These are the slides of the talk I gave at the workshop in Gainesville, Florida, on Jan. 19, 2013

1. Approximate Bayesian Computation as a new
empirical Bayes?
Christian P. Robert
Winter Workshop on Simulation, Gainesville, Jan. 19, 2013
Joint work with J.-M. Cornuet, J.-M. Marin,
K.L. Mengersen, N. Pillai, P. Pudlo and J. Rousseau

2. Advertisment
MCMSki IV to be held in Chamonix Mt Blanc, France, from
Monday, Jan. 6 to Wed., Jan. 8, 2014
All aspects of MCMC++ theory and methodology
Parallel (invited and contributed) sessions: call for proposals on
website http://www.pages.drexel.edu/ mwl25/mcmski/

3. Outline
Unavailable likelihoods
ABC methods
ABC as an inference machine
ABCel
Conclusion and perspectives

4. Intractable likelihood
Case of a well-defined statistical model where the likelihood
function
(θ|y) = f (y1 , . . . , yn |θ)
is (really!) not available in closed form
can (easily!) be neither completed nor demarginalised
cannot be estimated by an unbiased estimator
c Prohibits direct implementation of a generic MCMC algorithm
like Metropolis–Hastings

5. Intractable likelihood
Case of a well-defined statistical model where the likelihood
function
(θ|y) = f (y1 , . . . , yn |θ)
is (really!) not available in closed form
can (easily!) be neither completed nor demarginalised
cannot be estimated by an unbiased estimator
c Prohibits direct implementation of a generic MCMC algorithm
like Metropolis–Hastings

6. The abc alternative
Approximations to the original B problem
Degrading the precision down to a tolerance ε
Replacing the likelihood with a non-parametric approximation
Summarising the data with insufficient statistics

7. The abc alternative
Approximations to the original B problem
Degrading the precision down to a tolerance ε
Replacing the likelihood with a non-parametric approximation
Summarising the data with insufficient statistics

8. The abc alternative
Approximations to the original B problem
Degrading the precision down to a tolerance ε
Replacing the likelihood with a non-parametric approximation
Summarising the data with insufficient statistics

9. Different worries about abc
Impact on B inference
a mere computational issue (that will eventually end up being
solved by more powerful computers, &tc, even if too costly in
the short term, as for regular Monte Carlo methods)
an inferential issue (opening opportunities for new inference
machine, with legitimity different than for classical B
approach)
a Bayesian conundrum (how closely related to the/a B
approach? a new type of empirical Bayes inference?)

10. Different worries about abc
Impact on B inference
a mere computational issue (that will eventually end up being
solved by more powerful computers, &tc, even if too costly in
the short term, as for regular Monte Carlo methods)
an inferential issue (opening opportunities for new inference
machine, with legitimity different than for classical B
approach)
a Bayesian conundrum (how closely related to the/a B
approach? a new type of empirical Bayes inference?)

11. Different worries about abc
Impact on B inference
a mere computational issue (that will eventually end up being
solved by more powerful computers, &tc, even if too costly in
the short term, as for regular Monte Carlo methods)
an inferential issue (opening opportunities for new inference
machine, with legitimity different than for classical B
approach)
a Bayesian conundrum (how closely related to the/a B
approach? a new type of empirical Bayes inference?)

12. Econom’ections
Similar exploration of simulation-based and approximation
techniques in Econometrics
Simulated method of moments
Method of simulated moments
Simulated pseudo-maximum-likelihood
Indirect inference
[Gouri´roux & Monfort, 1996]
e
even though motivation is partly-defined models rather than
complex likelihoods

13. Econom’ections
Similar exploration of simulation-based and approximation
techniques in Econometrics
Simulated method of moments
Method of simulated moments
Simulated pseudo-maximum-likelihood
Indirect inference
[Gouri´roux & Monfort, 1996]
e
even though motivation is partly-defined models rather than
complex likelihoods

14. Indirect inference
^
Minimise [in θ] a distance between estimators β based on a
pseudo-model for genuine observations and for observations
simulated under the true model and the parameter θ.
[Gouri´roux, Monfort, & Renault, 1993;
e
Smith, 1993; Gallant & Tauchen, 1996]

15. Indirect inference (PML vs. PSE)
Example of the pseudo-maximum-likelihood (PML)
^
β(y) = arg max log f (yt |β, y1:(t−1) )
β
t
leading to
arg min ||β(yo ) − β(y1 (θ), . . . , yS (θ))||2
^ ^
θ
when
ys (θ) ∼ f (y|θ) s = 1, . . . , S

16. Indirect inference (PML vs. PSE)
Example of the pseudo-score-estimator (PSE)
2
∂ log f
^
β(y) = arg min (yt |β, y1:(t−1) )
β
t
∂β
leading to
arg min ||β(yo ) − β(y1 (θ), . . . , yS (θ))||2
^ ^
θ
when
ys (θ) ∼ f (y|θ) s = 1, . . . , S

17. Consistent indirect inference
“...in order to get a unique solution the dimension of
the auxiliary parameter β must be larger than or equal to
the dimension of the initial parameter θ. If the problem is
just identified the different methods become easier...”
Consistency depending on the criterion and on the asymptotic
identifiability of θ
[Gouri´roux & Monfort, 1996, p. 66]
e
Which connection [if any] with the B perspective?

18. Consistent indirect inference
“...in order to get a unique solution the dimension of
the auxiliary parameter β must be larger than or equal to
the dimension of the initial parameter θ. If the problem is
just identified the different methods become easier...”
Consistency depending on the criterion and on the asymptotic
identifiability of θ
[Gouri´roux & Monfort, 1996, p. 66]
e
Which connection [if any] with the B perspective?

19. Consistent indirect inference
“...in order to get a unique solution the dimension of
the auxiliary parameter β must be larger than or equal to
the dimension of the initial parameter θ. If the problem is
just identified the different methods become easier...”
Consistency depending on the criterion and on the asymptotic
identifiability of θ
[Gouri´roux & Monfort, 1996, p. 66]
e
Which connection [if any] with the B perspective?

