Laplace's method in Bayesian inverse problems

0.14 MB·

by  Philipp Wacker

Laplace’s method in Bayesian inverse problems
Philipp Wacker
Department of Mathematics
7
1 University of Erlangen-Nuremberg
0
2
January 30, 2017

n
a
J
Abstract
7
2 InaBayesianinverseproblemsetting,thesolutionconsistsofaposteriormeasureobtained
by combining prior belief, information about the forward operator, and noisy observational
]
data. This measure is most often given in terms of a density with respect to a reference
R
measure in a high-dimensional (or infinite-dimensional) Banach space. Although Monte Carlo
P
samplingmethodsprovideaway ofqueryingtheposterior, thenecessity ofevaluating thefor-
.
h wardoperatormanytimes(whichwill oftenbeacostlyPDEsolver) prohibitsthisinpractice.
t Forthisreason, many practitioners choose a suitable Gaussian approximation of theposterior
a
m measure, in a procedure called Laplace’s method. Oncegenerated, this Gaussian measure is a
lot easier to sample from and properties like moments are immediately acquired. This paper
[
derives Laplace’s approximation of the posterior measure attributed to the inverse problem

1 explicitly as the posterior measure of a second-order approximation of the data-misfit func-
v tional, specifically in the infinite-dimensional setting. By use of a reverse Cauchy-Schwarz
9 inequalitywe areable toexplicitly boundtheHellinger distancebetween theposterior and its
8 approximation.
9
7
0 1 Introduction
.
1
0 We consider an inverse problem
7 y =G(u)+η (1)
1
: where G:X Y is a (possibly) nonlinear mapping between Hilbert spaces X,Y and η N(0,Γ)
v → ∼
isadditivenoise. Thechallengeconsistsofinferringthevalueofufromthenoisy(andmaybelower-
i
X dimensional) observation y. This is generally an ill-defined problem, so some sort of regularization
r is needed.
a
In a Bayesian approach, we assume that u µ = N(0,C ), i.e. we have a Gaussian prior on
0 0
∼
the variable u. For simplicity we assume that the mean is 0, but this assumption can be dropped
withslightmodifications. Thisactsasaregularizationandmakestheinverseproblemwell-defined:
Standard theory (see [15]) yields the posterior measure µ on u given an observation y under mild
assumptions on the forward operator G:
dµ exp( Φ(u))
(u)= − , (2)
dµ exp( Φ(u))dµ (u)
0 0
−
R
1
where Φ(u) = ky−G(u)k2C0. Especially in higher dimensions, the posterior is often approximated
2
by a suitable Gaussian, in order to make computation feasible. For a beautiful example how this
is done in practice, see [1] in the setting of optimal experimental design of an infinite-dimensional
inverse problem, or in a finite-dimensional context in [10]. This is called Laplace’s method and is
the focus of this work. This is done by defining the functional I(u) = Φ(u)+ u2 . The maximum
2σ2
a posteriori point is
u :=arginf I(u) (3)
map u
and the Laplace approximation is defined as
ν =N(u ,HI(u )−1) (4)
map map
This means that the Laplace approximation is the Gaussian measure centered at the Maximum A
Posterioripoint,withcovarianceoperatormatchingthe“local”covariancestructureoftheposterior
measure. In finite dimensions, its density is exactly the normalized exponential of the measure’s
lognormallocalquadraticapproximation. Foragoodexplanation(andasageneralrecommendation
for a truly enjoyable book) of Laplace’s method in finite dimensions, see [12]. A treatise about
approximation of measures by Gaussian measures can be found in [13], although they employ the
Kullback-Leibler divergence (or relative entropy) as a notion of distance between measures. [8] is
an extensive study of various Gaussian approximation methods (including Laplace’s method) in
the context of reservoir modelling. Standard results about Laplace’s method are recorded in [17],
[3] Newer results on Laplace’s method can be found in [9]. Gaussian approximations in a different
context have been treated in [2], [14] in the case of diffusion processes and in [11] in the context
of molecular dynamics. [16] presents an approach running “anti-parallely” to Laplace’s method:
Insteadofapproximatingthe posteriormeasuredirectly, they approximatethe forwardoperatoror
the negativelog-likelihood. They,too,boundtheHellingerdistancebetweenthetrue posteriorand
the resulting approximation.
It can be shownthat µ=ν if G is linear and the prior measure was Gaussianin the first place.
