Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Concentration and robustness of
discrepancy-based ABC
Pierre Alquier

One World ABC Seminar – April 28, 2022

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Contents

1

Some problems with the likelihood and how to fix themSome problems with the likelihood
Minimum Distance Estimation (MDE)2

A Bayesian( ?) point of view
1st approach : “generalized posteriors”
2nd approach : ABC

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Contents

1

Some problems with the likelihood and how to fix themSome problems with the likelihood
Minimum Distance Estimation (MDE)2

A Bayesian( ?) point of view
1st approach : “generalized posteriors”
2nd approach : ABC

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)The Maximum Likelihood Estimator (MLE)
Let X1 , . . . , Xn be i.i.d in X from a probability distribution P0 .
Statistical inference :propose a model (Pθ , θ ∈ Θ), assume P0 = Pθ0 .
compute θ̂n = θ̂n (X1 , . . . , Xn ).
Letting pθ denote the density of Pθ , thenθ̂nMLE = arg max Ln (θ), where Ln (θ) =
θ∈Θ

n
Y

pθ (Xi ).

i=1

2

Example : P(m,σ) = N (m, σ ) then
m̂ =

n
n
1X
1X
Xi and σ̂ 2 =(Xi − m̂)2 .
n i=1
n i=1

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

MLE not unique / not consistent
Example :pθ (x) =exp(−|x − θ|)p,
2 π|x − θ|

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

MLE not unique / not consistent
Example :pθ (x) =exp(−|x − θ|)p,
2 π|x − θ|P
exp (− ni=1 |Xi − θ|)
Ln (θ) = √.
Q p(2 π)n ni=1 |Xi − θ|

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

MLE fails in the presence of outliers
What is an outlier ?
Huber proposed the contamination model : with probability ε,
Xi is not drawn from Pθ0 but from Q that can be anything :P0 = (1 − ε)Pθ0 + εQ.
Example : Pθ = Unif [0, θ], thenn

1 Y
Ln (θ) = n1{0≤Xi ≤θ} ⇒ θ̂ = max Xi .
1≤i≤n
θ i=1
In the case of the following contamination, the MLE isextremely far from the truth :P0 = (1 − ε).Unif [0, 1] + ε.N (1010 , 1)...
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Minimum Distance Estimationn

1X
Empirical distribution : P̂n :=δX .
n i=1 i
Minimum Distance Estimation (MDE)
Let d(·, ·) be a metric on probability distributions.
θ̂d := arg min d Pθ , P̂n .
θ∈Θ
Wolfowitz, J. (1957). The minimum distance method. The Annals of Mathematical Statistics.

Idea : MDE with an adequate d leads to robust estimation.
Bickel, P. J. (1976). Another look at robustness : a review of reviews and some newdevelopments. Scandinavian Journal of Statistics. Discussion by Sture Holm.
Parr, W. C. & Schucany, W. R. (1980). Minimum distance and robust estimation. JASA.
Yatracos, Y. G. (1985). Rates of convergence of minimum distance estimators and Kolmogorov’s
entropy. Annals of Statistics.
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Integral Probability Semimetrics
Integral Probability Semimetrics (IPS)Let F be a set of real-valued, measurable functions and put
dF (P, Q) = sup EX ∼P [f (X )] − EX ∼Q [f (X )] .
f ∈FMüller, A. (1997). Integral probability metrics and their generating classes of functions. Applied
Probability.

assumptions required in order to ensure thatdF (P, Q) = 0 ⇒ P = Q (that is, dF is a metric).
assumptions required in order to ensure that dF < +∞.
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Non-asymptotic bound for MDE
Theorem 1
X1 , . . . , Xn i.i.d from P0 ,for any f ∈ F, supx∈X |f (x)| ≤ 1.
i
Then hE dF (Pθ̂d , P0 ) ≤ inf dF (Pθ , P0 ) + 4.Radn (F).
Fθ∈Θ

Rademacher complexity"
Radn (F) := sup EY1 ,...,Yn ∼P E1 ,...,n
P

#
n
1X
i f (Yi ) .
supf ∈F ni=1where 1 , . . . , n are i.i.d Rademacher variables :P(1 = 1) = P(1 = −1) = 1/2.
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Example 1 : set of indicators
1A (x) =1 if x ∈ A,
0 if x ∈/ A.