20. Approximate Bayesian computation
Unavailable likelihoods
ABC methods
Genesis of ABC
ABC basics
Advances and interpretations
ABC as knn
ABC as an inference machine
ABCel
Conclusion and perspectives

21. Genetic background of ABC
skip genetics
ABC is a recent computational technique that only requires being
able to sample from the likelihood f (·|θ)
This technique stemmed from population genetics models, about
15 years ago, and population geneticists still contribute
significantly to methodological developments of ABC.
[Griffith & al., 1997; Tavar´ & al., 1999]
e

22. Demo-genetic inference
Each model is characterized by a set of parameters θ that cover
historical (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 of
polymorphism (DNA sample) y observed at the present time
Problem:
most of the time, we cannot calculate the likelihood of the
polymorphism data f (y|θ)...

23. Demo-genetic inference
Each model is characterized by a set of parameters θ that cover
historical (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 of
polymorphism (DNA sample) y observed at the present time
Problem:
most of the time, we cannot calculate the likelihood of the
polymorphism data f (y|θ)...

24. Neutral model at a given microsatellite locus, in a closed
panmictic population at equilibrium
Mutations according to
the Simple stepwise
Mutation Model
(SMM)
• date of the mutations ∼
Poisson process with
intensity θ/2 over the
branches
• MRCA = 100
• independent mutations:
±1 with pr. 1/2
Sample of 8 genes

25. Neutral model at a given microsatellite locus, in a closed
panmictic population at equilibrium
Kingman’s genealogy
When time axis is
normalized,
T (k) ∼ Exp(k(k − 1)/2)
Mutations according to
the Simple stepwise
Mutation Model
(SMM)
• date of the mutations ∼
Poisson process with
intensity θ/2 over the
branches
• MRCA = 100
• independent mutations:
±1 with pr. 1/2

26. Neutral model at a given microsatellite locus, in a closed
panmictic population at equilibrium
Kingman’s genealogy
When time axis is
normalized,
T (k) ∼ Exp(k(k − 1)/2)
Mutations according to
the Simple stepwise
Mutation Model
(SMM)
• date of the mutations ∼
Poisson process with
intensity θ/2 over the
branches
• MRCA = 100
• independent mutations:
±1 with pr. 1/2

27. Neutral model at a given microsatellite locus, in a closed
panmictic population at equilibrium
Kingman’s genealogy
When time axis is
normalized,
T (k) ∼ Exp(k(k − 1)/2)
Mutations according to
the Simple stepwise
Mutation Model
(SMM)
• date of the mutations ∼
Poisson process with
intensity θ/2 over the
branches
Observations: leafs of the tree
• MRCA = 100
^
θ=?
• independent mutations:
±1 with pr. 1/2

28. Much more interesting models. . .
several independent locus
Independent gene genealogies and mutations
different populations
linked by an evolutionary scenario made of divergences,
admixtures, migrations between populations, etc.
larger sample size
usually between 50 and 100 genes
MRCA
τ2
τ1
A typical evolutionary scenario: POP 0 POP 1 POP 2

29. Intractable likelihood
Missing (too missing!) data structure:
f (y|θ) = f (y|G , θ)f (G |θ)dG
G
cannot be computed in a manageable way...
The genealogies are considered as nuisance parameters
This modelling clearly differs from the phylogenetic perspective
where the tree is the parameter of interest.

30. Intractable likelihood
Missing (too missing!) data structure:
f (y|θ) = f (y|G , θ)f (G |θ)dG
G
cannot be computed in a manageable way...
The genealogies are considered as nuisance parameters
This modelling clearly differs from the phylogenetic perspective
where the tree is the parameter of interest.

31. A?B?C?
A stands for approximate
[wrong likelihood /
picture]
B stands for Bayesian
C stands for computation
[producing a parameter
sample]

32. A?B?C?
A stands for approximate
[wrong likelihood /
picture]
B stands for Bayesian
C stands for computation
[producing a parameter
sample]

33. A?B?C?
ESS=155.6 ESS=75.93 ESS=76.87
4
2.0
3
Density
Density
Density
1.0
2
1.0
1
0.0
0.0
0
A stands for approximate −0.5 0.0
θ
ESS=91.54
0.5 1.0 −0.4 −0.2 0.0
θ
ESS=108.4
0.2 0.4 −0.4 −0.2
θ
ESS=85.13
0.0 0.2
4
0.0 1.0 2.0 3.0
0.0 1.0 2.0 3.0
3
[wrong likelihood /
Density
Density
Density
2
1
0
picture] −0.6 −0.4 −0.2
θ
ESS=149.1
0.0 0.2 −0.4 0.0
θ
ESS=96.31
0.2 0.4 0.6 −0.2 0.0
θ
ESS=83.77
0.2 0.4 0.6
4
2.0
3
2.0
Density
Density
Density
2
1.0
1.0
B stands for Bayesian
1
0.0
0.0
0
−0.5 0.0 0.5 1.0 −0.4 0.0 0.2 0.4 0.6 −0.6 −0.4 −0.2 0.0 0.2 0.4
θ
ESS=155.7 θ
ESS=92.42 θ
ESS=95.01
0.0 1.0 2.0 3.0
2.0
3.0
C stands for computation
Density
Density
Density
1.0
1.5
0.0
0.0
[producing a parameter −0.5
θ
0.0
ESS=139.2
0.5 −0.4 0.0
θ
ESS=99.33
0.2 0.4 0.6 −0.4
θ
0.0
ESS=87.28
0.2 0.4 0.6
2.0
0.0 1.0 2.0 3.0
3
sample]
Density
Density
Density
2
1.0
1
0.0
0
−0.6 −0.2 0.2 0.6 −0.4 −0.2 0.0 0.2 0.4 −0.2 0.0 0.2 0.4 0.6

34. How Bayesian is aBc?
Could we turn the resolution into a Bayesian answer?
ideally so (not meaningfull: requires ∞-ly powerful computer
approximation error unknown (w/o costly simulation)
true Bayes for wrong model (formal and artificial)
true Bayes for noisy model (much more convincing)
true Bayes for estimated likelihood (back to econometrics?)
illuminating the tension between information and precision

35. Untractable likelihood
Back to stage zero: what can we do
when a likelihood function f (y|θ) is
well-defined but impossible / too
costly to compute...?
MCMC cannot be implemented!
shall we give up Bayesian
inference altogether?!