Heuristically, the approximation is bad when the posterior measure is multimodal or has different
tail properties than a Gaussian.
We are interested in deriving concrete error bounds for the approximation quality µ ν. The
≈
Hellinger distance between probability measures lends itself to this cause. Given two measures
µ,ν which are absolutely continuous w.r.t. another measure µ , the Hellinger distance (which is
0
independent of the choice of µ ) between µ and ν is
0
2
1 dµ dν dµ dν
(d (µ,ν))2 = dµ =1 dµ (5)
H 0 0
2 ·Z sdµ0 −sdµ0! −Z sdµ0sdµ0
The main conclusionof this paper is recordedinformally in the followingclaims and statedand
proven later.
Claim 1. While the posterior measure is given via its density w.r.t. the prior by
dµ exp( Φ(u))
(u)= − ,
dµ exp( Φ(u))dµ (u)
0 0
−
Laplace’s method yields a Gaussian approxiRmation of the form ν = N(umap,HI(umap)−1) and its
density w.r.t. the prior is
dν exp( TΦ(u))
(u)= − . (6)
dµ exp( TΦ(u))dµ (u)
0 0
−
R
2
Here, TΦ(u)=Φ(u )+DΦ(u )(u u )+ 1HΦ(u )[u u ,u u ], the second order
map map − map 2 map − map − map
Taylor approximation of the data-misfit functional Φ.
Claim 2. If there is K (0,1) such that
∈
Φ(u) TΦ(u) exp( I(u ))
map
exp exp K − ,
(cid:13)(cid:13) (cid:18)− 2 (cid:19)− (cid:18)− 2 (cid:19)(cid:13)(cid:13)L2(X,µ0) ≤ · 4 det(C01/2·HI(umap)·C01/2)
(cid:13) (cid:13) q
(cid:13) (cid:13)
then the Hellinger distance between the posterior and its Laplace’s approximation can be bounded:
K
d (µ,ν)
H
≤ 1+(1 K)2
−
The paper is organized as follows: First thpe more intuitive one-dimensional case is presented
andrepresentation(6)is derived. Theapproachtakencannotbe generalizedtothe infinite dimen-
sional case, as we use densities with respect to a Lebesgue measure. Then the infinite-dimensional
equivalent is proven, where we show the equality (6) directly by means of characteristic functions.
The following section shows that the problem of bounding the Hellinger distance can be reduced
to a reverseCauchy-Schwarzinequality. After recording a few elementary results about reverseCS
inequalities,weimmediatelyobtainlemma 2andalsoamorepractical(butlesstight)versionofit.
2 The Laplace approximation in one dimension
We recall
dµ exp( Φ(u))
(u)= − ,
dµ exp( Φ(u))dµ (u)
0 0
−
andfurthermorefortheLebesguemeasureλRinonedimension,itsµ’sLaplaceapproximation,which
is aGaussiancenteredatthe maximuma posterioripoint, withvarianceequalto the inverseofthe
second derivative of I at this point:
dν I′′(u ) I′′(u )
= map exp map (u u )2 , (7)
map
dλ 2π − 2 · −
r (cid:18) (cid:19)
hence
dν dν I′′(u ) u2
= dλ = I′′(u ) σ exp map (u u )2+ . (8)
dµ dµ0 map · · − 2 · − map 2σ2
0 dλ (cid:18) (cid:19)
p
As we mentioned, second-order Taylor approximations will play a leading role, hence we define
I′′(u )
R(u)=I(u) I(u ) map (u u )2, (9)
map map
− − 2 −
the error term of the second order Taylor approximation in u . Note that the first-order term
map
vanishes because of u being a minimum of I, which we assume to be C2. Interestingly, R(u) is
map
3
also an error term for the second order approximation of Φ, albeit in a slightly different way:
u2 u 2 Φ′′(u ) 1
R(u)=Φ(u)+ Φ(u ) map map (u u )2 (u u )2
2σ2 − map − 2σ2 − 2 − map − σ2 − map
u (u u )
= map· σ2− map +Φ′(umap)(u−umap)
Φ′′(u )
+ Φ(u) Φ(u ) Φ′(u )(u u ) map (u u )2
map map map map
− − − − 2 −
(cid:20) (cid:21)
=Φ(u) T(2) Φ(u).
