ABC-Gibbs

Component-wise approximate Bayesian
computation via Gibbs-like steps
Christian P. Robert(1,2)
arXiv:1905.13599 with Grégoire Clarté(1)
, Robin Ryder(1)
, Julien Stoehr(1)
(1) Université Paris-Dauphine, (2) University of Warwick
Université Paris-Dauphine
February 28, 2020

Approximate Bayesian computation (ABC)
ABC is a computational method which stemmed from population ge-
netics models about 20 years ago to deal with generative intractable
distribution.
[Tavaré et al., 1997; Beaumont et al., 2002]
Settings of interest: the likelihood function f(x | θ) does not admit a
closed form as a function of θ and/or is computationally too costly.
1. Model relying on a latent process z ∈ Z
f(x | θ) =
Z
f(y, z | θ)µ(dz).
2. Model with intractable normalising constant
f(x | θ) =
1
Z(θ)
q(x | θ), where Z(θ) =
X
q(x | θ)µ(dx).

Bayesian settings: the target is π(θ | xobs
) ∝ π(θ)f(xobs
| θ).
Algorithm: Vanilla ABC
Input: observed dataset xobs
,
number of iterations N,
threshold ε, summary
statistic s.
for i = 1, . . . , N do
θi ∼ π(·)
xi ∼ f(· | θi)
end
return θi d(s(xobs
), s(xi)) ≤ ε
s(xobs)
ε
(θi, S(xi))

Bayesian settings: the target is π(θ | xobs
) ∝ π(θ)f(xobs
| θ).
Algorithm: Vanilla ABC
,
number of iterations N,
threshold ε, summary
statistic s.
for i = 1, . . . , N do
θi ∼ π(·)
xi ∼ f(· | θi)
end
return θi d(s(xobs
), s(xi)) ≤ ε
s(xobs)
ε
(θi, S(xi))
Ouput: distributed according to πε(θ | s, xobs
)
π(θ)Pθ d(S(xobs
), S(x)) < ε ∝ π(θ | d(S(xobs
), S(x)) < ε)

Two limiting situations:
π∞(θ | s, xobs
) ∝ π(θ) and π0(θ | s, xobs
) ∝ π(θ | s(xobs
))= π(θ | xobs
)

Two limiting situations:
π∞(θ | s, xobs
) ∝ π(θ) and π0(θ | s, xobs
) ∝ π(θ | s(xobs
))= π(θ | xobs
)
Some difﬁculties raised by the vanilla version:
Calibration of the threshold ε: by regression or k-nearest neigh-
bour postprocessing
[Beaumont et al., 2002; Wilkinson, 2013; Biau et al., 2013]
Selection of the summary statistic s: advances with semi-automatic
procedures like pilot-run ABC or random forests
[Fearnhead and Prangle, 2012; Prangle et al., 2014; Raynal et al.,
2018]
Simulating from the prior is often poor in efﬁciency: versions
modifying proposal distribution on θ to increase density of x’s near
y (MCMC, SMC, PMC)
[Marjoram et al., 2003; Toni et al., 2008]

A ﬁrst example : hierarchical moving average model
α
µ1 µ2 µn. . .
x1 x2 xn. . .
σ
σ1 σ2 σn. . .
First parameter hierarchy:
α = (α1, α2, α3) ∼ E(1)⊗3
independently for each i ∈ {1, . . . , n},
(βi,1, βi,2, βi,3) ∼ Dir(α1, α2, α3)
µi = (βi,1 − βi,2, 2(βi,1 + βi,2) − 1)
Second parameter hierarchy:
σ = (σ1, σ2) ∼ C+
(1)⊗2
.
independently for each i ∈ {1, . . . , n},
σi ∼ IG(σ1, σ2)
Model for xi: independently for each i ∈ {1, . . . , n}, xi ∼ MA2(µi, σi),
i.e., for all j in N
xi,j = yj + µi,1yj−1 + µi,2yj−2 , with yj ∼ N(0, σ2
i )

A ﬁrst example : toy dataset
Settings: n = 5 times series of length T = 100 hierarchical model
with 13 parameters.
Figure: Histogram of ABC sam-
ple for µ1,1 along prior distribution
(black line)
Size of ABC reference table:
N = 5.5 · 106
.
ABC posterior sample size:
1000.