36. Untractable likelihood
Back to stage zero: what can we do
when a likelihood function f (y|θ) is
well-defined but impossible / too
costly to compute...?
MCMC cannot be implemented!
shall we give up Bayesian
inference altogether?!
or settle for an almost Bayesian
inference/picture...?

37. ABC methodology
Bayesian setting: target is π(θ)f (x|θ)
When likelihood f (x|θ) not in closed form, likelihood-free rejection
technique:
Foundation
For an observation y ∼ f (y|θ), under the prior π(θ), if one keeps
jointly 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; Tavar´ et al., 1997]
e

38. ABC methodology
Bayesian setting: target is π(θ)f (x|θ)
When likelihood f (x|θ) not in closed form, likelihood-free rejection
technique:
Foundation
For an observation y ∼ f (y|θ), under the prior π(θ), if one keeps
jointly 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; Tavar´ et al., 1997]
e

39. ABC methodology
Bayesian setting: target is π(θ)f (x|θ)
When likelihood f (x|θ) not in closed form, likelihood-free rejection
technique:
Foundation
For an observation y ∼ f (y|θ), under the prior π(θ), if one keeps
jointly 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; Tavar´ et al., 1997]
e

40. A as A...pproximative
When y is a continuous random variable, strict equality z = y is
replaced with a tolerance zone
ρ(y, z)
where ρ is a distance
Output distributed from
def
π(θ) Pθ {ρ(y, z) < } ∝ π(θ|ρ(y, z) < )
[Pritchard et al., 1999]

41. A as A...pproximative
When y is a continuous random variable, strict equality z = y is
replaced with a tolerance zone
ρ(y, z)
where ρ is a distance
Output distributed from
def
π(θ) Pθ {ρ(y, z) < } ∝ π(θ|ρ(y, z) < )
[Pritchard et al., 1999]

42. ABC algorithm
In most implementations, further degree of A...pproximation:
Algorithm 1 Likelihood-free rejection sampler
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

43. 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) .
...does it?!

44. 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) .
...does it?!

45. 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) .
...does it?!

46. 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
restricted posterior distribution:
π (θ|y) = π (θ, z|y)dz ≈ π(θ|η(y)) .
Not so good..!
skip convergence details!

47. Convergence of ABC
What happens when → 0?
For B ⊂ Θ, we have
A f (z|θ)dz f (z|θ)π(θ)dθ
,y B
π(θ)dθ = dz
B A ,y ×Θ
π(θ)f (z|θ)dzdθ A ,y A ,y ×Θ
π(θ)f (z|θ)dzdθ
B f (z|θ)π(θ)dθ m(z)
= dz
A ,y
m(z) A ,y ×Θ
π(θ)f (z|θ)dzdθ
m(z)
= π(B|z) dz
A ,y A ,y ×Θ π(θ)f (z|θ)dzdθ
which indicates convergence for a continuous π(B|z).

48. Convergence of ABC
What happens when → 0?
For B ⊂ Θ, we have
A f (z|θ)dz f (z|θ)π(θ)dθ
,y B
π(θ)dθ = dz
B A ,y ×Θ
π(θ)f (z|θ)dzdθ A ,y A ,y ×Θ
π(θ)f (z|θ)dzdθ
B f (z|θ)π(θ)dθ m(z)
= dz
A ,y
m(z) A ,y ×Θ
π(θ)f (z|θ)dzdθ
m(z)
= π(B|z) dz
A ,y A ,y ×Θ π(θ)f (z|θ)dzdθ
which indicates convergence for a continuous π(B|z).

49. Convergence (do not attempt!)
...and the above does not apply to insufficient statistics:
If η(y) is not a sufficient statistics, the best one can hope for is
π(θ|η(y)) , not π(θ|y)
If η(y) is an ancillary statistic, the whole information contained in
y is lost!, the “best” one can “hope” for is
π(θ|η(y)) = π(θ)
Bummer!!!

50. Convergence (do not attempt!)
...and the above does not apply to insufficient statistics:
If η(y) is not a sufficient statistics, the best one can hope for is
π(θ|η(y)) , not π(θ|y)
If η(y) is an ancillary statistic, the whole information contained in
y is lost!, the “best” one can “hope” for is
π(θ|η(y)) = π(θ)
Bummer!!!

51. Convergence (do not attempt!)
...and the above does not apply to insufficient statistics:
If η(y) is not a sufficient statistics, the best one can hope for is
π(θ|η(y)) , not π(θ|y)
If η(y) is an ancillary statistic, the whole information contained in
y is lost!, the “best” one can “hope” for is
π(θ|η(y)) = π(θ)
Bummer!!!

52. Convergence (do not attempt!)
...and the above does not apply to insufficient statistics:
If η(y) is not a sufficient statistics, the best one can hope for is
π(θ|η(y)) , not π(θ|y)
If η(y) is an ancillary statistic, the whole information contained in
y is lost!, the “best” one can “hope” for is
π(θ|η(y)) = π(θ)
Bummer!!!

53. Comments
Role of distance paramount (because = 0)
Scaling of components of η(y) is also determinant
matters little if “small enough”
representative of “curse of dimensionality”
small is beautiful!
the data as a whole may be paradoxically weakly informative
for ABC

54. ABC (simul’) advances
how approximative is ABC? ABC as knn
Simulating from the prior is often poor in efficiency
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]
...or by viewing the problem as a conditional density estimation
and by developing techniques to allow for larger
[Beaumont et al., 2002]
.....or even by including in the inferential framework [ABCµ ]
[Ratmann et al., 2009]

55. ABC (simul’) advances
how approximative is ABC? ABC as knn
Simulating from the prior is often poor in efficiency
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]
...or by viewing the problem as a conditional density estimation
and by developing techniques to allow for larger
[Beaumont et al., 2002]
.....or even by including in the inferential framework [ABCµ ]
[Ratmann et al., 2009]