− umap
Note that the terms in the second line vanish because of 0 = I′(umap) = Φ′(umap)+ uσm2ap and the
second term is exactly the error between Φ(u) and its second order Taylor polynom developed in
u . It holds that I′(u ) = 0 but in general Φ′(u ) = 0. With this definition of R, and
map map map
6
especially
I′′(u ) u2
map (u u )2+ = Φ(u)+I(u )+R(u),
− 2 · − map 2σ2 − map
we obtain for the density of ν w.r.t. µ :
0
dν
= I′′(u ) σ exp(I(u )) exp( Φ(u)+R(u)).
map map
dµ · · · −
0
p
An easy calculation shows
exp( I(u )) u2
map
− = exp µ (du)
σ I′′(umap) Z 2σ2−I(umap)− I′′(u2map)(u−umap)2! 0 (10)
p = exp( Φ(u)+R(u))µ (du)= exp( T(2) Φ(u))dµ (u).
− 0 − umap 0
Z Z
This also follows immediately from the normalization of dν/dµ and thus
0
dν exp( Φ(u)+R(u)) exp( TΦ(u))
(u)= − = − . (11)
dµ exp( Φ(u)+R(u))µ (du) exp( TΦ(u))µ (du)
0 0 0
− −
R R
3 The Laplace approximation in general Hilbert spaces
Inthissection,wewillshowclaim1inthesettingofageneral(possiblyinfinite-dimensional)Hilbert
space. In order to do this, we need a slight adaptation of a useful Gaussian integral calculation.
In Proposition 1.2.8 in [4], the authors prove the following: Let µ = N(0,Q) be a Gaus-
0
sian measure on a real Hilbert space H. Assume that M is a symmetric operator such that
Q1/2MQ1/2u,u < u,u for all 0=u H. Then for b H
h i h i 6 ∈ ∈
exp 1 (1 Q1/2MQ1/2)−1/2 Q1/2b 2
exp 1 Mu,u + b,u dµ (u)= 2 − · . (12)
ZH (cid:18)2h i h i(cid:19) 0 (cid:16) (cid:12)(cid:12) det(1−Q1/2MQ1/2) (cid:12)(cid:12) (cid:17)
We will need a generalization of this formula, which fopllows from analytical continuation of the
(real) Hilbert space’s inner product to its complex extension. Recall that this continuation will
not be positively definite anymore: λ a+b,λ c+d = λ λ a,c +λ b,c +λ a,d + b,d .
1 2 1 2 2 1
h i h i h i h i h i
The following lemma is stated without proof as it follows immediately from the bilinearity of the
analytical continuation stated above.
4
Lemma 1. Let µ = N(0,Q) be a Gaussian measure on a real Hilbert space H. Assume that
0
M is a symmetric operator such that Q1/2MQ1/2u,u < u,u for all 0 = u H. Then, with
h i h i 6 ∈
L:=Q1/2(1 Q1/2MQ1/2)−1Q1/2 and for b ,b H
1 2
− ∈
1
exp Mu,u + b +ib ,u dµ (u)
1 2 0
2h i h i
ZH (cid:18) (cid:19) (13)
exp 1 Lb ,b +i Lb ,b 1 Lb ,b
= 2h 1 2i h 1 2i− 2h 2 2i .
(cid:0) det(1 Q1/2MQ1/2) (cid:1)
−
Now we can prove our main result: p
Proposition 1. Consider the inverse problem 1 with prior µ and posterior µ given by
0
dµ exp( Φ(u))
= − .
dµ exp( Φ(u))dµ
0 0
−
The functional I(u)=Φ(u)+1 u 2 is assuRmed to be C2 in a neighborhood of u =arginfI(u).
2k kC0 map
Then the Laplace approximation of µ given by
ν =N(u ,HI(u )−1)
map map
is equivalently defined by
dν exp( TΦ(u))
= − ,
dµ exp( TΦ(u))dµ
0 0
−
where TΦ(u)=Φ(umap)+DΦ(umap)(u−umRap)+12HΦ(umap)[u−umap,u−umap] is the second order
Taylor approximation of Φ generated in u
map
Proof. This is done by comparing the Fourier transform (or characteristic function) of both repre-
sentations for ν. We calculate
e−I(umap)
exp( TΦ(u))dµ (u)= (14)
0
− detC1/2I′′(u )C1/2
Z map
and p
exp(i λ,u TΦ(u))dµ (u)= e−I(umap)·exp ihumap,λi− 12 ·I′′(umap)−1[λ,λ] (15)
0
Z h i− (cid:0)detC1/2I′′(umap)C1/2 (cid:1)
We show (15), as (14) follows from setting λ=0. Wepuse
u 2 1
TΦ(u)=R(u) Φ(u)= k kC I(umap) I′′(umap)[u umap,u umap].