A ﬁrst example : toy dataset
Settings: n = 5 times series of length T = 100 hierarchical model
with 13 parameters.
Figure: Histogram of ABC sam-
ple for µ1,1 along prior distribution
(black line)
Size of ABC reference table:
N = 5.5 · 106
.
ABC posterior sample size:
1000.
Not enough simulations to reach a decent threshold
Not enough time to produce enough simulations

The Gibbs Sampler
Our idea: combining ABC with Gibbs sampler in order to improve
ability to (more) efﬁciently explore Θ ⊂ Rn
when the number n of
parameters increases.

The Gibbs Sampler
Our idea: combining ABC with Gibbs sampler in order to improve
ability to (more) efﬁciently explore Θ ⊂ Rn
when the number n of
parameters increases.
The Gibbs Sampler produces a Markov chain with a target joint dis-
tribution π by alternatively sampling from each of its conditionals.
[Geman and Geman, 1984]
Algorithm: Gibbs sampler
, number of iterations N, starting point
θ(0)
= (θ
(0)
1 , . . . , θ
(0)
n ).
for i = 1, . . . , N do
for k = 1, . . . , n do
θ
(i)
k ∼ π · | θ
(i)
1 , . . . , θ
(i)
k−1, θ
(i−1)
k+1 , . . . , θ
(i−1)
n , xobs
end
end
return θ(0)
, . . . , θ(N)

Component-wise ABC [aka ABCG]
ABC Gibbs Sampler produces a Markov chain with each proposal an
ABC posterior conditional on different summaries sj and tolerance εj
Algorithm: Component-wise ABC
θ(0)
= (θ
(0)
1 , . . . , θ
(0)
n ), threshold ε = (ε1, . . . , εn), statistics
s1, . . . , sn.
for i = 1, . . . , N do
for j = 1, . . . , n do
θ
(i)
j ∼ πεj
(· | xobs
, sj, θ
(i)
1 , . . . , θ
(i)
j−1, θ
(i−1)
j+1 , . . . , θ
(i−1)
n )
end
end
return θ(0)
, . . . , θ(N)

Component-wise ABC [aka ABCG]
Algorithm: Component-wise ABC
θ(0)
= (θ
(0)
1 , . . . , θ
(0)
n ), threshold ε = (ε1, . . . , εn), statistics
s1, . . . , sn.
for i = 1, . . . , N do
for j = 1, . . . , n do
θ
(i)
j ∼ πεj
(· | xobs
, sj, θ
(i)
1 , . . . , θ
(i)
j−1, θ
(i−1)
j+1 , . . . , θ
(i−1)
n )
end
end
return θ(0)
, . . . , θ(N)
Questions:
Is there a limiting distribution ν∞
ε to the algorithm?
What is the nature of this limiting distribution?

Gibbs with ABC approximations
Souza Rodrigues et al. (arxiv:1906.04347) alternative ABC-ed Gibbs:
Further references to earlier occurrences of Gibbs versions of ABC
ABC version of Gibbs sampling with approximations to the con-
ditionals with no further corrections
Related to regression post-processing à la Beaumont et al. (2002)
used to designing approximate full conditional, possibly involv-
ing neural networks
Requires preliminary ABC step
Drawing from approximate full conditionals done exactly, possi-
bly via a bootstrapped version.

