Shanghai tutorial

From MCMC to ABC Methods


Christian P. Robert

Universit´ Paris-Dauphine, IuF, & CREST
e
http://www.ceremade.dauphine.fr/~xian

O’Bayes 11, Shanghai, June 10, 2011


Outline
Computational issues in Bayesian statistics

The Metropolis-Hastings Algorithm

The Gibbs Sampler

Approximate Bayesian computation

ABC for model choice


A typology of Bayes computational problems

(i). use of a complex parameter space, as for instance in
constrained parameter sets like those resulting from imposing
stationarity constraints in dynamic models;



(ii). use of a complex sampling model with an intractable
likelihood, as for instance in some latent variable or graphical
models or in inverse problems;



(iii). use of a huge dataset;



(iv). use of a complex prior distribution (which may be the
posterior distribution associated with an earlier sample);



(iv). use of a complex prior distribution (which may be the
posterior distribution associated with an earlier sample);
(v). use of a particular inferential procedure as for instance, Bayes
factors
π P (θ ∈ Θ0 | x) π(θ ∈ Θ0 )
B01 (x) = .
P (θ ∈ Θ1 | x) π(θ ∈ Θ1 )




Monte Carlo basics
Monte Carlo Methods based on Markov Chains
The Metropolis–Hastings algorithm
Random-walk Metropolis-Hastings algorithms

The Gibbs Sampler



Monte Carlo basics

General purpose

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

h(x)˜ (x)µ(dx)
π
Π(h) = h(x)π(x)µ(dx) =
π (x)µ(dx)
˜

when h(x)˜ (x)µ(dx) is intractable.
π

Monte Carlo basics

Monte Carlo 101

Generate an iid sample x1 , . . . , xN from π and estimate Π(h) by
N
ˆN
ΠM C (h) = N −1 h(xi ).
i=1

ˆN as
LLN: ΠM C (h) −→ Π(h)
If Π(h2 ) = h2 (x)π(x)µ(dx) < ∞,

√ L
CLT: ˆN
N ΠM C (h) − Π(h) N 0, Π [h − Π(h)]2 .

Monte Carlo basics

Monte Carlo 101

Generate an iid sample x1 , . . . , xN from π and estimate Π(h) by
N
ˆN
ΠM C (h) = N −1 h(xi ).
i=1

ˆN as
LLN: ΠM C (h) −→ Π(h)
If Π(h2 ) = h2 (x)π(x)µ(dx) < ∞,

√ L
CLT: ˆN
N ΠM C (h) − Π(h) N 0, Π [h − Π(h)]2 .

Caveat announcing MCMC

Often impossible or ineﬃcient to simulate directly from Π

Monte Carlo basics

Importance Sampling

For Q proposal distribution such that Q(dx) = q(x)µ(dx),
alternative representation

Π(h) = h(x){π/q}(x)q(x)µ(dx).

Monte Carlo basics

Importance Sampling

For Q proposal distribution such that Q(dx) = q(x)µ(dx),
alternative representation

Π(h) = h(x){π/q}(x)q(x)µ(dx).

Principle of importance
Generate an iid sample x1 , . . . , xN ∼ Q and estimate Π(h) by
N
ˆ IS
ΠQ,N (h) = N −1 h(xi ){π/q}(xi ).
i=1

Monte Carlo basics

Properties of importance

Then
ˆ as
LLN: ΠIS (h) −→ Π(h)
Q,N and if Q((hπ/q)2 ) < ∞,
√ L
CLT: ˆ Q,N
N (ΠIS (h) − Π(h)) N 0, Q{(hπ/q − Π(h))2 } .

Monte Carlo basics

Properties of importance

Then
ˆ as
LLN: ΠIS (h) −→ Π(h)
Q,N and if Q((hπ/q)2 ) < ∞,
√ L
CLT: ˆ Q,N
N (ΠIS (h) − Π(h)) N 0, Q{(hπ/q − Π(h))2 } .

Caveat
ˆ Q,N
If normalizing constant of π unknown, impossible to use ΠIS

Generic problem in Bayesian Statistics: π(θ|x) ∝ f (x|θ)π(θ).

Monte Carlo basics

Self-Normalised Importance Sampling

Self normalized version
N −1 N
ˆ Q,N
ΠSN IS (h) = {π/q}(xi ) h(xi ){π/q}(xi ).
i=1 i=1

Monte Carlo basics


N −1 N
ˆ Q,N
i=1 i=1

ˆ as
LLN : ΠSN IS (h) −→ Π(h)
Q,N
and if Π((1 + h2 )(π/q)) < ∞,
√ L
CLT : ˆ Q,N
N (ΠSN IS (h) − Π(h)) N 0, π {(π/q)(h − Π(h)}2 ) .

Monte Carlo basics


N −1 N
ˆ Q,N
i=1 i=1

ˆ as
LLN : ΠSN IS (h) −→ Π(h)
Q,N
and if Π((1 + h2 )(π/q)) < ∞,
√ L
CLT : ˆ Q,N
N (ΠSN IS (h) − Π(h)) N 0, π {(π/q)(h − Π(h)}2 ) .

c The quality of the SNIS approximation depends on the
choice of Q


Running Monte Carlo via Markov Chains (MCMC)

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

I= h(x)f (x)dx ,


Running Monte Carlo via Markov Chains (MCMC)

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

I= h(x)f (x)dx ,

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


Running Monte Carlo via Markov Chains (2)

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


Running Monte Carlo via Markov Chains (2)

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

Insures the convergence in distribution of (X (t) ) to a random
variable from f .
For a “large enough” T0 , X (T0 ) can be considered as
distributed from f
Produce a dependent sample X (T0 ) , X (T0 +1) , . . ., which is
generated from f , suﬃcient for most approximation purposes.