− − 2 − − 2 − −
and write J = HI(u ) and v = u for brevity. Note that for a bilinear operator K we will
map map
5
identify K(w,z)= Kw,z . Then
h i
exp(i λ,u TΦ(u))dµ (u)
0
h i−
Z
u 2 1
= exp i λ,u + k kC I(v) J[u v,u v] dµ (u)
0
h i 2 − − 2 − −
Z (cid:18) (cid:19)
1 1
=e−I(v) exp iλ,u + C−1u,u J(u v),u v dµ (u)
0
· h i 2h i− 2h − − i
Z (cid:18) (cid:19)
=e−I(v)−12hJv,vi exp Jv+iλ,u + 1 (C−1 J)u,u dµ0(u).
· h i 2h − i
Z (cid:18) (cid:19)
This is formula (13) with M = C−1 J and b = Jv, b = λ. In this case, 1 C1/2MC1/2 =
1 2
− −
1 C1/2(C−1 J)C1/2 =C1/2JC1/2 and thus (1 C1/2MC1/2)−1/2 =J−1/2C−1/2. Continuing,
− − −
exp 1 J1/2v 2+i J1/2v,J−1/2λ 1 J−1/2λ2
=e−I(v)−21hJv,vi 2| | h i− 2| |
· (cid:0) √detC1/2JC1/2 (cid:1)
exp i v,λ 1 J−1λ,λ
=e−I(v) h i− 2h i
· (cid:0)√detC1/2JC1/2 (cid:1)
This proves equation (15) and it follows that the characteristic function of the measure ν˜ defined
by dν˜ =1/Z exp( TΦ) fulfills
dµ0 · −
exp(i λ,u )dν˜(u) exp(i λ,u ) exp( TΦ(u))dµ (u)
νˆ˜(λ)= h i = h i · − 0
Z exp(i λ,u )dµ (u)
R R h i 0
1
=exp i u ,λ + HI(u )−R1(u),u ,
map map
h i 2h i
(cid:26) (cid:27)
i.e. ν˜=N(u ,HI(u )−1) as claimed.
map map
As in the one-dimensional setting, we conclude
dµ exp( Φ(u))
= −
dµ exp( Φ(u))dµ (u)
0 0
−
and R
dν exp( TΦ(u))
= − .
dµ exp( TΦ(u))dµ (u)
0 0
−
With these expressions, we derive a bRound on the Hellinger distance between µ and ν in the
next section.
4 Hellinger distance
There are many notions of metrics and semi-metrics between measures, notably total variation,
Hellinger, Wasserstein, Prokhorov and Kullback-Leibler. A survey of these and more probability
metrics, including a detailed exposition of their relations is [7].
6
WechoosetheHellingerdistance,mainlybecauseofitsgoodanalyticproperties,itsconsistency
with the total variation metric and because of the following lemma which allows us to bound the
difference between expectations under the different measures in question:
Lemma 2 (part of Lemma 7.14 in [5]). Let µ,µ′ be two probability measures which are absolutely
continuous w.r.t. another measureν on a Banach space (X, ). Assumethat f :X E, where
X
k·k →
(E, ) has second moments with respect to both µ and µ′. Then
k·k
Eµf Eµ′f 2 Eµ f 2+Eµ′ f 2 d (µ,µ′).
H
k − k≤ k k k k ·
q
Recall that
dµ dν
d (µ,ν)2 =1 (u) (u)dµ (u).
H 0
−Z sdµ0 sdµ0
Thus, with the results of the preceding sections,
exp( Φ(u)+TΦ(u))dµ (u)
d (µ,ν)2 =1 − 2 0
H
−
exp( RΦ(u))dµ (u) exp( TΦ(u))dµ (u)
0 0
− − (16)
qR exp 1Φ ,expqR1TΦ
=1 −2 −2 L2(X,µ0)
− exp(cid:10)−21Φ(cid:0) L2(X(cid:1),µ0)·(cid:0)exp −(cid:1)21(cid:11)TΦ L2(X,µ0)
(cid:13) (cid:0) (cid:1)(cid:13) (cid:13) (cid:0) (cid:1)(cid:13)
Remark 1. From positivity of the(cid:13)exponential(cid:13)and the Ca(cid:13)uchy-Schwarz(cid:13)-Bunyakowskyinequality it
can easily be seen that d (µ,ν) [0,1].
