼ Np(0, Ip)

where σ2 corresponds to the discretization step

Unfortunately, the discretized chain may be transient, for instancewhen

limx→±∞

∣∣σ2∇ log f(x)|x|−1∣∣ > 1

Extensions

Discretization

Instead, consider the sequence

x(t+1) = x(t) +σ2

2∇ log f(x(t)) + σεt, εt ∼ Np(0, Ip)

where σ2 corresponds to the discretization stepUnfortunately, the discretized chain may be transient, for instancewhen

limx→±∞

∣∣σ2∇ log f(x)|x|−1∣∣ > 1

Extensions

MH correction

Accept the new value Yt with probability

f(x(t))·exp

−∥∥∥Yt − x(t) − σ2

2 ∇ log f(x(t))∥∥∥2/2σ2

−∥∥∥x(t) − Yt − σ2

2 ∇ log f(Yt)∥∥∥2/2σ2 ∧ 1 .

Choice of the scaling factor σShould lead to an acceptance rate of 0.574 to achieve optimalconvergence rates (when the components of x are uncorrelated)

[Roberts & Rosenthal, 1998; Girolami & Calderhead, 2011]

Extensions

Optimizing the Acceptance Rate

Problem of choosing the transition q kernel from a practical pointof viewMost common solutions:

(a) a fully automated algorithm like ARMS;[Gilks & Wild, 1992]

(b) an instrumental density g which approximates f, such thatf/g is bounded for uniform ergodicity to apply;

(c) a random walk

In both cases (b) and (c), the choice of g is critical,

Extensions

Case of the random walk

Different approach to acceptance ratesA high acceptance rate does not indicate that the algorithm ismoving correctly since it indicates that the random walk is movingtoo slowly on the surface of f.

If x(t) and yt are close, i.e. f(x(t)) ' f(yt) y is accepted withprobability

(f(yt)

f(x(t)), 1

)' 1 .

For multimodal densities with well separated modes, the negativeeffect of limited moves on the surface of f clearly shows.

Extensions

Case of the random walk

Different approach to acceptance ratesA high acceptance rate does not indicate that the algorithm ismoving correctly since it indicates that the random walk is movingtoo slowly on the surface of f.If x(t) and yt are close, i.e. f(x(t)) ' f(yt) y is accepted withprobability

(f(yt)

f(x(t)), 1

)' 1 .

For multimodal densities with well separated modes, the negativeeffect of limited moves on the surface of f clearly shows.

Extensions

Case of the random walk (2)

If the average acceptance rate is low, the successive values of f(yt)tend to be small compared with f(x(t)), which means that therandom walk moves quickly on the surface of f since it oftenreaches the “borders” of the support of f

Extensions

Rule of thumb

In small dimensions, aim at an average acceptance rate of50%. In large dimensions, at an average acceptance rate of25%.

[Gelman,Gilks and Roberts, 1995]

warnin: rule to be taken with a pinch of salt!

Extensions

Rule of thumb

In small dimensions, aim at an average acceptance rate of50%. In large dimensions, at an average acceptance rate of25%.

[Gelman,Gilks and Roberts, 1995]

warnin: rule to be taken with a pinch of salt!

Extensions

Role of scale

Example (Noisy AR(1))

Hidden Markov chain from a regular AR(1) model,

xt+1 = ϕxt + εt+1 εt ∼ N(0, τ2)

and observablesyt|xt ∼ N(x2t ,σ

The distribution of xt given xt−1, xt+1 and yt is

exp−1

(xt −ϕxt−1)

2 + (xt+1 −ϕxt)2 +

σ2(yt − x

Extensions

Role of scale

Example (Noisy AR(1))

Hidden Markov chain from a regular AR(1) model,

xt+1 = ϕxt + εt+1 εt ∼ N(0, τ2)

and observablesyt|xt ∼ N(x2t ,σ

The distribution of xt given xt−1, xt+1 and yt is

exp−1

(xt −ϕxt−1)

2 + (xt+1 −ϕxt)2 +

σ2(yt − x

Extensions

Role of scale

Example (Noisy AR(1) continued)

For a Gaussian random walk with scale ω small enough, therandom walk never jumps to the other mode. But if the scale ω issufficiently large, the Markov chain explores both modes and give asatisfactory approximation of the target distribution.

Extensions

Role of scale

Markov chain based on a random walk with scale ω = .1.

Extensions

Role of scale

Markov chain based on a random walk with scale ω = .5.