ABC within Gibbs: Hierarchical models
α
µ1 µ2 µn. . .
x1 x2 xn. . .
Hierarchical Bayes models: often allow for
simpliﬁed conditional distributions thanks to
partial independence properties, e.g.,
xj | µj ∼ π(xj | µj), µj | α
i.i.d.
∼ π(µj | α), α ∼ π(α).
Algorithm: Component-wise ABC sampler for hierarchical model
, number of iterations N, thresholds εα
and εµ, summary statistics sα and sµ.
for i = 1, . . . , N do
for j = 1, . . . , n do
µ
(i)
j ∼ πεµ
(· | xobs
j , sµ, α(i−1)
)
end
α(i)
∼ πεα
(· | µ(i)
, sα)
end

Assumption: n = 1.
Theorem (Clarté et al. [2019])
Assume there exists a non-empty convex set C with positive prior measure
such that
κ1 = inf
sα(µ)∈C
π(Bsα(µ), α/4) > 0 ,
κ2 = inf
α
inf
sα(µ)∈C
πεµ (Bsα(µ),3 α/2 | xobs
, sµ, α) > 0 ,
κ3 = inf
α
πεµ (sα(µ) ∈ C | xobs
, sµ, α) > 0 ,
Then the Markov chain converges geometrically in total variation distance to
a stationary distribution ν∞
ε , with geometric rate 1 − κ1κ2κ2
3.
If the prior on α is deﬁned on a compact set, then the assumptions
are satisﬁed.

Compatibility issue: ν∞
0 is the limiting distribution associated to
Gibbs conditionals with different acceptance events, e.g., different statis-
tics:
π(α)π(sα(µ) | α) and π(µ)f(sµ(xobs
) | α, µ)
Conditionals may then be incompatible and limiting distribution
not genuine posterior [incoherent use of data]
unknown [except for a speciﬁc version]
possibly far from a genuine posterior
Proposition (Clarté et al. [2019])
If sα is jointly sufﬁcient, when the precision ε goes to zero, ABC within
Gibbs and ABC have the same limiting distribution.

Hierarchical models: toy example
Model:
α ∼ U([0 ; 20]),
(µ1, . . . , µn) | α ∼ N(α, 1)⊗n
,
(xi,1, . . . , xi,K) | µi ∼ N (µi, 0.1)
⊗K
.
Numerical experiment:
n = 20, K = 10,
Pseudo observation generated for α = 1.7,
Algorithms runs for a constant budget: Ntot = N × Nε = 21000.
We look at the estimates for µ1 whose value for the pseudo obser-
vations is 3.04.

Illustration:
Assumptions of convergence theorem hold
Tolerance as quantile of distances at each call, i..e, selection of sim-
ulation with smallest distance out of Nα = Nµ = 30
Summary statistic as empirical mean (sufﬁcient):
Setting when SMC-ABC fails as well

Figure: comparison of the sampled densities of µ1 (left) and α (right)
[dot-dashed line as true posterior]
0
1
2
3
4
0 2 4 6
0.0
0.5
1.0
1.5
2.0
−4 −2 0 2 4
Method ABC Gibbs Simple ABC

Figure: posterior densities of µ1 and α for 100 replicas of 3 ABC al-
gorithms [true marginal posterior as dashed line]
ABC−GibbsSimpleABCSMC−ABC
0 2 4 6
0
1
2
3
4
0
1
2
3
4
0
10
20
30
40
mu1
density
ABC−GibbsSimpleABCSMC−ABC
0 1 2 3 4 5
0.0
0.5
1.0
1.5
2.0
2.5
0.0
0.5
1.0
1.5
2.0
0
5
10
15
20
hyperparameter
density

Hierarchical models: moving average example
Pseudo observations: xobs
1 generated for µ1 = (−0.06, −0.22).
0
1
2
3
−1.0 −0.5 0.0 0.5 1.0
value
density
type
ABCGibbs
ABCsimple
prior
1st parameter, 1st coordinate
−1.0
−0.5
0.0
0.5
1.0
−1.0 −0.5 0.0 0.5 1.0
b1
b2
0.2
0.4
0.6
0.8
level
1st parameter simple
−1.0
−0.5
0.0
0.5
1.0
−1.0 −0.5 0.0 0.5 1.0
b1
b2
2.5
5.0
7.5
10.0
level
1st parameter gibbs
Separation from the prior for identical number of simulations.