H
∈
We would like to bound the Hellinger distance, i.e. optimally we would like to prove something
like d (µ,ν) ǫ for some ǫ > 0. This amounts to a reverse Cauchy-Schwarz inequality, or a
H
≤
statement of the kind
he−Φ/2,e−(TΦ)/2iL2(X,µ0) >(1−ε2)·ke−Φ/2kL2(X,µ0)·ke−(TΦ)/2kL2(X,µ0). (17)
In the next section, we present a few elementary results about this kind of inequality.
5 A reverse Cauchy-Schwarz inequality
Inthissection,H willalwaysbeaHilbertspacewithinnerproduct andnorm . Thestudyof
h·i k·k
ofreverseCauchy-Schwarz-Bunyakowskyinequalitiesisanactivefieldofresearchbyitself. Werefer
to [6] for further reading,althoughwe will need only a verybasic form ofa reverseCSB inequality.
Lemma 3. Let f,g H and D >0 with f g 2 D ( f 2+ g 2). Then
∈ k − k ≤ · k k k k
1 D
f,g − ( f 2+ g 2) (1 D) f g . (18)
h i≥ 2 · k k k k ≥ − ·k k·k k
7
Proof.
1 D 1 D
f,g − f 2 − g 2
h i− 2 ·k k − 2 ·k k
1 1 1 1 D 1 D
= f 2+ g 2 f g 2 − f 2 − g 2
2k k 2k k − 2k − k − 2 ·k k − 2 ·k k
1
= D ( f 2+ g 2) f g 2 0.
2 · k k k k −k − k ≥
(cid:2) (cid:3)
The last inequality in (18) is just Young’s inequality in R.
Remark 2. Lemma3canbethoughtofasareverseYoung’sinequality(whichgivesforallf,g H
∈
that f,g 1 f 2+ 1 f 2 and thus if the conditions on f,g as in lemma 3 are fulfilled, we can
h i ≤ 2k k 2k k
bound
1 D 1
− ( f 2+ g 2) f,g ( f 2+ g 2).
2 · k k k k ≤h i≤ 2 · k k k k
Lemma 4. Let f,g H and K (0,1) with f g K f . Then
∈ ∈ k − k≤ ·k k
1 K 1 K
f,g − f 2+ g 2 2 − f g .
h i≥ 1+(1 K)2 · k k k k ≥ · 1+(1 K)2 ·k k·k k
(cid:20) − (cid:21) (cid:20) − (cid:21)
(cid:0) (cid:1)
Proof.
f g 2 f g 2 f g 2
k − k k − k = k − k
f 2+ g 2 ≤ f 2+( f f g )2 2 f 2 2 f f g + f g 2
k k k k k k k k−k − k k k − k kk − k k − k
f g 2 (2 2K) f 2
k − k =1 − k k
≤ 2 f 2 2K f 2+ f g 2 − (2 2K) f 2+ f g 2
k k − k k k − k − k k k − k
2(1 K)
1 −
≤ − 1+(1 K)2
−
And using lemma 3 (with D =1 2(1−K) ), we obtain the result.
− 1+(1−K)2
Remark 3. This is another form of a reverse Young’s inequality, but with a different prerequisite:
If f g K f , we have
k − k≤ ·k k
1 K2 1
1 ( f 2+ g 2) f,g ( f 2+ g 2) (19)
2 · − 1+(1 K)2 · k k k k ≤h i≤ 2 · k k k k
(cid:18) − (cid:19)
6 Conditions for good approximation
Fromlemma4andtheexpressionford (µ,ν)in(16)(the Hellingerdistancebetweentheposterior
H
measure µ and its Laplace approximation ν) we immediately obtain the following:
Proposition 2. With the assumptions of proposition 1, if for some K (0,1)
∈
Φ(u) TΦ(u) exp( I(u ))
map
exp exp K − , (20)
(cid:13)(cid:13) (cid:18)− 2 (cid:19)− (cid:18)− 2 (cid:19)(cid:13)(cid:13)L2(X,µ0) ≤ · 4 det(C01/2·HI(umap)·C01/2)
(cid:13) (cid:13) q
(cid:13) (cid:13)
8
then
K
d (µ,ν) (21)
H
≤ 1+(1 K)2
−
Proof. We use lemma 4, where H = L2(X,µp), f = exp( TΦ/2) and g = exp( Φ/2). Note
0
− −
that f 2 = exp( I(u ))/ det(C1/2 HI(u ) C1/2) due to (14). Then we can set 1 ε2 =
k kH − map 0 · map · 0 −
2 1−K , hence ε= qK in equation (17).