56. ABC (simul’) advances
how approximative is ABC? ABC as knn
Simulating from the prior is often poor in efficiency
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]
...or by viewing the problem as a conditional density estimation
and by developing techniques to allow for larger
[Beaumont et al., 2002]
.....or even by including in the inferential framework [ABCµ ]
[Ratmann et al., 2009]

57. ABC (simul’) advances
how approximative is ABC? ABC as knn
Simulating from the prior is often poor in efficiency
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]
...or by viewing the problem as a conditional density estimation
and by developing techniques to allow for larger
[Beaumont et al., 2002]
.....or even by including in the inferential framework [ABCµ ]
[Ratmann et al., 2009]

58. ABC-NP
Better usage of [prior] simulations by
adjustement: instead of throwing away
θ such that ρ(η(z), η(y)) > , replace
θ’s with locally regressed transforms
θ∗ = θ − {η(z) − η(y)}T β
^
[Csill´ry et al., TEE, 2010]
e
^
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]

59. ABC-NP (regression)
Also found in the subsequent literature, e.g. in Fearnhead-Prangle (2012) :
weight directly simulation by
Kδ {ρ(η(z(θ)), η(y))}
or
S
1
Kδ {ρ(η(zs (θ)), η(y))}
S
s=1
[consistent estimate of f (η|θ)]
Curse of dimensionality: poor estimate when d = dim(η) is large...

60. ABC-NP (regression)
Also found in the subsequent literature, e.g. in Fearnhead-Prangle (2012) :
weight directly simulation by
Kδ {ρ(η(z(θ)), η(y))}
or
S
1
Kδ {ρ(η(zs (θ)), η(y))}
S
s=1
[consistent estimate of f (η|θ)]
Curse of dimensionality: poor estimate when d = dim(η) is large...

61. ABC-NP (density estimation)
Use of the kernel weights
Kδ {ρ(η(z(θ)), η(y))}
leads to the NP estimate of the posterior expectation
i θi Kδ {ρ(η(z(θi )), η(y))}
i Kδ {ρ(η(z(θi )), η(y))}
[Blum, JASA, 2010]

62. ABC-NP (density estimation)
Use of the kernel weights
Kδ {ρ(η(z(θ)), η(y))}
leads to the NP estimate of the posterior conditional density
i
˜
Kb (θi − θ)Kδ {ρ(η(z(θi )), η(y))}
i Kδ {ρ(η(z(θi )), η(y))}
[Blum, JASA, 2010]

63. ABC-NP (density estimations)
Other versions incorporating regression adjustments
i
˜
Kb (θ∗ − θ)Kδ {ρ(η(z(θi )), η(y))}
i
i Kδ {ρ(η(z(θi )), η(y))}
In all cases, error
E[^ (θ|y)] − g (θ|y) = cb 2 + cδ2 + OP (b 2 + δ2 ) + OP (1/nδd )
g
c
var(^ (θ|y)) =
g (1 + oP (1))
nbδd

64. ABC-NP (density estimations)
Other versions incorporating regression adjustments
i
˜
Kb (θ∗ − θ)Kδ {ρ(η(z(θi )), η(y))}
i
i Kδ {ρ(η(z(θi )), η(y))}
In all cases, error
E[^ (θ|y)] − g (θ|y) = cb 2 + cδ2 + OP (b 2 + δ2 ) + OP (1/nδd )
g
c
var(^ (θ|y)) =
g (1 + oP (1))
nbδd
[Blum, JASA, 2010]

65. ABC-NP (density estimations)
Other versions incorporating regression adjustments
i
˜
Kb (θ∗ − θ)Kδ {ρ(η(z(θi )), η(y))}
i
i Kδ {ρ(η(z(θi )), η(y))}
In all cases, error
E[^ (θ|y)] − g (θ|y) = cb 2 + cδ2 + OP (b 2 + δ2 ) + OP (1/nδd )
g
c
var(^ (θ|y)) =
g (1 + oP (1))
nbδd
[standard NP calculations]

66. ABC as knn
[Biau et al., 2012, arxiv:1207.6461]
Practice of ABC: determine tolerance as a quantile on observed
distances, say 10% or 1% quantile,
= N = qα (d1 , . . . , dN )
Interpretation of ε as nonparametric bandwidth only
approximation of the actual practice
[Blum & Fran¸ois, 2010]
c
ABC is a k-nearest neighbour (knn) method with kN = N N
[Loftsgaarden & Quesenberry, 1965]

67. ABC as knn
[Biau et al., 2012, arxiv:1207.6461]
Practice of ABC: determine tolerance as a quantile on observed
distances, say 10% or 1% quantile,
= N = qα (d1 , . . . , dN )
Interpretation of ε as nonparametric bandwidth only
approximation of the actual practice
[Blum & Fran¸ois, 2010]
c
ABC is a k-nearest neighbour (knn) method with kN = N N
[Loftsgaarden & Quesenberry, 1965]

68. ABC as knn
[Biau et al., 2012, arxiv:1207.6461]
Practice of ABC: determine tolerance as a quantile on observed
distances, say 10% or 1% quantile,
= N = qα (d1 , . . . , dN )
Interpretation of ε as nonparametric bandwidth only
approximation of the actual practice
[Blum & Fran¸ois, 2010]
c
ABC is a k-nearest neighbour (knn) method with kN = N N
[Loftsgaarden & Quesenberry, 1965]

69. ABC consistency
Provided
kN / log log N −→ ∞ and kN /N −→ 0
as N → ∞, for almost all s0 (with respect to the distribution of
S), with probability 1,
kN
1
ϕ(θj ) −→ E[ϕ(θj )|S = s0 ]
kN
j=1
[Devroye, 1982]
Biau et al. (2012) also recall pointwise and integrated mean square error
consistency results on the corresponding kernel estimate of the
conditional posterior distribution, under constraints
p
kN → ∞, kN /N → 0, hN → 0 and hN kN → ∞,