Hierarchical models: moving average example
Real dataset: measures of 8GHz daily ﬂux intensity emitted by 7
stellar objects from the NRL GBI website: http://ese.nrl.navy.
mil/.
[Lazio et al., 2008]
0
1
2
3
−1.0 −0.5 0.0 0.5 1.0
value
density
type
ABCGibbs
ABCsimple
prior
1st parameter, 1st coordinate
−1.0
−0.5
0.0
0.5
1.0
−1.0 −0.5 0.0 0.5 1.0
b1
b2
0.2
0.4
0.6
level
1st parameter simple
−1.0
−0.5
0.0
0.5
1.0
−1.0 −0.5 0.0 0.5 1.0
b1
b2
2
4
6
8
level
1st parameter gibbs
Separation from the prior for identical number of simulations.

Hierarchical models: g&k example
Model: g-and-k distribution (due to Tukey) deﬁned through inverse
cdf: easy to simulate but with no closed-form pdf:
A + B 1 + 0.8
1 − exp(−gΦ−1
(r)
1 + exp(−gΦ−1(r)
1 + Φ−1
(r)2 k
Φ−1
(r)
Note:: MCMC feasible in this setting (Pierre Jacob)
α
A1
A2
...
An
x1
x2
xn
...
B
g
k

Hierarchical models: g&k example
Assumption: B, g and k known, inference on α and Ai solely.
1 2 3 4 Hyperparameter
−7.5 −7.0 −6.5 −6.0 −5.5 −5.0−7.5 −7.0 −6.5 −6.0 −5.5 −5.0−7.5 −7.0 −6.5 −6.0 −5.5 −5.0−7.5 −7.0 −6.5 −6.0 −5.5 −5.0−7.5 −7.0 −6.5 −6.0 −5.5 −5.0
0
2
4
6
8
value
density
Method ABC Gibbs ABC−SMC vanilla ABC

ABC within Gibbs: general case
A general two-parameter model:
(θ1, θ2)
x
Algorithm: ABC within Gibbs
for i = 1, . . . , N do
θ
(i)
2 ∼ πε2
(· | θ
(i−1)
1 , s2, xobs
)
θ
(i)
1 ∼ πε1
(· | θ
(i)
2 , s1, xobs
)
end
return (θ
(i)
1 , θ
(i)
2 )i=2,...,N

A general two-parameter model:
(θ1, θ2)
x
Algorithm: ABC within Gibbs
for i = 1, . . . , N do
θ
(i)
2 ∼ πε2
(· | θ
(i−1)
1 , s2, xobs
)
θ
(i)
1 ∼ πε1
(· | θ
(i)
2 , s1, xobs
)
end
return (θ
(i)
1 , θ
(i)
2 )i=2,...,N
Assume that there exists 0 < κ < 1/2 such that
sup
θ1, ˜θ1
πε2
(· | xobs
, s2, θ1) − πε2
(· | xobs
, s2, ˜θ1) TV = κ.
The Markov chain then converges geometrically in total variation distance
to a stationary distribution ν∞
ε , with geometric rate 1 − 2κ.

Additional assumption: θ1 and θ2 are a priori independent
Assume that
κ1 = inf
θ1,θ2
π(Bs1(xobs),ε1
| θ1, θ2) > 0 ,
κ2 = inf
θ1,θ2
π(Bs2(xobs), 2
| θ1, θ2) > 0 ,
κ3 = sup
θ1, ˜θ1,θ2
π(· | θ1, θ2) − π(· | ˜θ1, θ2) TV < 1/2 .
Then the Markov chain converges in total variation distance to a stationary
distribution ν∞
ε with geometric rate 1 − κ1κ2(1 − 2κ3).