Image from Wikipedia.

Reminder - Vapnik-Chervonenkis dimension
Assume that F = {1A , A ∈ A} for some A ⊆ P(X ),SF (x1 , . . . , xn ) := {(f (x1 ), . . . , f (xn )), f ∈ F},
VC(F) := max {n : ∃x1 , . . . , xn , |SF (x1 , . . . , xn )| = 2n } .
Theorem (Bartlett and Mendelson)r
2.VC(F) log(n + 1)
Radn (F) ≤.
nBartlett, P. L. & Mendelson, S. (2002). Rademacher and Gaussian complexities : Risk bounds andstructural results. JMLR.
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Example 1 : KS and TV distances
Two classical examples :A = {all measurable sets in X }, then dF (·, ·) is the total
variation distance TV(·, ·).
VC(F) = +∞ when |X | = +∞,in general, Radn (F) 9 0.

X = R, A = {(−∞, x] , x ∈ R}, then dF (·, ·) is the
Kolmogorov-Smirnov distance KS(·, ·).
KS distance was actually proposed by S. Holm for robust
estimation,
VC(F) = 1.

r
2 log(n + 1)
E KS(Pθ̂KS , P0 ) ≤ inf KS(Pθ , P0 ) + 4.
.
θ∈ΘnPierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Example 2 : Maximum Mean Discrepancy (MMD)
Let (H, h·, ·iH ) be a RKHS with kernelk(x, y ) = hφ(x), φ(y )iH .
If kφ(x)kH = k(x, x) ≤ 1 then EX ∼P [φ(X )] is well-defined .
The map P 7→ EX ∼P [φ(X )] is one-to-one if k is characteristic.
For example, k(x, y ) = exp(−kx − y k2 /γ 2 ) works.

Definition - MMD
MMDk (P, Q) =sup

EX ∼P [f (X )] − EX ∼Q [f (X )]f ∈H
kf kH ≤ 1

= EX ∼P [φ(X )] − EX ∼Q [φ(X )].
H

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Example 2 : MMDr
F = {f ∈ H : kf kH ≤ 1} ⇒ Radn (F) ≤supx k(x, x).
nTheorem 2
For k bounded by 1 and characteristic,i
2
E MMDk (Pθ̂MMD , P0 ) ≤ inf MMDk (Pθ , P0 ) + √ .
k
θ∈Θ
n
h

Joint work with Badr-Eddine ChériefAbdellatif (Oxford).
Chérief-Abdellatif, B.-E. and Alquier, P. Finite Sample
Properties of Parametric MMD Estimation : Robustness
to Misspecification and Dependence. Bernoulli, 2022.

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Example 2 : MMD
We actually have2
MMD2k (Pθ , P̂n ) = EX ,X 0 ∼Pθ [k(X , X 0 )]−n
X

EX ∼Pθ [k(Xi , X )]
n i=1
1 X
+ 2
k(Xi , Xj )n 1≤i,j≤n

and so∇θ MMD2k (Pθ , P̂n )("= 2EX ,X 0 ∼Pθ

#
)
n
X
1
k(X , X 0 ) −k(Xi , X ) ∇θ [log pθ (X )]n i=1that can be approximated by sampling from Pθ .
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Example 2 : MMD
Dziugaite, G. K., Roy, D. M., & Ghahramani, Z. (2015). Training generative neural networks via
maximum mean discrepancy optimization. UAI 2015.

define the estimator and used it to train GANs.

Briol, F. X., Barp, A., Duncan, A. B., & Girolami, M. (2019). Statistical Inference for Generative
Models with Maximum Mean Discrepancy. Preprint arXiv :1906.05944.

assumptions ⇒√n(θ̂MMDk − θ0 )Pierre Alquier, RIKEN AIP

N (0, V0 (k)).