70. ABC consistency
Provided
kN / log log N −→ ∞ and kN /N −→ 0
as N → ∞, for almost all s0 (with respect to the distribution of
S), with probability 1,
kN
1
ϕ(θj ) −→ E[ϕ(θj )|S = s0 ]
kN
j=1
[Devroye, 1982]
Biau et al. (2012) also recall pointwise and integrated mean square error
consistency results on the corresponding kernel estimate of the
conditional posterior distribution, under constraints
p
kN → ∞, kN /N → 0, hN → 0 and hN kN → ∞,

71. Rates of convergence
Further assumptions (on target and kernel) allow for precise
(integrated mean square) convergence rates (as a power of the
sample size N), derived from classical k-nearest neighbour
regression, like
4
when m = 1, 2, 3, kN ≈ N (p+4)/(p+8) and rate N − p+8
4
when m = 4, kN ≈ N (p+4)/(p+8) and rate N − p+8 log N
4
when m > 4, kN ≈ N (p+4)/(m+p+4) and rate N − m+p+4
[Biau et al., 2012, arxiv:1207.6461]
Drag: Only applies to sufficient summary statistics

72. Rates of convergence
Further assumptions (on target and kernel) allow for precise
(integrated mean square) convergence rates (as a power of the
sample size N), derived from classical k-nearest neighbour
regression, like
4
when m = 1, 2, 3, kN ≈ N (p+4)/(p+8) and rate N − p+8
4
when m = 4, kN ≈ N (p+4)/(p+8) and rate N − p+8 log N
4
when m > 4, kN ≈ N (p+4)/(m+p+4) and rate N − m+p+4
[Biau et al., 2012, arxiv:1207.6461]
Drag: Only applies to sufficient summary statistics

73. ABC inference machine
Unavailable likelihoods
ABC methods
ABC as an inference machine
Error inc.
Exact BC and approximate
targets
summary statistic
ABCel
Conclusion and perspectives

74. How Bayesian is aBc..?
maybe a convergent method of inference (meaningful?
sufficient? foreign?)
approximation error unknown (w/o simulation)
pragmatic Bayes (there is no other solution!)
many calibration issues (tolerance, distance, statistics)
the NP side should be incorporated into the whole B picture
the approximation error should also be part of the B inference
to ABCel

75. ABCµ
Idea Infer about the error as well as about the parameter:
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 y when z ∼ f (z|θ)
Warning! Replacement of ξ( |y, θ) with a non-parametric kernel
approximation.
[Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS]

76. ABCµ
Idea Infer about the error as well as about the parameter:
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 y when z ∼ f (z|θ)
Warning! Replacement of ξ( |y, θ) with a non-parametric kernel
approximation.
[Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS]

77. ABCµ
Idea Infer about the error as well as about the parameter:
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 y when z ∼ f (z|θ)
Warning! Replacement of ξ( |y, θ) with a non-parametric kernel
approximation.
[Ratmann, Andrieu, Wiuf and Richardson, 2009, PNAS]

78. ABCµ details
Multidimensional distances ρk (k = 1, . . . , K ) and errors
k = ρk (ηk (z), ηk (y)), with
1
k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) =
^ K [{ k −ρk (ηk (zb ), ηk (y))}/hk ]
Bhk
b
then used in replacing ξ( |y, θ) with mink ξk ( |y, θ)
^
ABCµ involves acceptance probability
π(θ , ) q(θ , θ)q( , ) mink ξk ( |y, θ )
^
π(θ, ) q(θ, θ )q( , ) mink ξk ( |y, θ)
^

79. ABCµ details
Multidimensional distances ρk (k = 1, . . . , K ) and errors
k = ρk (ηk (z), ηk (y)), with
1
k ∼ ξk ( |y, θ) ≈ ξk ( |y, θ) =
^ K [{ k −ρk (ηk (zb ), ηk (y))}/hk ]
Bhk
b
then used in replacing ξ( |y, θ) with mink ξk ( |y, θ)
^
ABCµ involves acceptance probability
π(θ , ) q(θ , θ)q( , ) mink ξk ( |y, θ )
^
π(θ, ) q(θ, θ )q( , ) mink ξk ( |y, θ)
^

80. Wilkinson’s exact BC (not exactly!)
ABC approximation error (i.e. non-zero tolerance) replaced with
exact simulation from a controlled approximation to the target,
convolution of true posterior with kernel function
π(θ)f (z|θ)K (y − z)
π (θ, z|y) = ,
π(θ)f (z|θ)K (y − z)dzdθ
with K kernel parameterised by bandwidth .
[Wilkinson, 2008]
Theorem
The ABC algorithm based on the assumption of a randomised
observation y = y + ξ, ξ ∼ K , and an acceptance probability of
˜
K (y − z)/M
gives draws from the posterior distribution π(θ|y).

81. Wilkinson’s exact BC (not exactly!)
ABC approximation error (i.e. non-zero tolerance) replaced with
exact simulation from a controlled approximation to the target,
convolution of true posterior with kernel function
π(θ)f (z|θ)K (y − z)
π (θ, z|y) = ,
π(θ)f (z|θ)K (y − z)dzdθ
with K kernel parameterised by bandwidth .
[Wilkinson, 2008]
Theorem
The ABC algorithm based on the assumption of a randomised
observation y = y + ξ, ξ ∼ K , and an acceptance probability of
˜
K (y − z)/M
gives draws from the posterior distribution π(θ|y).

82. How exact a BC?
Pros
Pseudo-data from true model and observed data from noisy
model
Interesting perspective in that outcome is completely
controlled
Link with ABCµ and assuming y is observed with a
measurement error with density K
Relates to the theory of model approximation
[Kennedy & O’Hagan, 2001]
Cons
Requires K to be bounded by M
True approximation error never assessed

83. How exact a BC?
Pros
Pseudo-data from true model and observed data from noisy
model
Interesting perspective in that outcome is completely
controlled
Link with ABCµ and assuming y is observed with a
measurement error with density K
Relates to the theory of model approximation
[Kennedy & O’Hagan, 2001]
Cons
Requires K to be bounded by M
True approximation error never assessed

84. Noisy ABC for HMMs
Specific case of a hidden Markov model
Xt+1 ∼ Qθ (Xt , ·)
Yt+1 ∼ gθ (·|xt )
where only y0 is observed.
1:n
[Dean, Singh, Jasra, & Peters, 2011]
Use of specific constraints, adapted to the Markov structure:
y1 ∈ B(y1 , ) × · · · × yn ∈ B(yn , )
0 0