The Gibbs Sampler

General Principles

A very specific simulation algorithm based on the targetdistribution f:

1. Uses the conditional densities f1, . . . , fp from f

2. Start with the random variable X = (X1, . . . ,Xp)

3. Simulate from the conditional densities,

Xi|x1, x2, . . . , xi−1, xi+1, . . . , xp

∼ fi(xi|x1, x2, . . . , xi−1, xi+1, . . . , xp)

for i = 1, 2, . . . ,p.

The Gibbs Sampler

General Principles

Xi|x1, x2, . . . , xi−1, xi+1, . . . , xp

∼ fi(xi|x1, x2, . . . , xi−1, xi+1, . . . , xp)

for i = 1, 2, . . . ,p.

The Gibbs Sampler

General Principles

Xi|x1, x2, . . . , xi−1, xi+1, . . . , xp

∼ fi(xi|x1, x2, . . . , xi−1, xi+1, . . . , xp)

for i = 1, 2, . . . ,p.

The Gibbs Sampler

General Principles

Gibbs code

Algorithm (Gibbs sampler)

Given x(t) = (x(t)1 , . . . , x

(t)p ), generate

1. X(t+1)1 ∼ f1(x1|x

(t)2 , . . . , x

(t)p );

2. X(t+1)2 ∼ f2(x2|x

(t+1)1 , x

(t)3 , . . . , x

(t)p ),

p. X(t+1)p ∼ fp(xp|x

(t+1)1 , . . . , x

(t+1)p−1 )

X(t+1) → X ∼ f

The Gibbs Sampler

General Principles

Properties

The full conditionals densities f1, . . . , fp are the only densities usedfor simulation. Thus, even in a high dimensional problem, all ofthe simulations may be univariate

The Gibbs Sampler

General Principles

toy example: iid N(µ,σ2) variates

When Y1, . . . ,Yniid∼ N(y|µ,σ2) with both µ and σ unknown, the

posterior in (µ,σ2) is conjugate outside a standard familly

But...

µ|Y0:n,σ2 ∼ N

(µ∣∣∣ 1n∑n

i=1 Yi,σ2

σ2|Y1:n,µ ∼ IG(σ2

∣∣n2 − 1,

∑ni=1(Yi − µ)

assuming constant (improper) priors on both µ and σ2

I Hence we may use the Gibbs sampler for simulating from theposterior of (µ,σ2)

The Gibbs Sampler

General Principles

toy example: iid N(µ,σ2) variates

When Y1, . . . ,Yniid∼ N(y|µ,σ2) with both µ and σ unknown, the

posterior in (µ,σ2) is conjugate outside a standard familly

But...

µ|Y0:n,σ2 ∼ N

(µ∣∣∣ 1n∑n

i=1 Yi,σ2

σ2|Y1:n,µ ∼ IG(σ2

∣∣n2 − 1,

∑ni=1(Yi − µ)

assuming constant (improper) priors on both µ and σ2

I Hence we may use the Gibbs sampler for simulating from theposterior of (µ,σ2)

The Gibbs Sampler

General Principles

toy example: R code

Gibbs Sampler for Gaussian posterior

n = length(Y);

S = sum(Y);

mu = S/n;

for (i in 1:500)

S2 = sum((Y-mu)^2);

sigma2 = 1/rgamma(1,n/2-1,S2/2);

mu = S/n + sqrt(sigma2/n)*rnorm(1);

The Gibbs Sampler

General Principles

Example of results with n = 10 observations from theN(0, 1) distribution

Number of Iterations 1

, 2, 3, 4, 5, 10, 25, 50, 100, 500

The Gibbs Sampler

General Principles

Number of Iterations 1, 2

, 3, 4, 5, 10, 25, 50, 100, 500

The Gibbs Sampler

General Principles

Number of Iterations 1, 2, 3

, 4, 5, 10, 25, 50, 100, 500

The Gibbs Sampler

General Principles

Number of Iterations 1, 2, 3, 4

, 5, 10, 25, 50, 100, 500

The Gibbs Sampler

General Principles

Number of Iterations 1, 2, 3, 4, 5

, 10, 25, 50, 100, 500

The Gibbs Sampler

General Principles

Number of Iterations 1, 2, 3, 4, 5, 10

, 25, 50, 100, 500

The Gibbs Sampler

General Principles

Number of Iterations 1, 2, 3, 4, 5, 10, 25

, 50, 100, 500

The Gibbs Sampler

General Principles

Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50

, 100, 500

The Gibbs Sampler

General Principles

Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50, 100

The Gibbs Sampler

General Principles

Number of Iterations 1, 2, 3, 4, 5, 10, 25, 50, 100, 500

The Gibbs Sampler

General Principles

Limitations of the Gibbs sampler

Formally, a special case of a sequence of 1-D M-H kernels, all withacceptance rate uniformly equal to 1.The Gibbs sampler

1. limits the choice of instrumental distributions

2. requires some knowledge of f

3. is, by construction, multidimensional

4. does not apply to problems where the number of parametersvaries as the resulting chain is not irreducible.