Basics
The algorithm uses the target density

f

and a conditional density
q(y|x)
called the instrumental (or proposal) distribution


The MH algorithm

Algorithm (Metropolis–Hastings)
Given x(t) ,
1. Generate Yt ∼ q(y|x(t) ).
2. Take

Yt with prob. ρ(x(t) , Yt ),
X (t+1) =
x(t) with prob. 1 − ρ(x(t) , Yt ),

where
f (y) q(x|y)
ρ(x, y) = min ,1 .
f (x) q(y|x)


Features

Independent of normalizing constants for both f and q(·|x)
(ie, those constants independent of x)
Never move to values with f (y) = 0
The chain (x(t) )t may take the same value several times in a
row, even though f is a density wrt Lebesgue measure
The sequence (yt )t is usually not a Markov chain


Convergence properties

Under irreducibility,
1. The M-H Markov chain is reversible, with
invariant/stationary density f since it satisﬁes the detailed
balance condition
f (y) K(y, x) = f (x) K(x, y)


Convergence properties

Under irreducibility,
1. The M-H Markov chain is reversible, with
invariant/stationary density f since it satisﬁes the detailed
balance condition
f (y) K(y, x) = f (x) K(x, y)
2. As f is a probability measure, the chain is positive recurrent


Convergence properties (2)
4. If
q(y|x) > 0 for every (x, y), (2)
the chain is irreducible


4. If
5. For M-H, f -irreducibility implies Harris recurrence


4. If
5. For M-H, f -irreducibility implies Harris recurrence
6. Thus, for M-H satisfying (1) and (2)
(i) For h, with Ef |h(X)| < ∞,
T
1
lim h(X (t) ) = h(x)df (x) a.e. f.
T →∞ T t=1

(ii) and
lim K n (x, ·)µ(dx) − f =0
n→∞
TV
for every initial distribution µ, where K n (x, ·) denotes the
kernel for n transitions.


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 Markov
chain is a random walk if we take g to be symmetric g(x) = g(−x)


Algorithm (Random walk Metropolis)
Given x(t)
1. Generate Yt ∼ g(y − x(t) )
2. Take

Y f (Yt )
(t+1) t with prob. min 1, ,
X = f (x(t) )

x(t) otherwise.


Optimizing the Acceptance Rate

Problem of choice of the transition kernel from a practical point of
view
Most common alternatives:
1. an instrumental density g which approximates f , such that
f /g is bounded for uniform ergodicity to apply;
2. a random walk
In both cases, the choice of g is critical,


Case of the random walk

Diﬀerent approach to acceptance rates
A high acceptance rate does not indicate that the algorithm is
moving correctly since it indicates that the random walk is moving
too slowly on the surface of f .


Case of the random walk

Diﬀerent approach to acceptance rates
A high acceptance rate does not indicate that the algorithm is
moving correctly since it indicates that the random walk is moving
too slowly on the surface of f .
If x(t) and yt are close, i.e. f (x(t) ) f (yt ) y is accepted with
probability
f (yt )
min ,1 1.
f (x(t) )
For multimodal densities with well separated modes, the negative
eﬀect of limited moves on the surface of f clearly shows.


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 the
random walk moves quickly on the surface of f since it often
reaches the “borders” of the support of f


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 the
random walk moves quickly on the surface of f since it often
reaches the “borders” of the support of f
In small dimensions, aim at an average acceptance rate of 50%. In
large dimensions, at an average acceptance rate of 25%.
[Gelman,Gilks and Roberts, 1995]


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

xt+1 = ϕxt + t+1 t ∼ N (0, τ 2 )

and observables
yt |xt ∼ N (x2 , σ 2 )
t


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

xt+1 = ϕxt + t+1 t ∼ N (0, τ 2 )

and observables
yt |xt ∼ N (x2 , σ 2 )
t

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

−1 τ2
exp (xt − ϕxt−1 )2 + (xt+1 − ϕxt )2 + (yt − x2 )2
t .
2τ 2 σ2


Example (Noisy AR(1) continued)
For a Gaussian random walk with scale ω small enough, the
random walk never jumps to the other mode. But if the scale ω is
suﬃciently large, the Markov chain explores both modes and give a
satisfactory approximation of the target distribution.


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


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


MA(2)

xt = t − θ1 t−1 − θ2 t−2

Since the constraints on (ϑ1 , ϑ2 ) are well-deﬁned, use of a ﬂat
prior over the triangle as prior.
Simple representation of the likelihood

library(mnormt)
ma2like=function(theta){
n=length(y)
sigma = toeplitz(c(1 +theta[1]^2+theta[2]^2,
theta[1]+theta[1]*theta[2],theta[2],rep(0,n-3)))
dmnorm(y,rep(0,n),sigma,log=TRUE)
}


Basic RWHM for MA(2)

Algorithm 1 RW-HM-MA(2) sampler
set ω and ϑ(1)
for i = 2 to T do
˜ (i−1) (i−1)
generate ϑj ∼ U (ϑj − ω, ϑj + ω)
set p = 0 and ϑ (i) = ϑ(i−1)
˜
if ϑ within the triangle then
˜
p = exp(ma2like(ϑ) − ma2like(ϑ(i−1) ))
end if
if U < p then
˜
ϑ(i) = ϑ
end if
end for


Outcome
Result with a simulated sample of 100 points and ϑ1 = 0.6,
ϑ2 = 0.2 and scale ω = 0.2