1+(1−K)2 √1+(1−K)2
h i
The following corollary uses assumptions which can be checked more easily but is less strict.
Corollary 1. Define
σ I′′(u ) Φ(u) TΦ(u)2
K2 := map exp( min Φ(u),TΦ(u) ) min | − | ,1 µ (du).
0
exp( I(u ) · − { } · 4
p− map Z (cid:26) (cid:27)
Then we have
K
d (µ,ν) .
H
≤ 1+(1 K)2
−
Proof. This is due to the elementary inequalitype−x e−y e−(x∧y) (x y 1) from which we
| − |≤ · | − |∧
obtain
e−Φ(u)/2 e−TΦ(u)/2 2dµ e−(Φ(u)∧TΦ(u)) (Φ(u) TΦ(u)2/4 1)dµ
0 0
| − | ≤ · | − | ∧
Z Z
=K2 e−TΦ(u)dµ .
0
·
Z
References
[1] A. Alexanderian, N. Petra, G. Stadler, and O. Ghattas, A fast and scalable method
for a-optimal design of experiments for infinite-dimensional bayesian nonlinear inverse prob-
lems, SIAM Journal on Scientific Computing, 38 (2016), pp. A243–A272.
[2] C. Archambeau, D. Cornford, M. Opper, and J. Shawe-Taylor, Gaussian process
approximations of stochastic differential equations., Gaussian Processes in Practice, 1 (2007),
pp. 1–16.
[3] K. W. Breitung, Asymptotic approximations for probability integrals, Springer, 2006.
[4] G. Da Prato and J. Zabczyk, Second order partial differential equations in Hilbert spaces,
vol. 293, Cambridge University Press, 2002.
[5] M. Dashti and A. M. Stuart, The bayesian approach to inverse problems, arXiv preprint
arXiv:1302.6989,(2013).
9
[6] S. S. Dragomir, Reverses of schwarz inequality in inner product spaces with applications,
Mathematische Nachrichten, 288 (2015), pp. 730–742.
[7] A. L. Gibbs and F. E. Su, On choosing and bounding probability metrics, International
statistical review, 70 (2002), pp. 419–435.
[8] M. A. Iglesias, K. J. Law, and A. M. Stuart, Evaluation of gaussian approximations for
data assimilation in reservoir models, Computational Geosciences, 17 (2013), pp. 851–885.
[9] V. N. Kolokoltsov and T. M. Lapinski, Laplace approximation with estimated error and
application to probability, arXiv preprint arXiv:1502.03266,(2015).
[10] Q.Long,M.Scavino,R.Tempone, andS.Wang,Fastestimationofexpectedinformation
gains for bayesian experimental designs based on laplace approximations, Computer Methods
in Applied Mechanics and Engineering, 259 (2013), pp. 24–39.
[11] Y. Lu, A. M. Stuart, and H. Weber, Gaussian approximations for transition paths in
molecular dynamics, arXiv preprint arXiv:1604.06594,(2016).
[12] D. J. MacKay, Information theory, inference and learning algorithms, Cambridge university
press, 2003.
[13] F. Pinski, G. Simpson, A. Stuart, and H. Weber, Kullback–leibler approximation for
probability measures on infinite dimensional spaces, SIAM Journalon Mathematical Analysis,
47 (2015), pp. 4091–4122.
[14] D. Sanz-Alonso and A. M. Stuart, Gaussian approximations of small noise diffusions in
kullback-leibler divergence, arXiv preprint arXiv:1605.05878,(2016).
[15] A. M. Stuart, Inverse problems: a bayesian perspective, Acta Numerica, 19(2010),pp. 451–
559.
[16] A. M. Stuart and A. L. Teckentrup, Posterior consistency for gaussian process approx-
imations of bayesian posterior distributions, arXiv preprint arXiv:1603.02004,(2016).
[17] R. Wong, Asymptotic approximations of integrals, SIAM, 2001.
10