This document outlines the agenda for the second part of a lecture on Approximate Bayesian Computation (ABC). It begins with a discussion of simulation-based methods in econometrics like simulated method of moments. Next, it discusses the genetic origins and applications of ABC in population genetics, including coalescent theory. The document then covers using indirect inference to provide summary statistics for ABC and estimating demographic parameters from genetic data when the likelihood is intractable.
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Monash University short course, part II
1. MCMC and likelihood-free methods Part/day II: ABC
MCMC and likelihood-free methods
Part/day II: ABC
Christian P. Robert
Universit´ Paris-Dauphine, IuF, & CREST
e
Monash University, EBS, July 18, 2012
2. MCMC and likelihood-free methods Part/day II: ABC
Outline
simulation-based methods in Econometrics
Genetics of ABC
Approximate Bayesian computation
ABC for model choice
ABC model choice consistency
3. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
Econ’ections
simulation-based methods in
Econometrics
Genetics of ABC
Approximate Bayesian computation
ABC for model choice
ABC model choice consistency
4. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
Usages of simulation in Econometrics
Similar exploration of simulation-based techniques in Econometrics
Simulated method of moments
Method of simulated moments
Simulated pseudo-maximum-likelihood
Indirect inference
[Gouri´roux & Monfort, 1996]
e
5. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
Simulated method of moments
o
Given observations y1:n from a model
yt = r (y1:(t−1) , t , θ) , t ∼ g (·)
simulate 1:n , derive
yt (θ) = r (y1:(t−1) , t , θ)
and estimate θ by
n
arg min (yto − yt (θ))2
θ
t=1
6. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
Simulated method of moments
o
Given observations y1:n from a model
yt = r (y1:(t−1) , t , θ) , t ∼ g (·)
simulate 1:n , derive
yt (θ) = r (y1:(t−1) , t , θ)
and estimate θ by
n n 2
arg min yto − yt (θ)
θ
t=1 t=1
7. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
Method of simulated moments
Given a statistic vector K (y ) with
Eθ [K (Yt )|y1:(t−1) ] = k(y1:(t−1) ; θ)
find an unbiased estimator of k(y1:(t−1) ; θ),
˜
k( t , y1:(t−1) ; θ)
Estimate θ by
n S
arg min K (yt ) − ˜ t
k( s , y1:(t−1) ; θ)/S
θ
t=1 s=1
[Pakes & Pollard, 1989]
8. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
Indirect inference
ˆ
Minimise (in θ) the distance between estimators β based on
pseudo-models 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]
9. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
Indirect inference (PML vs. PSE)
Example of the pseudo-maximum-likelihood (PML)
ˆ
β(y) = arg max log f (yt |β, y1:(t−1) )
β
t
leading to
ˆ ˆ
arg min ||β(yo ) − β(y1 (θ), . . . , yS (θ))||2
θ
when
ys (θ) ∼ f (y|θ) s = 1, . . . , S
10. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
Indirect inference (PML vs. PSE)
Example of the pseudo-score-estimator (PSE)
2
ˆ ∂ log f
β(y) = arg min (yt |β, y1:(t−1) )
β
t
∂β
leading to
ˆ ˆ
arg min ||β(yo ) − β(y1 (θ), . . . , yS (θ))||2
θ
when
ys (θ) ∼ f (y|θ) s = 1, . . . , S
11. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
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...
12. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
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
13. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
AR(2) vs. MA(1) example
true (AR) model
yt = t −θ t−1
and [wrong!] auxiliary (MA) model
yt = β1 yt−1 + β2 yt−2 + ut
R code
x=eps=rnorm(250)
x[2:250]=x[2:250]-0.5*x[1:249]
simeps=rnorm(250)
propeta=seq(-.99,.99,le=199)
dist=rep(0,199)
bethat=as.vector(arima(x,c(2,0,0),incl=FALSE)$coef)
for (t in 1:199)
dist[t]=sum((as.vector(arima(c(simeps[1],simeps[2:250]-propeta[t]*
simeps[1:249]),c(2,0,0),incl=FALSE)$coef)-bethat)^2)
14. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
AR(2) vs. MA(1) example
One sample:
0.8
0.6
distance
0.4
0.2
0.0
−1.0 −0.5 0.0 0.5 1.0
15. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
AR(2) vs. MA(1) example
Many samples:
6
5
4
3
2
1
0
0.2 0.4 0.6 0.8 1.0
16. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
Choice of pseudo-model
Pick model such that
ˆ
1. β(θ) not flat
(i.e. sensitive to changes in θ)
ˆ
2. β(θ) not dispersed (i.e. robust agains changes in ys (θ))
[Frigessi & Heggland, 2004]
17. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
ABC using indirect inference (1)
We present a novel approach for developing summary statistics
for use in approximate Bayesian computation (ABC) algorithms by
using indirect inference(...) In the indirect inference approach to
ABC the parameters of an auxiliary model fitted to the data become
the summary statistics. Although applicable to any ABC technique,
we embed this approach within a sequential Monte Carlo algorithm
that is completely adaptive and requires very little tuning(...)
[Drovandi, Pettitt & Faddy, 2011]
c Indirect inference provides summary statistics for ABC...
18. MCMC and likelihood-free methods Part/day II: ABC
simulation-based methods in Econometrics
ABC using indirect inference (2)
...the above result shows that, in the limit as h → 0, ABC will
be more accurate than an indirect inference method whose auxiliary
statistics are the same as the summary statistic that is used for
ABC(...) Initial analysis showed that which method is more
accurate depends on the true value of θ.
[Fearnhead and Prangle, 2012]
c Indirect inference provides estimates rather than global inference...
19. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
Genetics of ABC
simulation-based methods in
Econometrics
Genetics of ABC
Approximate Bayesian computation
ABC for model choice
ABC model choice consistency
20. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
Genetic background of ABC
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
21. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
Population genetics
[Part derived from the teaching material of Raphael Leblois, ENS Lyon, November 2010]
Describe the genotypes, estimate the alleles frequencies,
determine their distribution among individuals, populations
and between populations;
Predict and understand the evolution of gene frequencies in
populations as a result of various factors.
c Analyses the effect of various evolutive forces (mutation, drift,
migration, selection) on the evolution of gene frequencies in time
and space.
22. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
A[B]ctors
Les mécanismes de l’évolution
Towards an explanation of the mechanisms of evolutionary
changes, mathematical models ofl’évolution ont to be developed
•! Les mathématiques de evolution started
in the years 1920-1930.
commencé à être développés dans les
années 1920-1930
Ronald A Fisher (1890-1962) Sewall Wright (1889-1988) John BS Haldane (1892-1964)
23. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
Le modèle
Wright-Fisher model de Wright-Fisher
•! En l’absence de mutation et de
A population of constant
sélection, les fréquences
size, in which individuals
alléliques dérivent (augmentent
reproduce at the same time.
et diminuent) inévitablement
jusqu’à Each gene in a generation is
la fixation d’un allèle
a copy of a gene of the
•! La dérive conduit generation.
previous donc à la
perte de variation génétique à
l’intérieur des populations mutation
In the absence of
and selection, allele
frequencies derive inevitably
until the fixation of an
allele.
24. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
Coalescent theory
[Kingman, 1982; Tajima, Tavar´, &tc]
e
!"#$%&'(('")**+$,-'".'"/010234%'".'5"*$*%()23$15"6"
Coalescence theory interested in the genealogy of a sample of
genes7**+$,-'",()5534%'" common ancestor of the sample.