Outcome
ϑ2 = 0.2 and scale ω = 0.5


Outcome
ϑ2 = 0.2 and scale ω = 2.0

The Gibbs Sampler

The Gibbs Sampler

The Gibbs Sampler
General Principles
Slice sampling
Convergence

The Gibbs Sampler
General Principles

General Principles

A very speciﬁc simulation algorithm based on the target
distribution f :
1. Uses the conditional densities f1 , . . . , fp from f

The Gibbs Sampler
General Principles

General Principles

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

The Gibbs Sampler
General Principles

General Principles

distribution 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

Algorithm (Gibbs sampler)
(t) (t)
Given x(t) = (x1 , . . . , xp ), generate
(t+1) (t) (t)
1. X1 ∼ f1 (x1 |x2 , . . . , xp );
(t+1) (t+1) (t) (t)
2. X2 ∼ f2 (x2 |x1 , x3 , . . . , xp ),
...
(t+1) (t+1) (t+1)
p. Xp ∼ fp (xp |x1 , . . . , xp−1 )

X(t+1) → X ∼ f

The Gibbs Sampler
General Principles

Properties

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

The Gibbs Sampler
General Principles

Properties

The full conditionals densities f1 , . . . , fp are the only densities used
for simulation. Thus, even in a high dimensional problem, all of
the simulations may be univariate
The Gibbs sampler is not reversible with respect to f . However,
each of its p components is. Besides, it can be turned into a
reversible sampler, either using the Random Scan Gibbs sampler
see section or running instead the (double) sequence

f1 · · · fp−1 fp fp−1 · · · f1

The Gibbs Sampler
General Principles

Limitations of the Gibbs sampler

Formally, a special case of a sequence of 1-D M-H kernels, all with
acceptance rate uniformly equal to 1.
The Gibbs sampler
1. limits the choice of instrumental distributions

The Gibbs Sampler
General Principles


The Gibbs sampler
2. requires some knowledge of f

The Gibbs Sampler
General Principles


The Gibbs sampler
3. is, by construction, multidimensional

The Gibbs Sampler
General Principles


The Gibbs sampler
3. is, by construction, multidimensional
4. does not apply to problems where the number of parameters
varies as the resulting chain is not irreducible.

The Gibbs Sampler
General Principles

A wee mixture problem
4
3
2
µ2

1
0
−1

−1 0 1 2 3 4

µ1

Gibbs started at random

The Gibbs Sampler
General Principles

A wee mixture problem

Gibbs stuck at the wrong mode
4
3

3
2

2
µ2

1

µ2

1
0

0
−1

−1

−1 0 1 2 3 4

µ1

Gibbs started at random −1 0 1 2 3

µ1

The Gibbs Sampler
Slice sampling

Slice sampler as generic Gibbs

If f (θ) can be written as a product
k
fi (θ),
i=1

The Gibbs Sampler
Slice sampling

Slice sampler as generic Gibbs

If f (θ) can be written as a product
k
fi (θ),
i=1

it can be completed as
k
I0≤ωi ≤fi (θ) ,
i=1

leading to the following Gibbs algorithm:

The Gibbs Sampler
Slice sampling

Algorithm (Slice sampler)
Simulate
(t+1)
1. ω1 ∼ U[0,f1 (θ(t) )] ;
...
(t+1)
k. ωk ∼ U[0,fk (θ(t) )] ;
k+1. θ(t+1) ∼ UA(t+1) , with
(t+1)
A(t+1) = {y; fi (y) ≥ ωi , i = 1, . . . , k}.

The Gibbs Sampler
Slice sampling

Example of results with a truncated N (−3, 1) distribution
0.010
0.008
0.006
y

0.004
0.002
0.000

0.0 0.2 0.4 0.6 0.8 1.0

x

Number of Iterations 2

The Gibbs Sampler
Slice sampling

0.010
0.008
0.006
y

0.004
0.002
0.000

0.0 0.2 0.4 0.6 0.8 1.0

x

Number of Iterations 2, 3

The Gibbs Sampler
Slice sampling

0.010
0.008
0.006
y

0.004
0.002
0.000

0.0 0.2 0.4 0.6 0.8 1.0

x

Number of Iterations 2, 3, 4

The Gibbs Sampler
Slice sampling

0.010
0.008
0.006
y

0.004
0.002
0.000

0.0 0.2 0.4 0.6 0.8 1.0

x

Number of Iterations 2, 3, 4, 5

The Gibbs Sampler
Slice sampling

0.010
0.008
0.006
y

0.004
0.002
0.000

0.0 0.2 0.4 0.6 0.8 1.0

x

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

The Gibbs Sampler
Slice sampling

0.010
0.008
0.006
y

0.004
0.002
0.000

0.0 0.2 0.4 0.6 0.8 1.0

x

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

The Gibbs Sampler
Slice sampling

0.010
0.008
0.006
y

0.004
0.002
0.000

0.0 0.2 0.4 0.6 0.8 1.0

x

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

The Gibbs Sampler
Slice sampling

Good slices, tough slices

The slice sampler usually enjoys good theoretical properties (like
geometric ergodicity and even uniform ergodicity under bounded f
and bounded X ).
As k increases, the determination of the set A(t+1) may get
increasingly complex.

The Gibbs Sampler
Convergence

Properties of the Gibbs sampler

Theorem (Convergence)
For
(Y1 , Y2 , · · · , Yp ) ∼ g(y1 , . . . , yp ),
if either
[Positivity condition]
(i) g (i) (y > 0 for every i = 1, · · · , p, implies that
i)
g(y1 , . . . , yp ) > 0, where g (i) denotes the marginal distribution
of Yi , or
(ii) the transition kernel is absolutely continuous with respect to g,
then the chain is irreducible and positive Harris recurrent.