85. Noisy ABC for HMMs
Specific case of a hidden Markov model
Xt+1 ∼ Qθ (Xt , ·)
Yt+1 ∼ gθ (·|xt )
where only y0 is observed.
1:n
[Dean, Singh, Jasra, & Peters, 2011]
Use of specific constraints, adapted to the Markov structure:
y1 ∈ B(y1 , ) × · · · × yn ∈ B(yn , )
0 0

86. Noisy ABC-MLE
Idea: Modify instead the data from the start
0
(y1 + ζ1 , . . . , yn + ζn )
[ see Fearnhead-Prangle ]
noisy ABC-MLE estimate
arg max Pθ Y1 ∈ B(y1 + ζ1 , ), . . . , Yn ∈ B(yn + ζn , )
0 0
θ
[Dean, Singh, Jasra, & Peters, 2011]

87. Consistent noisy ABC-MLE
Degrading the data improves the estimation performances:
Noisy ABC-MLE is asymptotically (in n) consistent
under further assumptions, the noisy ABC-MLE is
asymptotically normal
increase in variance of order −2
likely degradation in precision or computing time due to the
lack of summary statistic [curse of dimensionality]

88. Which summary?
Fundamental difficulty of the choice of the summary statistic when
there is no non-trivial sufficient statistics
Loss of statistical information balanced against gain in data
roughening
Approximation error and information loss remain unknown
Choice of statistics induces choice of distance function
towards standardisation

89. Which summary?
Fundamental difficulty of the choice of the summary statistic when
there is no non-trivial sufficient statistics
Loss of statistical information balanced against gain in data
roughening
Approximation error and information loss remain unknown
Choice of statistics induces choice of distance function
towards standardisation

90. Which summary for model choice?
Depending on the choice of η(·), the Bayes factor based on this
insufficient statistic,
η π1 (θ1 )f1η (η(y)|θ1 ) dθ1
B12 (y) = ,
π2 (θ2 )f2η (η(y)|θ2 ) dθ2
is either consistent or inconsistent
[X, Cornuet, Marin, & Pillai, 2012]
Consistency only depends on the range of Ei [η(y)] under both
models
[Marin, Pillai, X, & Rousseau, 2012]

91. Which summary for model choice?
Depending on the choice of η(·), the Bayes factor based on this
insufficient statistic,
η π1 (θ1 )f1η (η(y)|θ1 ) dθ1
B12 (y) = ,
π2 (θ2 )f2η (η(y)|θ2 ) dθ2
is either consistent or inconsistent
[X, Cornuet, Marin, & Pillai, 2012]
Consistency only depends on the range of Ei [η(y)] under both
models
[Marin, Pillai, X, & Rousseau, 2012]

92. Semi-automatic ABC
Fearnhead and Prangle (2012) study ABC and the selection of the
summary statistic in close proximity to Wilkinson’s proposal
ABC considered as inferential method and calibrated as such
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

93. Summary [of F&P/statistics)
optimality of the posterior expectation
E[θ|y]
of the parameter of interest as summary statistics η(y)!
[requires iterative process]
use of the standard quadratic loss function
(θ − θ0 )T A(θ − θ0 )
recent extension to model choice, optimality of Bayes factor
B12 (y)
[F&P, ISBA 2012, Kyoto]

94. Summary [of F&P/statistics)
optimality of the posterior expectation
E[θ|y]
of the parameter of interest as summary statistics η(y)!
[requires iterative process]
use of the standard quadratic loss function
(θ − θ0 )T A(θ − θ0 )
recent extension to model choice, optimality of Bayes factor
B12 (y)
[F&P, ISBA 2012, Kyoto]

95. Summary [about summaries]
Choice of summary statistics is paramount for ABC
validation/performances
At best, ABC approximates π(. | η(y))
Model selection feasible with ABC [with caution!]
For estimation, consistency if {θ; µ(θ) = µ0 } = θ0 when
Eθ [η(y)] = µ(θ)
For testing consistency if
{µ1 (θ1 ), θ1 ∈ Θ1 } ∩ {µ2 (θ2 ), θ2 ∈ Θ2 } = ∅
[Marin et al., 2011]

96. Summary [about summaries]
Choice of summary statistics is paramount for ABC
validation/performances
At best, ABC approximates π(. | η(y))
Model selection feasible with ABC [with caution!]
For estimation, consistency if {θ; µ(θ) = µ0 } = θ0 when
Eθ [η(y)] = µ(θ)
For testing consistency if
{µ1 (θ1 ), θ1 ∈ Θ1 } ∩ {µ2 (θ2 ), θ2 ∈ Θ2 } = ∅
[Marin et al., 2011]

97. Summary [about summaries]
Choice of summary statistics is paramount for ABC
validation/performances
At best, ABC approximates π(. | η(y))
Model selection feasible with ABC [with caution!]
For estimation, consistency if {θ; µ(θ) = µ0 } = θ0 when
Eθ [η(y)] = µ(θ)
For testing consistency if
{µ1 (θ1 ), θ1 ∈ Θ1 } ∩ {µ2 (θ2 ), θ2 ∈ Θ2 } = ∅
[Marin et al., 2011]

98. Summary [about summaries]
Choice of summary statistics is paramount for ABC
validation/performances
At best, ABC approximates π(. | η(y))
Model selection feasible with ABC [with caution!]
For estimation, consistency if {θ; µ(θ) = µ0 } = θ0 when
Eθ [η(y)] = µ(θ)
For testing consistency if
{µ1 (θ1 ), θ1 ∈ Θ1 } ∩ {µ2 (θ2 ), θ2 ∈ Θ2 } = ∅
[Marin et al., 2011]

99. Summary [about summaries]
Choice of summary statistics is paramount for ABC
validation/performances
At best, ABC approximates π(. | η(y))
Model selection feasible with ABC [with caution!]
For estimation, consistency if {θ; µ(θ) = µ0 } = θ0 when
Eθ [η(y)] = µ(θ)
For testing consistency if
{µ1 (θ1 ), θ1 ∈ Θ1 } ∩ {µ2 (θ2 ), θ2 ∈ Θ2 } = ∅
[Marin et al., 2011]

