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

Robust estimation via
Minimum Distance Estimation
Pierre AlquierStatistical Estimation and Deep Learning in UQ for PDEs
Vienna, May 2022

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Maximum Mean Discrepancy (MMD)
Details on the computation of the MMD estimator
Complex models

3

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

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Maximum Mean Discrepancy (MMD)
Details on the computation of the MMD estimator
Complex models

3

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

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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
n
n
1X
1X
2
m̂ =
Xi and σ̂ =(Xi − m̂)2 .
n i=1
n i=1
Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation

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

Some problems with the likelihood
Minimum Distance Estimation (MDE)

Minimum Distance Estimation
Minimum Distance Estimation (MDE)
Let d(·, ·) be a metric on probability distributions.
n


1X
δX .
θ̂d := arg min d Pθ , P̂n where P̂n :=n i=1 iθ∈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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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
sup
i f (Yi ) .
f ∈F ni=1where 1 , . . . , n are i.i.d Rademacher variables :P(1 = 1) = P(1 = −1) = 1/2.
Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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,hi2
E MMDk (Pθ̂MMD , P0 ) ≤ inf MMDk (Pθ , P0 ) + √ .
kθ∈ΘnJoint 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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation

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

Some problems with the likelihood
Minimum Distance Estimation (MDE)

MDE and robustness : Huber’s contamination
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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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
Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

Contents
1

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

2

Maximum Mean Discrepancy (MMD)
Details on the computation of the MMD estimator
Complex models

3

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

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

Reminderk(x, x 0 ) = hφ(x), φ(x 0 )iH , k bounded by 1 and characteristic.
Reminder - 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

Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

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

Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

Generative Adversarial Networks (GAN, 1/2)
Generative model X ∼ Pθ :U ∼ Unif[0, 1]d ,X = Fθ (U) where Fθ is someNN with weights θ.

Dziugaite, G. K., Roy, D. M. & Ghahramani, Z. (2015). Training generative neural networks via
maximum mean discrepancy optimization. UAI.
Li, Y., Swersky, K. & Zemel, R. (2015). Generative Moment Matching Networks. ICML.

→ proposed to minimize the MMD to learn θ.
Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

GAN (2/2)
Results from Dziugaite et al. (2015).

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

Inference for Systems of SDEs (1/2)

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

dXt = b(Xt , θ1 )dt + σ(Xt , θ2 )dWt
easy to sample from the model with a given θ = (θ1 , θ2 ),they propose a method to approximate the gradient of
the MMD criterion.

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

Inference for Systems of SDEs (2/2)
Example in a (stochastic) Lotka-Volterra model.

Results from Briol et al. (2019) : compare MMD minimization
to Wasserstein minimization.

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

Regression
problem with regression : we want to specify and estimate
a parametric model Pθ(X ) for Y |X . MMD requires tospecify a model for (X , Y ).
naturalPn idea : estimate the distribution of X by1
i=1 δXi and use the MMD procedure on Pθ(X ) .
nthe previous theory shows directly that we estimate the
distribution of (X , Y ) consistently.
it is far more difficult to prove that we estimate the
distribution of Y |X .
Joint work with M. Gerber (Bristol).
Alquier, P. and Gerber, M. (2020). Universal Robust Regression via
Maximum Mean Discrepancy. Preprint arXiv.
Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

Copulasanother semi-parametricmodel, MMD approach
can be adapted.
asymptotic theory + R
package.
With B.-E. Chérief-Abdellatif (Oxford), J.-D. Fermanian(ENSAE Paris), A. Derumigny (TU Delft).
Alquier, P., Chérief-Abdellatif, B.-E., Derumigny, A. andFermanian, J.-D. Estimation of copulas via Maximum
Mean Discrepancy. JASA, to appear.

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

Example : Gaussian copulas

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

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

Details on the computation of the MMD estimator
Complex models

Example : other models

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Maximum Mean Discrepancy (MMD)
Details on the computation of the MMD estimator
Complex models

3

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

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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.

computation via variational approximations.

Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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
1π, set of functions F and threshold . Put P̂n = n 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

Minimum Distance Estimation

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

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

F
Properties of π̂n,(·)
In a forthcoming joint paperwith S. Legramanti (Universityof Bergamo) and D. Durante(Bocconi University, Milan) we
study 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

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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
2.3d
4
n
n
F


+
Rad
(F)
+
≥
1
−
.
π̂n,
d
(P
,
P
)
≤

+
F
θ
0
∗
n
∗ +n
3
n
cMn
Pierre Alquier, RIKEN AIP

Minimum Distance Estimation

Some problems with the likelihood and how to fix them
Maximum Mean Discrepancy (MMD)
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

Minimum Distance Estimation