The Gibbs Sampler
Convergence

Properties of the Gibbs sampler (2)

Consequences
(i) If h(y)g(y)dy < ∞, then

T
1
lim h1 (Y (t) ) = h(y)g(y)dy a.e. g.
nT →∞ T
t=1

(ii) If, in addition, (Y (t) ) is aperiodic, then

lim K n (y, ·)µ(dx) − f =0
n→∞
TV

for every initial distribution µ.

The Gibbs Sampler
Convergence

Hammersley-Cliﬀord theorem

An illustration that conditionals determine the joint distribution
Theorem
If the joint density g(y1 , y2 ) have conditional distributions
g1 (y1 |y2 ) and g2 (y2 |y1 ), then

g2 (y2 |y1 )
g(y1 , y2 ) = .
g2 (v|y1 )/g1 (y1 |v) dv

[Hammersley & Cliﬀord, circa 1970]

The Gibbs Sampler
Convergence

General HC decomposition

Under the positivity condition, the joint distribution g satisﬁes
p
g j (y j |y 1 , . . . , y j−1 ,y j+1
, . . . , y p)
g(y1 , . . . , yp ) ∝
g j (y j |y 1 , . . . , y j−1 ,y , . . . , y p)
j=1 j+1

for every permutation on {1, 2, . . . , p} and every y ∈ Y .





The Gibbs Sampler

ABC basics
Alphabet soup
Calibration of ABC


ABC basics

Untractable likelihoods

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

f (y|θ) = f (y, z|θ) dz
Z

is impossible or too costly because of the dimension of z
c MCMC cannot be implemented!

ABC basics

Illustrations

Example
Stochastic volatility model: for Highest weight trajectories

t = 1, . . . , T,

0.4
0.2
yt = exp(zt ) t , zt = a+bzt−1 +σηt ,

0.0
−0.2
T very large makes it diﬃcult to
−0.4
include z within the simulated 0 200 400

t
600 800 1000

parameters

ABC basics

Illustrations

Example
Potts model: if y takes values on a grid Y of size k n and

f (y|θ) ∝ exp θ Iyl =yi
l∼i

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

ABC basics

Illustrations

Example
Inference on CMB: in cosmology, study of the Cosmic Microwave
Background via likelihoods immensely slow to computate (e.g
WMAP, Plank), because of numerically costly spectral transforms
[Data is a Fortran program]
[Kilbinger et al., 2010, MNRAS]

ABC basics

Illustrations

Example
Phylogenetic tree: in population
genetics, reconstitution of a common
ancestor from a sample of genes via
a phylogenetic tree that is close to
impossible to integrate out
[100 processor days with 4
parameters]
[Cornuet et al., 2009, Bioinformatics]

ABC basics

The ABC method

Bayesian setting: target is π(θ)f (x|θ)

ABC basics

The ABC method

When likelihood f (x|θ) not in closed form, likelihood-free rejection
technique:

ABC basics

The ABC method

When likelihood f (x|θ) not in closed form, likelihood-free rejection
technique:
ABC algorithm
For an observation y ∼ f (y|θ), under the prior π(θ), keep jointly
simulating
θ ∼ π(θ) , z ∼ f (z|θ ) ,
until the auxiliary variable z is equal to the observed value, z = y.

[Tavar´ et al., 1997]
e

ABC basics

Why does it work?!

The proof is trivial:

f (θi ) ∝ π(θi )f (z|θi )Iy (z)
z∈D
∝ 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 is replaced
with a tolerance condition,

(y, z) ≤

where is a distance

ABC basics

A as approximative

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

(y, z) ≤

where is a distance
Output distributed from

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

ABC basics

ABC algorithm

Algorithm 2 Likelihood-free rejection sampler 2
for i = 1 to N do
repeat
generate θ from the prior distribution π(·)
generate z from the likelihood f (·|θ )
until ρ{η(z), η(y)} ≤
set 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:

π(θ)f (z|θ)IA ,y (z)
π (θ, z|y) = ,
A ,y ×Θ π(θ)f (z|θ)dzdθ

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

ABC basics

Output

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

π(θ)f (z|θ)IA ,y (z)
π (θ, z|y) = ,
A ,y ×Θ π(θ)f (z|θ)dzdθ

where A ,y = {z ∈ D|ρ(η(z), η(y)) < }.
The idea behind ABC is that the summary statistics coupled with a
small tolerance should provide a good approximation of the
posterior distribution:

π (θ|y) = π (θ, z|y)dz ≈ π(θ|y) .

ABC basics

MA example

Back to the MA(q) model
q
xt = t + ϑi t−i
i=1

Simple prior: uniform over the inverse [real and complex] roots in
q
Q(u) = 1 − ϑi ui
i=1

under the identiﬁability conditions

ABC basics

MA example

Back to the MA(q) model
q
xt = t+ ϑi t−i
i=1

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 ( t )−q<t≤T
3. producing a simulated series (xt )1≤t≤T

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 ( t )−q<t≤T
3. producing a simulated series (xt )1≤t≤T
Distance: basic distance between the series
T
ρ((xt )1≤t≤T , (xt )1≤t≤T ) = (xt − xt )2
t=1

or distance between summary statistics like the q autocorrelations
T
τj = xt xt−j
t=j+1

ABC basics

Comparison of distance impact

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

ABC basics

ABC advances

Simulating from the prior is often poor in eﬃciency

ABC basics

ABC advances

Either modify the proposal distribution on θ to increase the density
of x’s within the vicinity of y...
[Marjoram et al, 2003; Bortot et al., 2007; Sisson et al., 2007]