For both situations, a limiting distribution exists when the thresholds
go to 0.
Assume that
L0 = sup
ε2
sup
θ1, ˜θ1
πε2
(· | xobs
, s2, θ1) − π0(· | xobs
, s2, ˜θ1) TV < 1/2 ,
L1(ε1) = sup
θ2
πε1
(· | xobs
, s1, θ2) − π0(· | xobs
, s1, θ2) TV −−−−→
ε1→0
0 ,
L2(ε2) = sup
θ1
πε2
(· | xobs
, s2, θ1) − π0(· | xobs
, s2, θ1) TV −−−−→
ε2→0
0 .
Then
ν∞
ε − ν∞
0 TV ≤
L1(ε1) + L2(ε2)
1 − 2L0
−−−→
ε→0
0.

Compatibility issue: the general case inherits the compatibility issue
already noticed in the hierarchical setting.
Lemma (Clarté et al. [2019])
1. If sθ1
and sθ2
are conditionally sufﬁcient, the conditionals are compatible
and , when the precision goes to zero, ABC within Gibbs and ABC have
the same limiting distribution.
2. If π(θ1, θ2) = π(θ1)π(θ2) and sθ1
= sθ2
, when the precision goes to
zero, ABC within Gibbs and ABC have the same limiting distribution.

General case: g&k example
Figure: posterior densities for parameters µ1, . . . , µ4 in the doubly
hierarchical g & k model
µ1 µ2 µ3 µ4
−4 −3 −2 −1 0 −4 −3 −2 −1 0 −4 −3 −2 −1 0 −4 −3 −2 −1 0
0
1
2
3
4
5

General case: g&k example
Figure: posterior densities for α, B, g and k
α B g k
−3 −2 −1 0 0.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.000.00 0.25 0.50 0.75 1.00
0.0
2.5
5.0
7.5
10.0

General case: inverse problem
Example inspired by deterministic inverse problems:
difﬁcult to tackle with traditional methods
pseudo-likelihood extremely expensive to compute
requiring the use of surrogate models
Let y be solution of heat equation on a circle deﬁned for (τ, z) ∈]0, T]×
[0, 1[ by
∂τy(z, τ) = ∂z (θ(z)∂zy(z, τ)) ,
with
θ(z) =
n
j=1
θj1[(j−1)/n,j/n](z)
and boundary conditions y(z, 0) = y0(z) and y(0, τ) = y(1, τ)
[Kaipio and Fox, 2011]

Assume y0 known and parameter θ = (θ1, . . . , θn), with discretized
equation via first order finite elements of size 1/n for z and ∆ for τ.
Stepwise approximation of solution
^y(z, t) =
n
j=1
yj,tφj(z)
where, for j < n,
φj(z) = (1 − |nz − j|)1|nz−j|<1
and
φn(z) = (1 − nz)10<z<1/n + (nz − n + 1)11−1/n<z<1
and with yj,t defined by
yj,t+1 − yj,t
3∆
+
yj+1,t+1 − yj+1,t
6∆
+
yj−1,t+1 − yj−1,t
6∆
= yj,t+1(θj+1 + θj) − yj−1,t+1θj − yj+1,t+1θj+1.
Noisy version of this process, chosen as xj,t = N(^yj,t, σ2
).

In ABC-Gibbs, each parameter θm updated with summary statistics
observations at locations m−2, m−1, m, m+1 while ABC-SMC relies
on the whole data as statistic. In experiment, n = 20 and ∆ = 0.1, with
a prior θj ∼ U[0, 1], independently. Convergence theorem applies to
this setting.
Comparison with ABC, using same simulation budget, keeping the
total number of simulations constant at N · N = 8 · 106
.
Choice of ABC reference table size critical: for a ﬁxed computational
budget reach balance between quality of the approximations of the
conditionals (improved by increasing N ), and Monte-Carlo error and
convergence of the algorithm, (improved by increasing N).