100. Empirical likelihood (EL)
Unavailable likelihoods
ABC methods
ABC as an inference machine
ABCel
ABC and EL
Composite likelihood
Illustrations
Conclusion and perspectives

101. Empirical likelihood (EL)
Dataset x made of n independent replicates x = (x1 , . . . , xn ) of
some X ∼ F
Generalized moment condition model
EF h(X , φ) = 0,
where h is a known function, and φ an unknown parameter
Corresponding empirical likelihood
n
Lel (φ|x) = max pi
p
i=1
for all p such that 0 pi 1, i pi = 1, i pi h(xi , φ) = 0.
[Owen, 1988, Bio’ka, & Empirical Likelihood, 2001]

102. Empirical likelihood (EL)
Dataset x made of n independent replicates x = (x1 , . . . , xn ) of
some X ∼ F
Generalized moment condition model
EF h(X , φ) = 0,
where h is a known function, and φ an unknown parameter
Corresponding empirical likelihood
n
Lel (φ|x) = max pi
p
i=1
for all p such that 0 pi 1, i pi = 1, i pi h(xi , φ) = 0.
[Owen, 1988, Bio’ka, & Empirical Likelihood, 2001]

103. Convergence of EL [3.4]
Theorem 3.4 Let X , Y1 , . . . , Yn be independent rv’s with common
distribution F0 . For θ ∈ Θ, and the function h(X , θ) ∈ Rs , let
θ0 ∈ Θ be such that
Var(h(Yi , θ0 ))
is finite and has rank q > 0. If θ0 satisfies
E(h(X , θ0 )) = 0,
then
Lel (θ0 |Y1 , . . . , Yn )
−2 log → χ2
(q)
n−n
in distribution when n → ∞.
[Owen, 2001]

104. Convergence of EL [3.4]
“...The interesting thing about Theorem 3.4 is what is not there. It
^
includes no conditions to make θ a good estimate of θ0 , nor even
conditions to ensure a unique value for θ0 , nor even that any solution θ0
exists. Theorem 3.4 applies in the just determined, over-determined, and
under-determined cases. When we can prove that our estimating
^
equations uniquely define θ0 , and provide a consistent estimator θ of it,
then confidence regions and tests follow almost automatically through
Theorem 3.4.”.
[Owen, 2001]

105. Raw ABCel sampler
Act as if EL was an exact likelihood
[Lazar, 2003]
for i = 1 → N do
generate φi from the prior distribution π(·)
set the weight ωi = Lel (φi |xobs )
end for
return (φi , ωi ), i = 1, . . . , N
Output weighted sample of size N

106. Raw ABCel sampler
Act as if EL was an exact likelihood
[Lazar, 2003]
for i = 1 → N do
generate φi from the prior distribution π(·)
set the weight ωi = Lel (φi |xobs )
end for
return (φi , ωi ), i = 1, . . . , N
Performance evaluated through effective sample size
 2
N  N 
ESS = 1 ωi ωj
 
i=1 j=1

107. Raw ABCel sampler
Act as if EL was an exact likelihood
[Lazar, 2003]
for i = 1 → N do
generate φi from the prior distribution π(·)
set the weight ωi = Lel (φi |xobs )
end for
return (φi , ωi ), i = 1, . . . , N
More advanced algorithms can be adapted to EL:
E.g., adaptive multiple importance sampling (AMIS) of
Cornuet et al. to speed up computations
[Cornuet et al., 2012]

108. Moment condition in population genetics?
EL does not require a fully defined and often complex (hence
debatable) parametric model
Main difficulty
Derive a constraint
EF h(X , φ) = 0,
on the parameters of interest φ when X is made of the genotypes
of the sample of individuals at a given locus
E.g., in phylogeography, φ is composed of
dates of divergence between populations,
ratio of population sizes,
mutation rates, etc.
None of them are moments of the distribution of the allelic states
of the sample

109. Moment condition in population genetics?
EL does not require a fully defined and often complex (hence
debatable) parametric model
Main difficulty
Derive a constraint
EF h(X , φ) = 0,
on the parameters of interest φ when X is made of the genotypes
of the sample of individuals at a given locus
c h made of pairwise composite scores (whose zero is the pairwise
maximum likelihood estimator)

110. Pairwise composite likelihood
The intra-locus pairwise likelihood
j
2 (xk |φ) 2 (xk , xk |φ)
i
=
i<j
1 n
with xk , . . . , xk : allelic states of the gene sample at the k-th locus
The pairwise score function
j
φ log 2 (xk |φ) φ log 2 (xk , xk |φ)
i
=
i<j
Composite likelihoods are often much narrower than the
original likelihood of the model
Safe with EL because we only use position of its mode

111. Pairwise likelihood: a simple case
1
Assumptions 2 (δ|θ) =√ ρ (θ)|δ|
1 + 2θ
sample ⊂ closed, panmictic with
θ
population at equilibrium ρ(θ) = √
1 + θ + 1 + 2θ
marker: microsatellite
mutation rate: θ/2 Pairwise score function
∂θ log 2 (δ|θ) =
i j 1 |δ|
if xk et xk are two genes of the − + √
sample, 1 + 2θ θ 1 + 2θ
j
2 (xk , xk |θ)
i depends only on
i j
δ = xk − xk

112. Pairwise likelihood: a simple case
1
Assumptions 2 (δ|θ) =√ ρ (θ)|δ|
1 + 2θ
sample ⊂ closed, panmictic with
θ
population at equilibrium ρ(θ) = √
1 + θ + 1 + 2θ
marker: microsatellite
mutation rate: θ/2 Pairwise score function
∂θ log 2 (δ|θ) =
i j 1 |δ|
if xk et xk are two genes of the − + √
sample, 1 + 2θ θ 1 + 2θ
j
2 (xk , xk |θ)
i depends only on
i j
δ = xk − xk