ABC basics

ABC advances


...or by viewing the problem as a conditional density estimation
and by developing techniques to allow for larger
[Beaumont et al., 2002]

ABC basics

ABC advances


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

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

Alphabet soup

ABC-NP

Better usage of [prior] simulations by
adjustement: instead of throwing away
θ such that ρ(η(z), η(y)) > , replace
θs with locally regressed
ˆ
θ∗ = θ − {η(z) − η(y)}T β
[Csill´ry et al., TEE, 2010]
e

ˆ
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-MCMC

Markov chain (θ(t) ) created via the transition function

θ ∼ Kω (θ |θ(t) ) if x ∼ f (x|θ ) is such that x = y


π(θ )Kω (t) |θ )
θ (t+1)
= and u ∼ U (0, 1) ≤ π(θ(t) )K (θ |θ(t) ) ,

 (t) ω (θ
θ otherwise,

Alphabet soup

ABC-MCMC

Markov chain (θ(t) ) created via the transition function

θ ∼ Kω (θ |θ(t) ) if x ∼ f (x|θ ) is such that x = y


π(θ )Kω (t) |θ )
θ (t+1)
= and u ∼ U (0, 1) ≤ π(θ(t) )K (θ |θ(t) ) ,

 (t) ω (θ
θ otherwise,

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

Alphabet soup

ABC-MCMC (2)

Algorithm 3 Likelihood-free MCMC sampler
Use Algorithm 2 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] ,
π(θ )Kω (θ (t−1) |θ )
if u ≤ I
π(θ (t−1) Kω (θ |θ (t−1) ) A ,y (z ) then
set (θ(t) , z(t) ) = (θ , z )
else
(θ(t) , z(t) )) = (θ(t−1) , z(t−1) ),
end if
end for

Alphabet soup

ABCµ

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

Use of a joint density

f (θ, |y) ∝ ξ( |y, θ) × πθ (θ) × π ( )

where y is the data, and ξ( |y, θ) is the prior predictive density of
ρ(η(z), η(y)) given θ and x when z ∼ f (z|θ)

Alphabet soup

ABCµ

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

Use of a joint density

f (θ, |y) ∝ ξ( |y, θ) × πθ (θ) × π ( )

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

Alphabet soup

ABCµ details

Multidimensional distances ρk (k = 1, . . . , K) and errors
k = ρk (ηk (z), ηk (y)), with

ˆ 1
k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) = K[{ k −ρk (ηk (zb ), ηk (y))}/hk ]
Bhk
b

ˆ
then used in replacing ξ( |y, θ) with mink ξk ( |y, θ)

Alphabet soup

ABCµ details

Multidimensional distances ρk (k = 1, . . . , K) and errors
k = ρk (ηk (z), ηk (y)), with

ˆ 1
k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) = K[{ k −ρk (ηk (zb ), ηk (y))}/hk ]
Bhk
b

ˆ
then used in replacing ξ( |y, θ) with mink ξk ( |y, θ)
ABCµ involves acceptance probability

ˆ
π(θ , ) q(θ , θ)q( , ) mink ξk ( |y, θ )
ˆ
π(θ, ) q(θ, θ )q( , ) mink ξk ( |y, θ)

Alphabet soup

ABCµ multiple errors

[ c Ratmann et al., PNAS, 2009]

Alphabet soup

ABCµ for model choice

[ c Ratmann et al., PNAS, 2009]

Alphabet soup

Questions about ABCµ

For each model under comparison, marginal posterior on used to
assess the ﬁt of the model (HPD includes 0 or not).

Alphabet soup

Questions about ABCµ

For each model under comparison, marginal posterior on used to
assess the ﬁt of the model (HPD includes 0 or not).
Is the data informative about ? [Identiﬁability]
How is the prior π( ) impacting the comparison?
How is using both ξ( |x0 , θ) and π ( ) compatible with a
standard probability model? [remindful of Wilkinson]
Where is the penalisation for complexity in the model
comparison?
[X, Mengersen & Chen, 2010, PNAS]

Alphabet soup

A PMC version

Generate a sample at iteration t by
N
(t) (t−1) (t−1)
πt (θ ) ∝
ˆ ωj Kt (θ(t) |θj )
j=1

modulo acceptance of the associated xt , and use an importance
(t)
weight associated with an accepted simulation θi
(t) (t) (t)
ωi ∝ π(θi ) πt (θi ) .
ˆ

c Still likelihood free

Alphabet soup

The ABC-PMC algorithm
Given a decreasing sequence of approximation levels 1 ≥ ... ≥ T,

1. At iteration t = 1,
For i = 1, ..., N
(1) (1)
Simulate θi ∼ π(θ) and x ∼ f (x|θi ) until (x, y) < 1
(1)
Set ωi = 1/N
(1)
Take τ 2 as twice the empirical variance of the θi ’s
2. At iteration 2 ≤ t ≤ T ,
For i = 1, ..., N , repeat
(t−1) (t−1)
Pick θi from the θj ’s with probabilities ωj
(t) 2 (t)
generate θi |θi ∼ N (θi , σt ) and x ∼ f (x|θi )
until (x, y) < t
(t) (t) N (t−1) −1 (t) (t−1)
Set ωi ∝ π(θi )/ j=1 ωj ϕ σt θi − θj )
2 (t)
Take τt+1 as twice the weighted empirical variance of the θi ’s