The Gibbs Sampler

General Principles

The Gibbs Sampler

General Principles

The Gibbs Sampler

General Principles

The Gibbs Sampler

General Principles

A wee problem

−1 0 1 2 3 4

Gibbs started at random

Gibbs stuck at the wrong mode

−1 0 1 2 3

The Gibbs Sampler

General Principles

A wee problem

−1 0 1 2 3 4

Gibbs started at random

Gibbs stuck at the wrong mode

−1 0 1 2 3

The Gibbs Sampler

General Principles

Slice sampler as generic Gibbs

If f(θ) can be written as a product

k∏i=1

fi(θ),

it can be completed as

k∏i=1

I06ωi6fi(θ),

leading to the following Gibbs algorithm:

The Gibbs Sampler

General Principles

Slice sampler as generic Gibbs

If f(θ) can be written as a product

k∏i=1

fi(θ),

it can be completed as

k∏i=1

I06ωi6fi(θ),

leading to the following Gibbs algorithm:

The Gibbs Sampler

General Principles

Slice sampler (code)

Algorithm (Slice sampler)

Simulate

1. ω(t+1)1 ∼ U[0,f1(θ(t))]

k. ω(t+1)k ∼ U[0,fk(θ(t))]

k+1. θ(t+1) ∼ UA(t+1) , with

A(t+1) = y; fi(y) > ω(t+1)i , i = 1, . . . ,k.

The Gibbs Sampler

General Principles

Example of results with a truncated N(−3, 1) distribution

0.0 0.2 0.4 0.6 0.8 1.0

Number of Iterations 2

, 3, 4, 5, 10, 50, 100

The Gibbs Sampler

General Principles

0.0 0.2 0.4 0.6 0.8 1.0

Number of Iterations 2, 3

, 4, 5, 10, 50, 100

The Gibbs Sampler

General Principles

0.0 0.2 0.4 0.6 0.8 1.0

Number of Iterations 2, 3, 4

, 5, 10, 50, 100

The Gibbs Sampler

General Principles

0.0 0.2 0.4 0.6 0.8 1.0

Number of Iterations 2, 3, 4, 5

, 10, 50, 100

The Gibbs Sampler

General Principles

0.0 0.2 0.4 0.6 0.8 1.0

Number of Iterations 2, 3, 4, 5, 10

, 50, 100

The Gibbs Sampler

General Principles

0.0 0.2 0.4 0.6 0.8 1.0

Number of Iterations 2, 3, 4, 5, 10, 50

The Gibbs Sampler

General Principles

0.0 0.2 0.4 0.6 0.8 1.0

Number of Iterations 2, 3, 4, 5, 10, 50, 100

The Gibbs Sampler

ABC basics

Regular Bayesian computation issues

Recap’: 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 (inanalytic and numerical senses) and when the completion step

f(y|θ) =

f(y, z|θ) dz

ABC basics

Illustration

Phylogenetic 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

Illustration

demo-genetic inference

Genetic model of evolution from acommon ancestor (MRCA)characterized by a set of parametersthat cover historical, demographic, andgenetic factorsDataset of polymorphism (DNA sample)observed at the present time

!""#$%&'()*+,(-*.&(/+0$'"1)()&$/+2!,03 !1/+*%*'"4*+56(""4&7()&$/.+.1#+4*.+8-9':*.+

Différents scénarios possibles, choix de scenario par ABC

Le scenario 1a est largement soutenu par rapport aux

autres ! plaide pour une origine commune des

populations pygmées d’Afrique de l’Ouest Verdu et al. 2009

ABC basics

Illustration

Pygmies population demo-genetics

Pygmies populations: do theyhave a common origin? whenand how did they split fromnon-pygmies populations? werethere more recent interactionsbetween pygmies andnon-pygmies populations?