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Example 3 : Wasserstein
Another classical metric belongs to the IPS family :Wδ (P, Q) =sup

EX ∼P [f (X )] − EX ∼Q [f (X )]f :X →R
Lip(f ) ≤ 1

where Lip(f ) := supx6=y |f (x) − f (y )|/δ(x, y ).
In general, Radn (F) 9 0, so will not converge in fullgenerality as with MMD and KS.
However, nice results can be proven under additional
assumptions :Bernton, E., Jacob, P. E., Gerber, M. & Robert, C. P. (2019). On parameter estimation with the
Wasserstein distance. Information and Inference : A Journal of the IMA.
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

MDE and robustness
Reminderhi
E dF (Pθ̂d , P0 ) ≤ inf dF (Pθ , P0 ) + 4.Radn (F).
θ∈ΘF

Huber’s contamination model : P0 = (1 − ε)Pθ0 + εQ.
dF (Pθ0 , P0 ) = sup EX ∼Pθ0 f (X ) − (1 − ε)EX ∼Pθ0 f (X ) − εEX ∼Q f (X )f ∈F

= sup εEX ∼Pθ0 f (X ) − εEX ∼Q f (X )f ∈F

= ε.dF (Pθ0 , Q) ≤ 2εif for any f ∈ F, sup |f (x)| ≤ 1xCorollary - in Huber’s contamination modelhi
E dF (Pθ̂d , Pθ0 ) ≤ 4ε + 4.Radn (F).
F

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

Some problems with the likelihood
Minimum Distance Estimation (MDE)

MDE and robustness : toy experiment
Model : N (θ, 1), X1 , . . . , Xn i.i.d N (θ0 , 1), n = 100 and werepeat the exp. 200 times. Kernel k(x, y ) = exp(−|x − y |).

mean abs. error

θ̂MLE θ̂MMDk θ̂KS
0.081 0.094 0.088

Now, ε = 2% of the observations drawn from a Cauchy.
mean abs. error

0.276 0.095 0.088

Now, ε = 1% are replaced by 1, 000.
mean abs. error

Pierre Alquier, RIKEN AIP

10.008 0.088 0.082

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABCContents

1

Some problems with the likelihood and how to fix themSome problems with the likelihood
Minimum Distance Estimation (MDE)2

A Bayesian( ?) point of view
1st approach : “generalized posteriors”
2nd approach : ABC

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

Generalized posteriors
Posteriorπ(θ|X1 , . . . , Xn ) ∝ Ln (θ)π(θ).
Generalized posteriorπ̂β,Rn (θ) ∝ exp(−β.Rn (θ))π(θ).
old idea in ML (PAC-Bayes, forecasting with expert
advice...) and in statistics (Gibbs posteriors...)popularized / extended and studied by :Bissiri, P. G., Holmes, C. C. & Walker, S. G. (2016). A general framework for updating belief
distributions. JRSS-B.
Knoblauch, J., Jewson, J. & Damoulas, T. (2022). An Optimization-centric View on Bayes’
Rule : Reviewing and Generalizing Variational Inference. JMLR (to appear).
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

Generalizing the posterior with IPS
Generalized posterior with IPS
π̂β,Rn (θ) ∝ exp(−β.dF (Pθ , P̂n ))π(θ).
in the MMD case : non-asymptotic result in
Chérief-Abdellatif, B.-E. and Alquier, P. (2020). MMD-Bayes : Robust Bayesian Estimation via
Maximum Mean Discrepancy. Proceedings of AABI.

asymptotic results to come verysoon in a joint paper with
Takuo Matsubara (Newcastle)
and Jeremias Knoblauch (UCL).
both papers discuss computation via variational
approximations or MCMC.
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABCContents

1

Some problems with the likelihood and how to fix themSome problems with the likelihood
Minimum Distance Estimation (MDE)2