Alphabet soup

Sequential Monte Carlo

SMC is a simulation technique to approximate a sequence of
related probability distributions πn with π0 “easy” and πT as
target.
Iterated IS: particles moved from time n to time n via kernel Kn
and use of a sequence of extended targets πn˜
n
πn (z0:n ) = πn (zn )
˜ Lj (zj+1 , zj )
j=0

where the Lj ’s are backward Markov kernels [check that πn (zn ) is
a marginal]
[Del Moral, Doucet & Jasra, Series B, 2006]

Alphabet soup

Sequential Monte Carlo (2)

Algorithm 4 SMC sampler
(0)
sample zi ∼ γ0 (x) (i = 1, . . . , N )
(0) (0) (0)
compute weights wi = π0 (zi ))/γ0 (zi )
for t = 1 to N do
if ESS(w(t−1) ) < NT then
resample N particles z (t−1) and set weights to 1
end if
(t−1) (t−1)
generate zi ∼ Kt (zi , ·) and set weights to
(t) (t) (t−1)
(t) (t−1) πt (zi ))Lt−1 (zi ), zi ))
wi = Wi−1 (t−1) (t−1) (t)
πt−1 (zi ))Kt (zi ), zi ))

end for

Alphabet soup

ABC-SMC
[Del Moral, Doucet & Jasra, 2009]

True derivation of an SMC-ABC algorithm
Use of a kernel Kn associated with target π n and derivation of the
backward kernel
π n (z )Kn (z , z)
Ln−1 (z, z ) =
πn (z)

Update of the weights
M
m=1 IA (xm )
in
win ∝ wi(n−1) M
n

m=1 IA n−1
(xm
i(n−1) )

when xm ∼ K(xi(n−1) , ·)
in

Alphabet soup

ABC-SMCM

Modiﬁcation: Makes M repeated simulations of the pseudo-data z
given the parameter, rather than using a single [M = 1]
simulation, leading to weight that is proportional to the number of
accepted zi s
M
1
ω(θ) = Iρ(η(y),η(zi ))<
M
i=1

[limit in M means exact simulation from (tempered) target]

Alphabet soup

Properties of ABC-SMC

The ABC-SMC method properly uses a backward kernel L(z, z ) to
simplify the importance weight and to remove the dependence on
the unknown likelihood from this weight. Update of importance
weights is reduced to the ratio of the proportions of surviving
particles

Alphabet soup

Properties of ABC-SMC

The ABC-SMC method properly uses a backward kernel L(z, z ) to
simplify the importance weight and to remove the dependence on
the unknown likelihood from this weight. Update of importance
weights is reduced to the ratio of the proportions of surviving
particles
Adaptivity in ABC-SMC algorithm only found in on-line
construction of the thresholds t , slowly enough to keep a large
number of accepted transitions

Alphabet soup

Semi-automatic ABC

Fearnhead and Prangle (2010) study ABC and the selection of the
summary statistic in close proximity to Wilkinson’s proposal
ABC then considered from a purely inferential viewpoint and
calibrated for estimation purposes.
Use of a randomised (or ‘noisy’) version of the summary statistics

η (y) = η(y) + τ
˜

Derivation of a well-calibrated version of ABC, i.e. an algorithm
that gives proper predictions for the distribution associated with
this randomised summary statistic.

Alphabet soup

Summary statistics

Optimality of the posterior expectations of the parameters of
interest as summary statistics!

Alphabet soup

Summary statistics

Optimality of the posterior expectations of the parameters of
interest as summary statistics!
Use of the standard quadratic loss function

(θ − θ0 )T A(θ − θ0 ) .

Calibration of ABC

Which summary?

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

Calibration of ABC

Which summary?

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

Calibration of ABC

Which summary?

Starting from a large collection of summary statistics is available,
Joyce and Marjoram (2008) consider the sequential inclusion into
the ABC target, with a stopping rule based on a likelihood ratio
test.
Does not taking into account the sequential nature of the tests
Depends on parameterisation
Order of inclusion matters.





The Gibbs Sampler


Model choice
Gibbs random ﬁelds
Model choice via ABC
Illustrations
Generic ABC model choice

Model choice

Bayesian model choice

Several models M1 , M2 , . . . are considered simultaneously for a
dataset y and the model index M is part of the inference.
Use of a prior distribution. π(M = m), plus a prior distribution on
the parameter conditional on the value m of the model index,
πm (θ m )
Goal is to derive the posterior distribution of M , challenging
computational target when models are complex.

Model choice

Generic ABC for model choice

Algorithm 5 Likelihood-free model choice sampler (ABC-MC)
for t = 1 to T do
repeat
Generate m from the prior π(M = m)
Generate θ m from the prior πm (θ m )
Generate z from the model fm (z|θ m )
until ρ{η(z), η(y)} <
Set m(t) = m and θ (t) = θ m
end for

Model choice

ABC estimates
Posterior probability π(M = m|y) approximated by the frequency
of acceptances from model m
T
1
Im(t) =m .
T
t=1

Issues with implementation:
should tolerances be the same for all models?
should summary statistics vary across models (incl. their
dimension)?
should the distance measure ρ vary as well?

Model choice

ABC estimates
Posterior probability π(M = m|y) approximated by the frequency
of acceptances from model m
T
1
Im(t) =m .
T
t=1

Issues with implementation:
should tolerances be the same for all models?
should summary statistics vary across models (incl. their
dimension)?
should the distance measure ρ vary as well?
Extension to a weighted polychotomous logistic regression estimate
of π(M = m|y), with non-parametric kernel weights
[Cornuet et al., DIYABC, 2009]

Model choice

The Great ABC controversy
On-going controvery in phylogeographic genetics about the validity
of using ABC for testing

Against: Templeton, 2008,
2009, 2010a, 2010b, 2010c
argues that nested hypotheses
cannot have higher probabilities
than nesting hypotheses (!)