Figure: mean and variance of the ABC and ABC-Gibbs estimators of
θ1 as N increases [horizontal line shows true value]
0.5
0.6
0.7
0.8
0 10 20 30 40
0.000
0.002
0.004
0 10 20 30 40
Method ABC Gibbs Simple ABC

Figure: histograms of ABC-Gibbs and ABC outputs compared with
uniform prior
ABC Gibbs Simple ABC
0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00
0
1
2
3

i.i.d.
∼ π(µj | α) , α ∼ π(α)
that is,
prior simulations of α ∼ π(α) and of ˜µ ∼ π(˜µ | α) until
d(sα(µ), sα(˜µ)) < εα
simulation of µ from instrumental q(µ) and of auxiliary variables
˜µ and ˜x until both constraints satisﬁed

i.i.d.
∼ π(µj | α) , α ∼ π(α)
Resulting Gibbs sampler stationary for posterior proportional to
π(α, µ) q(sα(µ))
projection
f(sµ(xobs
) | µ)
projection
that is, for likelihood associated with sµ(xobs
) and prior distribution
proportional to π(α, µ)q(sα(µ)) [exact!]

Figure: histograms of ABC-Gibbs and ABC outputs compared with
uniform prior
ABC Gibbs Simple ABC
0.25 0.50 0.75 1.00 0.25 0.50 0.75 1.00
0
1
2
3

Take home messages
Under certain conditions to specify,

Take home messages
We provide theoretical guarantee on the convergence of ABC within
Gibbs.
• Result n°1: a limiting distribution ν∞
ε exists when the sample
size grows
• Result n°2: a limiting distribution ν∞
0 exists when the thresh-
old goes to 0
• Result n°3: ν∞
0 is the posterior distribution π(θ | s(xobs
)).
The method inherits issues from vanilla ABC, namely the
choice of the statistics [plus compatibility of the condition-
als].
In practice, ABC within Gibbs exhibits better performances than
vanilla ABC and SMC-ABC [even when conditions not satisﬁed]

ABC postdoc positions
2 post-doc positions open with the ABSint ANR research grant:
Focus on approximate Bayesian techniques like ABC, variational
Bayes, PAC-Bayes, Bayesian non-parametrics, scalable MCMC,
and related topics. A potential direction of research would be the
derivation of new Bayesian tools for model checking in such c
omplex environments.
Terms: up to 24 months, no teaching duty attached, primarily
located in Université Paris-Dauphine, with supported periods in
Oxford (J. Rousseau) [barring no-deal Brexit!] and visits to Mont-
pellier (J.-M. Marin).
No hard deadline.
If interested, send application to me: bayesianstatistics@gmail.com

ABC workshops
ABC in Grenoble, France, March 18-19 2020 [mirrored in War-
wick]
ABC in Longyearbyen, Svalbard/Spitzberg, April 12-13 2021

Bibliography I
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proximate Bayesian computation via Gibbs-like steps. arXiv preprint
arXiv:1905.13599, 2019.
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approximate Bayesian computation: semi-automatic approximate
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Bibliography II
J. P. Kaipio and C. Fox. The Bayesian Framework for Inverse Problems
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T. J. W. Lazio, E. B. Waltman, F. D. Ghigo, R. Fiedler, R. S. Foster, and
a. K. J. Johnston. A Dual-Frequency, Multiyear Monitoring Program
of Compact Radio Sources. The Astrophysical Journal Supplement Se-
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Bibliography III
S. Tavaré, D. J. Balding, R. C. Grifﬁths, and P. Donnelly. Inferring Coa-
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proximate Bayesian computation scheme for parameter inference
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tions in Genetics and Molecular Biology, 12(2):129–141, 2013.

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