!""#$%&'()*+,(-*.&(/+0$'"1)()&$/+2!,03 !1/+*%*'"4*+56(""4&7()&$/.+.1#+4*.+8-9':*.+

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

Why does it work?!

The proof is trivial:

f(θi) ∝∑z∈D

π(θi)f(z|θi)Iy(z)

∝ π(θi)f(y|θi)= π(θi|y) .

[Accept–Reject 101]

ABC basics

A as approximative

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

ρ(y, z) 6 ε

where ρ is a distance

Output distributed from

π(θ)Pθρ(y, z) < ε ∝ π(θ|ρ(y, z) < ε)

ABC basics

A as approximative

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

ρ(y, z) 6 ε

where ρ is a distanceOutput distributed from

π(θ)Pθρ(y, z) < ε ∝ π(θ|ρ(y, z) < ε)

ABC basics

ABC algorithm

Algorithm 1 Likelihood-free rejection sampler 2

for i = 1 to N dorepeat

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

until ρη(z),η(y) 6 εset θi = θ

end for

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

ABC basics

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|η(y)) .

ABC basics

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|η(y)) .

ABC basics

Pima Indian benchmark

−0.005 0.010 0.020 0.030

−0.05 −0.03 −0.01

−1.0 0.0 1.0 2.0

0.00.2

0.40.6

0.81.0

ityFigure: Comparison between density estimates of the marginals on β1(left), β2 (center) and β3 (right) from ABC rejection samples (red) andMCMC samples (black)

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

ABC basics

ABC-MCMC

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

θ(t+1) =

θ′ ∼ Kω(θ

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

and u ∼ U(0, 1) 6 π(θ′)Kω(θ(t)|θ′)π(θ(t))Kω(θ′|θ(t))

θ(t) otherwise,

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

ABC basics

ABC-MCMC

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

θ(t+1) =

θ′ ∼ Kω(θ

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

and u ∼ U(0, 1) 6 π(θ′)Kω(θ(t)|θ′)π(θ(t))Kω(θ′|θ(t))

θ(t) otherwise,

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

ABC basics

ABC-MCMC (2)

Algorithm 2 Likelihood-free MCMC sampler

Use Algorithm 1 to get (θ(0), z(0))for t = 1 to N do

Generate θ ′ from Kω(·|θ(t−1)

Generate z ′ from the likelihood f(·|θ ′),Generate u from U[0,1],

if u 6 π(θ ′)Kω(θ(t−1)|θ ′)π(θ(t−1)Kω(θ ′|θ(t−1))

IAε,y(z ′) then

set (θ(t), z(t)) = (θ ′, z ′)else(θ(t), z(t))) = (θ(t−1), z(t−1)),

end ifend for

ABC basics

Sequential Monte Carlo

SMC is a simulation technique to approximate a sequence ofrelated probability distributions πn with π0 “easy” and πT astarget.Iterated IS as PMC : particles moved from time n to time n viakernel Kn and use of a sequence of extended targets πn

πn(z0:n) = πn(zn)

n∏j=0

Lj(zj+1, zj)

where the Lj’s are backward Markov kernels [check that πn(zn) isa marginal]

[Del Moral, Doucet & Jasra, Series B, 2006]

ABC basics

Sequential Monte Carlo (2)

Algorithm 3 SMC sampler [Del Moral, Doucet & Jasra, Series B,2006]

sample z(0)i ∼ γ0(x) (i = 1, . . . ,N)

compute weights w(0)i = π0(z

(0)i ))/γ0(z

(0)i )

for t = 1 to N doif ESS(w(t−1)) < NT then

resample N particles z(t−1) and set weights to 1end ifgenerate z

(t−1)i ∼ Kt(z

(t−1)i , ·) and set weights to

w(t)i =W

(t−1)i−1

πt(z(t)i ))Lt−1(z

(t)i ), z

(t−1)i ))

πt−1(z(t−1)i ))Kt(z

(t−1)i ), z

(t)i ))

end for

ABC basics

ABC-SMC

[Del Moral, Doucet & Jasra, 2009]

True derivation of an SMC-ABC algorithmUse of a kernel Kn associated with target πεn and derivation of thebackward kernel

Ln−1(z, z′) =

πεn(z′)Kn(z

′, z)

πn(z)

Update of the weights

win ∝ wi(n−1)

∑Mm=1 IAεn (x

min)∑M

m=1 IAεn−1 (xmi(n−1))

when xmin ∼ K(xi(n−1), ·)

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