A Bayesian( ?) point of view
1st approach : “generalized posteriors”
2nd approach : ABC

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

ABC with IPS
What follows is based on a joint work with :Sirio Legramanti(University of Bergamo)

Pierre Alquier, RIKEN AIP

Daniele Durante(Bocconi University, Milan)

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

Reminder on ABC
Approximate Bayesian Computation (ABC)
input : sample X1n = (X1 , . . . , Xn ), model (Pθ , θ ∈ Θ), priorπ, statistic S, distance δ and threshold .
(i) sample θ ∼ π,(ii) sample Y1n = (Y1 , . . . , Yn ) i.i.d. from Pθ :if δ(S(X1n ), S(Y1n )) ≤  return θ,else goto (i).

how close is the distribution of the output to the posteriorπ(θ|X1 , . . . , Xn ) ?
reverse point of view : what are the properties of the“generalized posterior” we sample from ?
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

ABC with IPS
Here, we study the situation :P
S(x1 , . . . , xn ) = n1 ni=1 δxi the empirical distribution,
δ(P, Q) = dF (P, Q).
IPS–ABC
input : sample X1n = (X1 , . . . , Xn ), model (Pθ , θ ∈PΘ), prior
π, set of functions F and threshold . Put P̂n = n1 ni=1 δXi .
(i) sample θ ∼ π,(ii) sample Y1n = (Y1 , . . . , Yn ) i.i.d. from Pθ and put
PP̂nY = n1 ni=1 δYi ,if dF (P̂n , P̂nY ) ≤  return θ,else goto (i).
F
Notation : the output ϑ ∼ π̂n,(·).
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

F(·)
Properties of π̂n,

3 questions :
1

?

F
π̂n,
(θ) −−→ π(θ|X1n ).
&?

2

3

?

F
π̂n,
(θ) −−−→?

n→∞?
F
π̂n,n (·) −−−→ δθ0 if P0 = Pθ0 .
n→∞Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

Contraction of the ABC posterior∗ := inf dF (Pθ , P0 ).
θ∈Θ

Theorem 3
Assume :for all  > 0, π({θ : dF (Pθ , P0 ) ≤ ∗ + }) ≥ cd .
∀f ∈ F, supx∈X |f (x)| ≤ 1.
Radn (F) −−−→ 0.
n→∞Let n be any sequence such that n /Radn (F) → ∞ and n2n → ∞.
Then, with probability → 1 on the sample, for any Mn → ∞,
s


Mn
log
d
4
2.3d
n
n
F


π̂n,
+
Rad
(F)
+
≥
1
−
.
d
(P
,
P
)
≤

+
F
θ
0
∗
n
∗ +n
3
n
cMn
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

Example : MMD-ABC with bounded kernel
As an example, consider MMDk when k(x, x) ≤ 1, as
Park, M., Jitkrittum, W. & Sejdinovic, D. (2016). K2-ABC : Approximate Bayesian Computation
with kernel embeddings. AISTATS.

Corollary
Assume :for all  > 0, π({θ : dF (Pθ , P0 ) ≤ ∗ + }) ≥ cd .
√Let 1/ n  n  1. Then, with probability → 1 on the sample, for any
Mn → ∞,
q


Mn
1
+
log
d
d
n
F
 ≥ 1 − 2.3 .
MMDk (Pθ , P0 ) ≤ ∗ + 4n +
√
π̂n,
∗ +n
3
cMn
n

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

A result without Rademacher complexity
Theorem 4Assume :for all  > 0, π({θ : dF (Pθ , P0 ) ≤ ∗ + }) ≥ cd ,P a.s.

dF (P̂n , P0 ) −−0−−→ 0,
PY1n ∼Pθ (dF (P̂nY , Pθ ) > ) ≤ c(θ)fn () where fn () −−−→ 0.
n→∞Let n → 0 and fn (n ) → 0. Then, with probability → 1 on the sample,
for some C > 0 and any Mn → ∞,
 d 
n
C
4n
F
−1
+
f
≥1−
.
π̂n,
d
(P
,
P
)
≤