Model choice

The Great ABC controversy

On-going controvery in phylogeographic genetics about the validity
of using ABC for testing
Replies: Fagundes et al., 2008,
Against: Templeton, 2008, Beaumont et al., 2010, Berger et
2009, 2010a, 2010b, 2010c al., 2010, Csill`ry et al., 2010
e
argues that nested hypotheses point out that the criticisms are
cannot have higher probabilities addressed at [Bayesian]
than nesting hypotheses (!) model-based inference and have
nothing to do with ABC...



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

1
f (y) = exp − Vc (yc ) ,
Z
c∈C

where Z is the normalising constant, C is the set of cliques of G
and Vc is any function also called potential
U (y) = c∈C Vc (yc ) is the energy function



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

1
f (y) = exp − Vc (yc ) ,
Z
c∈C

where Z is the normalising constant, C is the set of cliques of G
and Vc is any function also called potential
U (y) = c∈C Vc (yc ) is the energy function

c Z is usually unavailable in closed form


Potts model
Potts model
Vc (y) is of the form

Vc (y) = θS(y) = θ δyl =yi
l∼i

where l∼i denotes a neighbourhood structure


Potts model
Potts model
Vc (y) is of the form

Vc (y) = θS(y) = θ δyl =yi
l∼i

where l∼i denotes a neighbourhood structure

In most realistic settings, summation
Zθ = exp{θ T S(x)}
x∈X

involves too many terms to be manageable and numerical
approximations cannot always be trusted
[Cucala, Marin, CPR & Titterington, 2009]


Bayesian Model Choice

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

exp{θ T S0 (x)}/Zθ 0 ,0 π0 (dθ 0 )
0
Bm0 /m1 (x) =
exp{θ T S1 (x)}/Zθ 1 ,1 π1 (dθ 1 )
1


Bayesian Model Choice

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

exp{θ T S0 (x)}/Zθ 0 ,0 π0 (dθ 0 )
0
Bm0 /m1 (x) =
exp{θ T S1 (x)}/Zθ 1 ,1 π1 (dθ 1 )
1

Use of Jeﬀreys’ scale to select most appropriate model


Neighbourhood relations

Choice to be made between M neighbourhood relations
m
i∼i (0 ≤ m ≤ M − 1)

with
Sm (x) = I{xi =xi }
m
i∼i

driven by the posterior probabilities of the models.


Model index

Formalisation via a model index M that appears as a new
parameter with prior distribution π(M = m) and
π(θ|M = m) = πm (θm )


Model index

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

P(M = m|x) ∝ fm (x|θm )πm (θm ) dθm π(M = m) ,
Θm


Sufficient statistics
By definition, if S(x) sufficient statistic for the joint parameters
(M, θ0 , . . . , θM −1 ),
P(M = m|x) = P(M = m|S(x)) .


(M, θ0 , . . . , θM −1 ),
P(M = m|x) = P(M = m|S(x)) .
For each model m, own suﬃcient statistic Sm (·) and
S(·) = (S0 (·), . . . , SM −1 (·)) also suﬃcient.


(M, θ0 , . . . , θM −1 ),
P(M = m|x) = P(M = m|S(x)) .
For each model m, own sufficient statistic Sm (·) and
S(·) = (S0 (·), . . . , SM −1 (·)) also sufficient.
For Gibbs random fields,
1 2
x|M = m ∼ fm (x|θm ) = fm (x|S(x))fm (S(x)|θm )
1
= f 2 (S(x)|θm )
n(S(x)) m
where
n(S(x)) = {˜ ∈ X : S(˜ ) = S(x)}
x x
c S(x) is therefore also sufficient for the joint parameters
[Specific to Gibbs random fields!]


ABC model choice Algorithm

ABC-MC
Generate m∗ from the prior π(M = m).
∗
Generate θm∗ from the prior πm∗ (·).
Generate x∗ from the model fm∗ (·|θm∗ ).
∗

Compute the distance ρ(S(x0 ), S(x∗ )).
Accept (θm∗ , m∗ ) if ρ(S(x0 ), S(x∗ )) < .
∗

Note When = 0 the algorithm is exact


ABC approximation to the Bayes factor

Frequency ratio:
ˆ
P(M = m0 |x0 ) π(M = m1 )
BF m0 /m1 (x0 ) = ×
ˆ
P(M = m1 |x0 ) π(M = m0 )
{mi∗ = m0 } π(M = m1 )
= × ,
{mi∗ = m1 } π(M = m0 )


ABC approximation to the Bayes factor

Frequency ratio:
ˆ
P(M = m0 |x0 ) π(M = m1 )
BF m0 /m1 (x0 ) = ×
ˆ
P(M = m1 |x0 ) π(M = m0 )
{mi∗ = m0 } π(M = m1 )
= × ,
{mi∗ = m1 } π(M = m0 )

replaced with

1 + {mi∗ = m0 } π(M = m1 )
BF m0 /m1 (x0 ) = ×
1 + {mi∗ = m1 } π(M = m0 )

to avoid indeterminacy (also Bayes estimate).

Illustrations

Toy example

iid Bernoulli model versus two-state ﬁrst-order Markov chain, i.e.
n
f0 (x|θ0 ) = exp θ0 I{xi =1} {1 + exp(θ0 )}n ,
i=1

versus
n
1
f1 (x|θ1 ) = exp θ1 I{xi =xi−1 } {1 + exp(θ1 )}n−1 ,
2
i=2

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

Illustrations

Toy example (2)

(left) Comparison of the true BF m0 /m1 (x0 ) with BF m0 /m1 (x0 )
(in logs) over 2, 000 simulations and 4.106 proposals from the
prior. (right) Same when using tolerance corresponding to the
1% quantile on the distances.