"
!! back in time to the " "!"7**+$,-'"8",$)('5,'1,'"9"
25. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
Common ancestor
Les différentes
lignées fusionnent
(coalescent) au fur
et à mesure que
l’on remonte vers le
Time of coalescence
passé
(T)
Modélisation du processus de dérive génétique
The different lineages mergeen “remontant dans lein the past.
when we go back temps”
jusqu’à l’ancêtre commun d’un échantillon de gènes 6
26. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
Sous l’hypothèse de neutralité des marqueurs génétiques étudiés,
les mutations sont indépendantes de la généalogie
Neutral généalogie ne dépend que des processus démographiques
i.e. la mutations
On construit donc la généalogie selon les paramètres
démographiques (ex. N),
puis on ajoute aassumption les
Under the posteriori of
mutations sur les différentes
neutrality, the mutations
branches,independentau feuilles de
are du MRCA of the
l’arbregenealogy.
On obtient ainsi des données de
We construct the genealogy
polymorphisme sous les modèles
according to the
démographiques et mutationnels
demographic parameters,
considérés
then we add a posteriori the
mutations. 20
27. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
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 can not calculate the likelihood of
the polymorphism data f (y|θ).
28. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
Untractable likelihood
Missing (too missing!) data structure:
f (y|θ) = f (y|G , θ)f (G |θ)dG
G
The genealogies are considered as nuisance parameters.
This problematic thus differs from the phylogenetic approach
where the tree is the parameter of interesst.
29. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
A genuine !""#$%&'()*+,(-*.&(/+0$'"1)()&$/+2!,03!
example of application
1/+*%*'"4*+56(""4&7()&$/.+.1#+4*.+8-9':*.+
94
Pygmies populations: do they have a common origin? Is there a
lot of exchanges between pygmies and non-pygmies populations?
30. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
!""#$%&'()*+,(-*.&(/+0$'"1)()&$/+2!,03!
Scenarios 1/+*%*'"4*+56(""4&7()&$/.+.1#+4*.+8-9':*.+
under competition
Différents scénarios possibles, choix de scenario par ABC
Verdu et al. 2009
96
31. 1/+*%*'"4*+56(""4&7()&$/.+.1#+4*.+
MCMC and likelihood-free methods Part/day II: ABC
!""#$%&'()*+,(-*.&(/+0$'"1)()&$/+2!,03
Genetics of ABC
1/+*%*'"4*+56(""4&7()&$/.+.1#+4*.+8-9':*.
Différentsresults
Simulation scénarios possibles, choix de scenari
Différents scénarios possibles, choix de scenario par ABC
c Scenario 1A is chosen. largement soutenu par rapport aux
Le scenario 1a est
autres ! plaide pour une origine commune des
Le scenario 1a est largement soutenu par
populations pygmées d’Afrique de l’Ouest rap
Verdu e
32. MCMC and likelihood-free methods Part/day II: ABC
Genetics of ABC
!""#$%&'()*+,(-*.&(/+0$'"1)()&$/+2!,03!
Most likely scenario
1/+*%*'"4*+56(""4&7()&$/.+.1#+4*.+8-9':*.+
Scénario
évolutif :
on « raconte »
une histoire à
partir de ces
inférences
Verdu et al. 2009
99
33. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Approximate Bayesian computation
simulation-based methods in
Econometrics
Genetics of ABC
Approximate Bayesian computation
ABC for model choice
ABC model choice consistency
34. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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!
35. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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!
36. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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
37. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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
38. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Illustrations
Example (Genesis)
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]
39. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
The ABC method
Bayesian setting: target is π(θ)f (x|θ)
40. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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 rejection
technique:
41. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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 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
42. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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]
43. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Earlier occurrence
‘Bayesian statistics and Monte Carlo methods are ideally
suited to the task of passing many models over one
dataset’
[Don Rubin, Annals of Statistics, 1984]
Note Rubin (1984) does not promote this algorithm for
likelihood-free simulation but frequentist intuition on posterior
distributions: parameters from posteriors are more likely to be
those that could have generated the data.
44. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
A as A...pproximative
When y is a continuous random variable, equality z = y is replaced
with a tolerance condition,
(y, z) ≤
where is a distance
45. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
A as A...pproximative
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) < )
[Pritchard et al., 1999]
46. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
ABC algorithm
Algorithm 1 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
47. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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)) < }.
48. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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) .
49. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Convergence of ABC (first attempt)
What happens when → 0?
50. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Convergence of ABC (first attempt)
What happens when → 0?
If f (·|θ) is continuous in y , uniformly in θ [!], given an arbitrary
δ > 0, there exists 0 such that < 0 implies
π(θ) f (z|θ)IA ,y (z) dz π(θ)f (y|θ)(1 δ)µ(B )
∈
A ,y ×Θ π(θ)f (z|θ)dzdθ Θ π(θ)f (y|θ)dθ(1 ± δ)µ(B )
51. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Convergence of ABC (first attempt)
What happens when → 0?
If f (·|θ) is continuous in y , uniformly in θ [!], given an arbitrary
δ > 0, there exists 0 such that < 0 implies
π(θ) f (z|θ)IA ,y (z) dz π(θ)f (y|θ)(1 δ)
XX )
µ(BX
∈
A ,y ×Θ π(θ)f (z|θ)dzdθ Θ π(θ)f (y|θ)dθ(1 ± δ)
µ(B )
XXX
52. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Convergence of ABC (first attempt)
What happens when → 0?
If f (·|θ) is continuous in y , uniformly in θ [!], given an arbitrary
δ 0, there exists 0 such that 0 implies
π(θ) f (z|θ)IA ,y (z) dz π(θ)f (y|θ)(1 δ)
XX )
µ(BX
∈
A ,y ×Θ π(θ)f (z|θ)dzdθ Θ π(θ)f (y|θ)dθ(1 ± δ) X
µ(B )
X
X
[Proof extends to other continuous-in-0 kernels K ]
53. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Convergence of ABC (second attempt)
What happens when → 0?
54. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Convergence of ABC (second attempt)
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).
55. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Probit modelling on Pima Indian women
Example (R benchmark)
200 Pima Indian women with observed variables
plasma glucose concentration in oral glucose tolerance test
diastolic blood pressure
diabetes pedigree function
presence/absence of diabetes
56. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Probit modelling on Pima Indian women
Example (R benchmark)
200 Pima Indian women with observed variables
plasma glucose concentration in oral glucose tolerance test
diastolic blood pressure
diabetes pedigree function
presence/absence of diabetes
Probability of diabetes function of above variables
P(y = 1|x) = Φ(x1 β1 + x2 β2 + x3 β3 ) ,
57. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Probit modelling on Pima Indian women
Example (R benchmark)
200 Pima Indian women with observed variables
plasma glucose concentration in oral glucose tolerance test
diastolic blood pressure
diabetes pedigree function
presence/absence of diabetes
Probability of diabetes function of above variables
P(y = 1|x) = Φ(x1 β1 + x2 β2 + x3 β3 ) ,
Test of H0 : β3 = 0 for 200 observations of Pima.tr based on a
g -prior modelling:
β ∼ N3 (0, n XT X)−1
58. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Probit modelling on Pima Indian women
Example (R benchmark)
200 Pima Indian women with observed variables
plasma glucose concentration in oral glucose tolerance test
diastolic blood pressure
diabetes pedigree function
presence/absence of diabetes
Probability of diabetes function of above variables
P(y = 1|x) = Φ(x1 β1 + x2 β2 + x3 β3 ) ,
Test of H0 : β3 = 0 for 200 observations of Pima.tr based on a
g -prior modelling:
β ∼ N3 (0, n XT X)−1
Use of importance function inspired from the MLE estimate
distribution
ˆ ˆ
β ∼ N (β, Σ)
59. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Pima Indian benchmark
80
100
1.0
80
60
0.8
60
0.6
Density
Density
Density
40
40
0.4
20
20
0.2
0.0
0
0
−0.005 0.010 0.020 0.030 −0.05 −0.03 −0.01 −1.0 0.0 1.0 2.0
Figure: Comparison between density estimates of the marginals on β1
(left), β2 (center) and β3 (right) from ABC rejection samples (red) and
MCMC samples (black)
.
60. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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 u i
i=1
under the identifiability conditions
61. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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)
62. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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 )−qt≤T
3. producing a simulated series (xt )1≤t≤T
63. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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 )−qt≤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
64. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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
65. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Comparison of distance impact
4
1.5
3
1.0
2
0.5
1
0.0
0
0.0 0.2 0.4 0.6 0.8 −2.0 −1.0 0.0 0.5 1.0 1.5
θ1 θ2
Evaluation of the tolerance on the ABC sample against both
distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
66. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Comparison of distance impact
4
1.5
3
1.0
2
0.5
1
0.0
0
0.0 0.2 0.4 0.6 0.8 −2.0 −1.0 0.0 0.5 1.0 1.5
θ1 θ2
Evaluation of the tolerance on the ABC sample against both
distances ( = 100%, 10%, 1%, 0.1%) for an MA(2) model
67. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
Homonomy
The ABC algorithm is not to be confused with the ABC algorithm
The Artificial Bee Colony algorithm is a swarm based meta-heuristic
algorithm that was introduced by Karaboga in 2005 for optimizing
numerical problems. It was inspired by the intelligent foraging
behavior of honey bees. The algorithm is specifically based on the
model proposed by Tereshko and Loengarov (2005) for the foraging
behaviour of honey bee colonies. The model consists of three
essential components: employed and unemployed foraging bees, and
food sources. The first two components, employed and unemployed
foraging bees, search for rich food sources (...) close to their hive.
The model also defines two leading modes of behaviour (...):
recruitment of foragers to rich food sources resulting in positive
feedback and abandonment of poor sources by foragers causing
negative feedback.
[Karaboga, Scholarpedia]
68. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
ABC advances
Simulating from the prior is often poor in efficiency
69. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
ABC advances
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]
70. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
ABC advances
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]
71. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
ABC basics
ABC advances
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]
72. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
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
(use with BIC)
θ∗ = θ − {η(z) − η(y)}T β
ˆ [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]
73. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
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)]
74. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
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...
75. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-NP (density estimation)
Use of the kernel weights
Kδ {ρ(η(z(θ)), η(y))}
leads to the NP estimate of the posterior expectation
i θi Kδ {ρ(η(z(θi )), η(y))}
i Kδ {ρ(η(z(θi )), η(y))}
[Blum, JASA, 2010]
76. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-NP (density estimation)
Use of the kernel weights
Kδ {ρ(η(z(θ)), η(y))}
leads to the NP estimate of the posterior conditional density
˜
Kb (θi − θ)Kδ {ρ(η(z(θi )), η(y))}
i
i Kδ {ρ(η(z(θi )), η(y))}
[Blum, JASA, 2010]
77. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-NP (density estimations)
Other versions incorporating regression adjustments
˜
Kb (θi∗ − θ)Kδ {ρ(η(z(θi )), η(y))}
i
i Kδ {ρ(η(z(θi )), η(y))}
78. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-NP (density estimations)
Other versions incorporating regression adjustments
˜
Kb (θi∗ − θ)Kδ {ρ(η(z(θi )), η(y))}
i
i Kδ {ρ(η(z(θi )), η(y))}
In all cases, error
E[ˆ (θ|y)] − g (θ|y) = cb 2 + cδ 2 + OP (b 2 + δ 2 ) + OP (1/nδ d )
g
c
var(ˆ (θ|y)) =
g (1 + oP (1))
nbδ d
[Blum, JASA, 2010]
79. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-NP (density estimations)
Other versions incorporating regression adjustments
˜
Kb (θi∗ − θ)Kδ {ρ(η(z(θi )), η(y))}
i
i Kδ {ρ(η(z(θi )), η(y))}
In all cases, error
E[ˆ (θ|y)] − g (θ|y) = cb 2 + cδ 2 + OP (b 2 + δ 2 ) + OP (1/nδ d )
g
c
var(ˆ (θ|y)) =
g (1 + oP (1))
nbδ d
[standard NP calculations]
83. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-NCH (2)
Why neural network?
fights curse of dimensionality
selects relevant summary statistics
provides automated dimension reduction
offers a model choice capability
improves upon multinomial logistic
[Blum Fran¸ois, 2009]
c
84. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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) ) ,
ω (θ
(t)
θ otherwise,
85. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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) ) ,
ω (θ
(t)
θ otherwise,
has the posterior π(θ|y ) as stationary distribution
[Marjoram et al, 2003]
86. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
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] ,
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
87. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Why does it work?