113. Pairwise likelihood: a simple case
1
Assumptions 2 (δ|θ) =√ ρ (θ)|δ|
1 + 2θ
sample ⊂ closed, panmictic with
θ
population at equilibrium ρ(θ) = √
1 + θ + 1 + 2θ
marker: microsatellite
mutation rate: θ/2 Pairwise score function
∂θ log 2 (δ|θ) =
i j 1 |δ|
if xk et xk are two genes of the − + √
sample, 1 + 2θ θ 1 + 2θ
j
2 (xk , xk |θ)
i depends only on
i j
δ = xk − xk

114. Pairwise likelihood: 2 diverging populations
MRCA Then 2 (δ|θ, τ) =
+∞
e−τθ
√ ρ(θ)|k| Iδ−k (τθ).
τ 1 + 2θ k=−∞
where
In (z) nth-order modified
POP a POP b Bessel function of the first
Assumptions kind
τ: divergence date of
pop. a and b
θ/2: mutation rate
i j
Let xk and xk be two genes
coming resp. from pop. a and
b
i j
Set δ = xk − xk .

115. Pairwise likelihood: 2 diverging populations
MRCA
A 2-dim score function
∂τ log 2 (δ|θ, τ) = −θ+
τ θ 2 (δ − 1|θ, τ) + 2 (δ + 1|θ, τ)
2 2 (δ|θ, τ)
POP a POP b ∂θ log 2 (δ|θ, τ) =
1 q(δ|θ, τ)
Assumptions −τ − + +
1 + 2θ 2 (δ|θ, τ)
τ 2 (δ − 1|θ, τ) + 2 (δ + 1|θ, τ)
τ: divergence date of
2 2 (δ|θ, τ)
pop. a and b
θ/2: mutation rate where
i j q(δ|θ, τ) :=
Let xk andxk be two genes ∞
e−τθ ρ (θ)
coming resp. from pop. a and √ |k|ρ(θ)|k| Iδ−k (τθ)
1 + 2θ ρ(θ)
b k=−∞
i j
Set δ = xk − xk .

116. Example: normal posterior
ABCel with two constraints
ESS=155.6 ESS=75.93 ESS=76.87
4
2.0
3
Density
Density
Density
1.0
2
1.0
1
0.0
0.0
0
−0.5 0.0 0.5 1.0 −0.4 −0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2
θ
ESS=91.54 θ
ESS=108.4 θ
ESS=85.13
4
0.0 1.0 2.0 3.0
0.0 1.0 2.0 3.0
3
Density
Density
Density
2
1
0
−0.6 −0.4 −0.2 0.0 0.2 −0.4 0.0 0.2 0.4 0.6 −0.2 0.0 0.2 0.4 0.6
θ
ESS=149.1 θ
ESS=96.31 θ
ESS=83.77
4
2.0
3
2.0
Density
Density
Density
2
1.0
1.0
1
0.0
0.0
−0.5 0.0 0.5 1.0 −0.4 0.0 0.2 0.4 0.6 0 −0.6 −0.4 −0.2 0.0 0.2 0.4
θ
ESS=155.7 θ
ESS=92.42 θ
ESS=95.01
0.0 1.0 2.0 3.0
2.0
3.0
Density
Density
Density
1.0
1.5
0.0
0.0
−0.5 0.0 0.5 −0.4 0.0 0.2 0.4 0.6 −0.4 0.0 0.2 0.4 0.6
θ
ESS=139.2 θ
ESS=99.33 θ
ESS=87.28
2.0
0.0 1.0 2.0 3.0
3
Density
Density
Density
2
1.0
1
0.0
0
−0.6 −0.2 0.2 0.6 −0.4 −0.2 0.0 0.2 0.4 −0.2 0.0 0.2 0.4 0.6
Sample sizes are of 25 (column 1), 50 (column 2) and 75 (column 3)
observations

117. Example: normal posterior
ABCel with three constraints
ESS=300.1 ESS=205.5 ESS=179.4
3.0
2.0
Density
Density
Density
1.0
1.5
1.0
0.0
0.0
0.0
−0.4 0.0 0.4 0.8 −0.6 −0.2 0.0 0.2 0.4 −0.2 0.0 0.2 0.4
θ
ESS=265.1 θ
ESS=250.3 θ
ESS=134.8
4
4
2.0
Density
Density
Density
3
3
2
1.0
2
1
1
0.0
0
0
−0.3 −0.2 −0.1 0.0 0.1 −0.6 −0.4 −0.2 0.0 0.2 −0.4 −0.2 0.0 0.1
θ
ESS=331.5 θ
ESS=167.4 θ
ESS=136.5
2.0
4
3
3
Density
Density
Density
1.0
2
2
1
1
0.0
0
−0.8 −0.4 0.0 0.4 −0.9 −0.7 −0.5 −0.3 0 −0.4 −0.2 0.0 0.2
θ
ESS=322.4 θ
ESS=202.7 θ
ESS=166
0.0 1.0 2.0 3.0
4
2.0
3
Density
Density
Density
2
1.0
1
0.0
0
−0.2 0.0 0.2 0.4 0.6 0.8 −0.4 −0.2 0.0 0.2 0.4 −0.4 −0.2 0.0 0.2
θ
ESS=263.7 θ
ESS=190.9 θ
ESS=165.3
3.0
3
2.0
Density
Density
Density
2
1.5
1.0
1
0.0
0.0
0
−1.0 −0.6 −0.2 −0.4 −0.2 0.0 0.2 0.4 0.6 −0.5 −0.3 −0.1 0.1
Sample sizes are of 25 (column 1), 50 (column 2) and 75 (column 3)
observations

118. Example: Superposition of gamma processes
Example of superposition of N renewal processes with waiting
times τij (i = 1, . . . , M, j = 1, . . .) ∼ G(α, β), when N is unknown.
Renewal processes
ζi1 = τi1 , ζi2 = ζi1 + τi2 , . . .
with observations made of first n values of the ζij ’s,
z1 = min{ζij }, z2 = min{ζij ; ζij > z1 }, . . .
ending with
zn = min{ζij ; ζij > zn−1 } .
[Cox & Kartsonaki, B’ka, 2012]