+
F
θ
0
∗
n
∗ +n
3
Mn
Mn

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

Example : MMD-ABC with unbounded kernel
Corollary
Assume :for all  > 0, π({θ : dF (Pθ , P0 ) ≤ ∗ + }) ≥ cd ,EZ ∼Q [k(Z , Z )] < +∞ for Q = P0 and any Q ∈ {Pθ , θ ∈ Θ}.
Let 1/n2d  n  1. Then, with probability → 1 on the sample, there isa C > 0 such that for any Mn → ∞,
s
!
4n
1 Mn
C
F
π̂n,∗ +n MMDk (Pθ , P0 ) ≤ ∗ +
+
≥1−
.
d
3
n n
Mn

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

Example : Wasserstein-ABC
Bernton, E., Jacob, P. E., Gerber, M. & Robert, C. P. (2019). Approximate Bayesian
Computation with the Wasserstein distance. JRSS-B.

considered ABC with the Wasserstein distance, and proved
Theorem 4 in this case (note that our Theorem 4 is a
restatement of their result for a general IPS).
However, it not easy to prove non-trivial bounds :PY1n ∼Pθ (dF (P̂nY , Pθ ) > ) ≤ c(θ)fn (),and the examples they cite require X to be a bounded space.
Weed, J. & Bach, F. (2019). Sharp asymptotic and finite-sample rates of convergence of
empirical measures in Wasserstein distance. Bernoulli.

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABCn → ∞, fixed 
Jiang, B., Wu, T.-Y. & Wong, W. H. (2018). Approximate Bayesian computation with
Kullback-Leibler divergence as data discrepancy. AISTATS.

provides a general result that can be directly used here :Theorem (simplified version)
Assume :Radn (F) → 0,then for any measurable B,
a.s.

F
π̂n,
(θ ∈ B) −−−→ π(θ ∈ B|dF (Pθ , P0 ) ≤ ∗ + ).
n→∞Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

 & 0, fixed nBernton, E., Jacob, P. E., Gerber, M. & Robert, C. P. (2019). Approximate Bayesian
Computation with the Wasserstein distance. JRSS-B.

provides a general result that can be directly used here :Theorem (simplified version)
Assume :Pθ has continuous, bounded densities pθ , and P0 = Pθ0 ,dF is continous, and is a metric,
then a.s. with respect to the sample, for any measurable B,Fπ̂n,(θ ∈ B) −−→ π(θ ∈ B|X1n ).
&0

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

Experimental results

2.5

2.0

α
0%

1.5

5%
10%

1.0

15%

0.5

0.0

MMD

Wasserstein

KL

Pierre Alquier, RIKEN AIP

Gamma``Exact`` Gaussian

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

Other current and future directions
go beyond IPS, see (among others) f -divergences orenergy statistics in
Frazier, D. T. (2020). Robust and efficient Approximate Bayesian Computation : A minimum
distance approach. Preprint arXiv.
Nguyen, H. D., Arbel, J., Lü, H. and Forbes, F. (2020). Approximate Bayesian computation via
the energy statistic. IEEE Access.

solve practical issues : choice of n , choice of k in MMDk .
semi-parametric models (with J.-D. Fermanian (ENSAE
Paris), A. Derumigny (TU Delft) and M. Gerber (Bristol).
Alquier, P., Chérief-Abdellatif,
B.-E., Derumigny, A. andFermanian, J.-D. Estimation of
copulas via Maximum Mean
Discrepancy. JASA, to appear.
Alquier, P. and Gerber, M. (2020).
Universal Robust Regression via
Maximum Mean Discrepancy.
Preprint arXiv.
Pierre Alquier, RIKEN AIP

Discrepancy-based ABC

Some problems with the likelihood and how to fix them
A Bayesian( ?) point of view

1st approach : “generalized posteriors”
2nd approach : ABC

La fin終わり
ありがとう ございます。

Pierre Alquier, RIKEN AIP

Discrepancy-based ABC