Acceptance probability does not involve calculating the likelihood
and
π (θ , z |y) q(θ (t−1) |θ )f (z(t−1) |θ (t−1) )
×
π (θ (t−1) , z(t−1) |y) q(θ |θ (t−1) )f (z |θ )
π(θ ) XX|θ IA ,y (z )
)
f (z X X
=
π(θ (t−1) ) f (z(t−1) |θ (t−1) ) IA ,y (z(t−1) )
q(θ (t−1) |θ ) f (z(t−1) |θ (t−1) )
×
q(θ |θ (t−1) ) X|θ
X )
f (z XX
88. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Why does it work?
Acceptance probability does not involve calculating the likelihood
and
π (θ , z |y) q(θ (t−1) |θ )f (z(t−1) |θ (t−1) )
×
π (θ (t−1) , z(t−1) |y) q(θ |θ (t−1) )f (z |θ )
π(θ ) XX|θ IA ,y (z )
)
f (z X X
= hh (
π(θ (t−1) ) ((hhhhh) IA ,y (z(t−1) )
f (z(t−1) |θ (t−1)
((( h ((
hh (
q(θ (t−1) |θ ) ((hhhhh)
f (z(t−1) |θ (t−1)
((
((( h
×
q(θ |θ (t−1) ) X|θ
X )
f (z XX
89. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Why does it work?
Acceptance probability does not involve calculating the likelihood
and
π (θ , z |y) q(θ (t−1) |θ )f (z(t−1) |θ (t−1) )
×
π (θ (t−1) , z(t−1) |y) q(θ |θ (t−1) )f (z |θ )
π(θ ) X|θ IA ,y (z )
X )
f (z X X
= hh (
π(θ (t−1) ) ((hhhhh) XX(t−1) )
f (z(t−1) |θ (t−1) IAX
(( X
((( h ,y (zX
X
hh (
q(θ (t−1) |θ ) ((hhhhh)
f (z(t−1) |θ (t−1)
((
((( h
×
q(θ |θ (t−1) ) X|θ
X )
f (z XX
π(θ )q(θ (t−1) |θ )
= IA ,y (z )
π(θ (t−1) q(θ |θ (t−1) )
90. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A toy example
Case of
1 1
x ∼ N (θ, 1) + N (−θ, 1)
2 2
under prior θ ∼ N (0, 10)
91. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A toy example
Case of
1 1
x ∼ N (θ, 1) + N (−θ, 1)
2 2
under prior θ ∼ N (0, 10)
ABC sampler
thetas=rnorm(N,sd=10)
zed=sample(c(1,-1),N,rep=TRUE)*thetas+rnorm(N,sd=1)
eps=quantile(abs(zed-x),.01)
abc=thetas[abs(zed-x)eps]
92. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A toy example
Case of
1 1
x ∼ N (θ, 1) + N (−θ, 1)
2 2
under prior θ ∼ N (0, 10)
ABC-MCMC sampler
metas=rep(0,N)
metas[1]=rnorm(1,sd=10)
zed[1]=x
for (t in 2:N){
metas[t]=rnorm(1,mean=metas[t-1],sd=5)
zed[t]=rnorm(1,mean=(1-2*(runif(1).5))*metas[t],sd=1)
if ((abs(zed[t]-x)eps)||(runif(1)dnorm(metas[t],sd=10)/dnorm(metas[t-1],sd=10))){
metas[t]=metas[t-1]
zed[t]=zed[t-1]}
}
93. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A toy example
x =2
94. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A toy example
x =2
0.30
0.25
0.20
0.15
0.10
0.05
0.00
−4 −2 0 2 4
θ
95. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A toy example
x = 10
96. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A toy example
x = 10
0.25
0.20
0.15
0.10
0.05
0.00
−10 −5 0 5 10
θ
97. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A toy example
x = 50
98. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A toy example
x = 50
0.10
0.08
0.06
0.04
0.02
0.00
−40 −20 0 20 40
θ
99. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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 y when z ∼ f (z|θ)
100. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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 y when z ∼ f (z|θ)
Warning! Replacement of ξ( |y, θ) with a non-parametric kernel
approximation.
101. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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)
102. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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)
103. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABCµ multiple errors
[ c Ratmann et al., PNAS, 2009]
104. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABCµ for model choice
[ c Ratmann et al., PNAS, 2009]
105. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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).
106. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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]
107. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-PRC
Another sequential version producing a sequence of Markov
(t) (t)
transition kernels Kt and of samples (θ1 , . . . , θN ) (1 ≤ t ≤ T )
108. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-PRC
Another sequential version producing a sequence of Markov
(t) (t)
transition kernels Kt and of samples (θ1 , . . . , θN ) (1 ≤ t ≤ T )
ABC-PRC Algorithm
(t−1)
1. Select θ at random from previous θi ’s with probabilities
(t−1)
ωi (1 ≤ i ≤ N).
2. Generate
(t) (t)
θi ∼ Kt (θ|θ ) , x ∼ f (x|θi ) ,
3. Check that (x, y ) , otherwise start again.
[Sisson et al., 2007]
109. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Why PRC?
Partial rejection control: Resample from a population of weighted
particles by pruning away particles with weights below threshold C ,
replacing them by new particles obtained by propagating an
existing particle by an SMC step and modifying the weights
accordinly.
[Liu, 2001]
110. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Why PRC?
Partial rejection control: Resample from a population of weighted
particles by pruning away particles with weights below threshold C ,
replacing them by new particles obtained by propagating an
existing particle by an SMC step and modifying the weights
accordinly.
[Liu, 2001]
PRC justification in ABC-PRC:
Suppose we then implement the PRC algorithm for some
c 0 such that only identically zero weights are smaller
than c
Trouble is, there is no such c...
111. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-PRC weight
(t)
Probability ωi computed as
(t) (t) (t) (t)
ωi ∝ π(θi )Lt−1 (θ |θi ){π(θ )Kt (θi |θ )}−1 ,
where Lt−1 is an arbitrary transition kernel.
112. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-PRC weight
(t)
Probability ωi computed as
(t) (t) (t) (t)
ωi ∝ π(θi )Lt−1 (θ |θi ){π(θ )Kt (θi |θ )}−1 ,
where Lt−1 is an arbitrary transition kernel.
In case
Lt−1 (θ |θ) = Kt (θ|θ ) ,
all weights are equal under a uniform prior.
113. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-PRC weight
(t)
Probability ωi computed as
(t) (t) (t) (t)
ωi ∝ π(θi )Lt−1 (θ |θi ){π(θ )Kt (θi |θ )}−1 ,
where Lt−1 is an arbitrary transition kernel.
In case
Lt−1 (θ |θ) = Kt (θ|θ ) ,
all weights are equal under a uniform prior.
Inspired from Del Moral et al. (2006), who use backward kernels
Lt−1 in SMC to achieve unbiasedness
114. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-PRC bias
Lack of unbiasedness of the method
116. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A mixture example (1)
Toy model of Sisson et al. (2007): if
θ ∼ U(−10, 10) , x|θ ∼ 0.5 N (θ, 1) + 0.5 N (θ, 1/100) ,
then the posterior distribution associated with y = 0 is the normal
mixture
θ|y = 0 ∼ 0.5 N (0, 1) + 0.5 N (0, 1/100)
restricted to [−10, 10].
Furthermore, true target available as
π(θ||x| ) ∝ Φ( −θ)−Φ(− −θ)+Φ(10( −θ))−Φ(−10( +θ)) .
118. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A PMC version
Use of the same kernel idea as ABC-PRC but with IS correction
( check PMC )
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
[Beaumont et al., 2009]
119. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Our 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
120. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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 as PMC: 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]
121. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Sequential Monte Carlo (2)
Algorithm 3 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
122. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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
m=1 IA n
(xin )
win ∝ wi(n−1) M m
(xi(n−1) )
m=1 IA n−1
m
when xin ∼ K (xi(n−1) , ·)
123. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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]
124. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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
Major assumption: the forward kernel K is supposed to be invariant
against the true target [tempered version of the true posterior]
125. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
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
Major assumption: the forward kernel K is supposed to be invariant
against the true target [tempered version of the true posterior]
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
126. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
A mixture example (2)
Recovery of the target, whether using a fixed standard deviation of
τ = 0.15 or τ = 1/0.15, or a sequence of adaptive τt ’s.
1.0
1.0
1.0
1.0
1.0
0.8
0.8
0.8
0.8
0.8
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0.2
0.0
0.0
0.0
0.0
0.0
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
θ θ θ θ θ
127. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Yet another ABC-PRC
Another version of ABC-PRC called PRC-ABC with a proposal
distribution Mt , S replications of the pseudo-data, and a PRC step:
PRC-ABC Algorithm
(t−1) (t−1)
1. Select θ at random from the θi ’s with probabilities ωi
(t) iid (t−1)
2. Generate θi ∼ Kt , x1 , . . . , xS ∼ f (x|θi ), set
(t) (t)
ωi = Kt {η(xs ) − η(y)} Kt (θi ) ,
s
(t)
and accept with probability p (i) = ωi /ct,N
(t) (t)
3. Set ωi ∝ ωi /p (i) (1 ≤ i ≤ N)
[Peters, Sisson Fan, Stat Computing, 2012]
128. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Yet another ABC-PRC
Another version of ABC-PRC called PRC-ABC with a proposal
distribution Mt , S replications of the pseudo-data, and a PRC step:
(t−1) (t−1)
Use of a NP estimate Mt (θ) = ωi ψ(θ|θi ) (with a
fixed variance?!)
S = 1 recommended for “greatest gains in sampler
performance”
ct,N upper quantile of non-zero weights at time t
129. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Wilkinson’s exact BC
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]
130. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
Wilkinson’s exact BC
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).
131. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
How exact a BC?
“Using to represent measurement error is
straightforward, whereas using to model the model
discrepancy is harder to conceptualize and not as
commonly used”
[Richard Wilkinson, 2008]
132. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
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
Requires a modification of the standard ABC algorithm
133. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC for HMMs
Specific case of a hidden Markov model
Xt+1 ∼ Qθ (Xt , ·)
Yt+1 ∼ gθ (·|xt )
0
where only y1:n is observed.
[Dean, Singh, Jasra, Peters, 2011]
134. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC for HMMs
Specific case of a hidden Markov model
Xt+1 ∼ Qθ (Xt , ·)
Yt+1 ∼ gθ (·|xt )
0
where only y1:n is observed.
[Dean, Singh, Jasra, Peters, 2011]
Use of specific constraints, adapted to the Markov structure:
0 0
y1 ∈ B(y1 , ) × · · · × yn ∈ B(yn , )
135. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-MLE for HMMs
ABC-MLE defined by
ˆ 0 0
θn = arg max Pθ Y1 ∈ B(y1 , ), . . . , Yn ∈ B(yn , )
θ
136. MCMC and likelihood-free methods Part/day II: ABC
Approximate Bayesian computation
Alphabet soup
ABC-MLE for HMMs
ABC-MLE defined by
ˆ 0 0
θn = arg max Pθ Y1 ∈ B(y1 , ), . . . , Yn ∈ B(yn , )
θ
Exact MLE for the likelihood same basis as Wilkinson!
0
pθ (y1 , . . . , yn )
corresponding to the perturbed process
(xt , yt + zt )1≤t≤n zt ∼ U(B(0, 1)
[Dean, Singh, Jasra, Peters, 2011]