Back to sufficiency

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


Back to sufficiency

If η1 (x) sufficient statistic for model m = 1 and parameter θ1 and
η2 (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
[X, Cornuet, Marin, and Pillai, 2011]


Limiting behaviour of B12 (T → ∞)

ABC approximation
T
t=1 Imt =1 Iρ{η(zt ),η(y)}≤
B12 (y) = T
,

where the (mt , z t )’s are simulated from the (joint) prior


Limiting behaviour of B12 (T → ∞)

ABC approximation
T
B12 (y) = T
,

where the (mt , z t )’s are simulated from the (joint) prior
As T go to inﬁnity, limit

Iρ{η(z),η(y)}≤ π1 (θ 1 )f1 (z|θ 1 ) dz dθ 1
B12 (y) =
Iρ{η(z),η(y)}≤ π2 (θ 2 )f2 (z|θ 2 ) dz dθ 2
η
Iρ{η,η(y)}≤ π1 (θ 1 )f1 (η|θ 1 ) dη dθ 1
= η ,
Iρ{η,η(y)}≤ π2 (θ 2 )f2 (η|θ 2 ) dη dθ 2
η η
where f1 (η|θ 1 ) and f2 (η|θ 2 ) distributions of η(z)


Limiting behaviour of B12 ( → 0)

When goes to zero,
η
η π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1
B12 (y) = η ,
π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2


Limiting behaviour of B12 ( → 0)

When goes to zero,
η
η π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1
B12 (y) = η ,
π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2

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


Limiting behaviour of B12 (under suﬃciency)

If η(y) suﬃcient statistic for both models,

fi (y|θ i ) = gi (y)fiη (η(y)|θ i )

Thus
η
Θ1 π(θ 1 )g1 (y)f1 (η(y)|θ 1 ) dθ 1
B12 (y) = η
Θ2 π(θ 2 )g2 (y)f2 (η(y)|θ 2 ) dθ 2
η
g1 (y) π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 g1 (y) η
= η = B (y) .
g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g2 (y) 12

[Didelot, Everitt, Johansen & Lawson, 2011]


Limiting behaviour of B12 (under sufficiency)

If η(y) sufficient statistic for both models,

fi (y|θ i ) = gi (y)fiη (η(y)|θ i )

Thus
η
Θ1 π(θ 1 )g1 (y)f1 (η(y)|θ 1 ) dθ 1
B12 (y) = η
Θ2 π(θ 2 )g2 (y)f2 (η(y)|θ 2 ) dθ 2
η
g1 (y) π1 (θ 1 )f1 (η(y)|θ 1 ) dθ 1 g1 (y) η
= η = B (y) .
g2 (y) π2 (θ 2 )f2 (η(y)|θ 2 ) dθ 2 g2 (y) 12

c No discrepancy only when cross-model sufficiency


Poisson/geometric example

Sample
x = (x1 , . . . , xn )
from either a Poisson P(λ) or from a geometric G(p) Then
n
S= yi = η(x)
i=1

suﬃcient statistic for either model but not simultaneously
Discrepancy ratio

g1 (x) S!n−S / i yi !
=
g2 (x) 1 n+S−1
S


Poisson/geometric discrepancy

η
Range of B12 (x) versus B12 (x) B12 (x): The values produced have
nothing in common.


Formal recovery

Creating an encompassing exponential family
T T
f (x|θ1 , θ2 , α1 , α2 ) ∝ exp{θ1 η1 (x) + θ1 η1 (x) + α1 t1 (x) + α2 t2 (x)}

leads to a suﬃcient statistic (η1 (x), η2 (x), t1 (x), t2 (x))


Formal recovery

T T


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


Formal recovery

T T


Only applies in genuine suﬃciency settings...
c Inability to evaluate loss brought by summary statistics


Meaning of the ABC-Bayes factor

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

η (θ0 + 1)2 −θ0
lim B12 (y) = e
n→∞ θ0


MA(q) divergence

Evolution [against ] of ABC Bayes factor, in terms of frequencies of
visits to models MA(1) (left) and MA(2) (right) when equal to
10, 1, .1, .01% quantiles on insuﬃcient autocovariance distances. Sample
of 50 points from a MA(2) with θ1 = 0.6, θ2 = 0.2. True Bayes factor
equal to 17.71.


MA(q) divergence

Evolution [against ] of ABC Bayes factor, in terms of frequencies of
visits to models MA(1) (left) and MA(2) (right) when equal to
10, 1, .1, .01% quantiles on insuﬃcient autocovariance distances. Sample
of 50 points from a MA(1) model with θ1 = 0.6. True Bayes factor B21
equal to .004.


A population genetics evaluation

Population genetics example with
3 populations
2 scenari
15 individuals
5 loci
single mutation parameter


A population genetics evaluation

Population genetics example with
3 populations
2 scenari
15 individuals
5 loci
single mutation parameter
24 summary statistics
2 million ABC proposal
importance [tree] sampling alternative


Stability of importance sampling


Comparison with ABC
Use of 24 summary statistics and DIY-ABC logistic correction


Comparison with ABC
Use of 15 summary statistics and DIY-ABC logistic correction


The only safe cases

Besides speciﬁc models like Gibbs random ﬁelds,
using distances over the data itself escapes the discrepancy...
[Toni & Stumpf, 2010; Sousa et 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 et al., 2009]

...and so does the use of more informal model fitting measures
[Ratmann et al., 2009, 2011]

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