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Efficien ea men o he model e o in he calib a ion
o compu e codes: he Comple e Maximum a Pos e io i
me hod
Oma Kahol, Pie o Ma co Congedo, Oli ie Le Mai e, Eno a Denimal Goy
To ci e his e sion:
Oma Kahol, Pie o Ma co Congedo, Oli ie Le Mai e, Eno a Denimal Goy. Efficien
ea men o he model e o in he calib a ion o compu e codes: he Comple e Max-
imum a Pos e io i me hod. In e na ional Jou nal o Unce ain y Quan i ica ion, 2025,
�10.1615/In .J.Unce ain yQuan i ica ion.2025056317�. �hal-05090880�
Efficien ea men o he model e o in he calib a ion
o compu e codes: he Comple e Maximum a Pos e io i
me hod
Oma Kahola,b,, Pie o Ma co Congedoa, Oli ie Le Maî ec, Eno a Denimal
Goya
aIn ia, Cen e de Ma héma iques Appliquées, Ecole poly echnique, IPP, Rou e de Saclay,
91128 Palaiseau Cedex, F ance
bDipa imen o di Scienze e Tecnologie Ae ospaziali, Poli ecnico di Milano, Via La Masa
34, 20156 Milano, I aly
cCNRS, In ia, Cen e de Ma héma iques Appliquées, Ecole poly echnique, IPP, Rou e de
Saclay, 91128 Palaiseau Cedex, F ance
Abs ac
Compu e models a e widely used o he p edic ion o complex physical phe-
nomena. Based on obse a ions o hese physical phenomena, i is possible
o calib a e he model pa ame e s. In mos cases, such compu e models a e
mis-speci ied, and he calib a ion p ocess mus be imp o ed by including a
model e o e m. The model e o hype pa ame e s a e, howe e , a ely
lea ned join ly wi h he model pa ame e s o educe he dimensionali y o
he p oblem. Sequen ial and non-sequen ial app oaches ha e been in o-
duced o es ima e he hype pa ame e s. The o me , such as he Kennedy
and O’Hagan (KOH) amewo k, es ima es he model e o hype pa ame e s
be o e calib a ing he model pa ame e s. The la e , such as he Full Max-
imum a Pos e io i (FMP), in oduces a unc ional dependence be ween he
model pa ame e s and he model e o hype pa ame e s. Despi e being mo e
eliable in some cases (bimodali y e.g.), he FMP me hod s ill ails o es i-
ma e co ec ly he pos e io dis ibu ion shape. This wo k p oposes a new
me hodology o ea ing he model e o e m in compu e code calib a ion.
I builds upon he KOH and FMP amewo k. Called he Comple e Maxi-
mum a Pos e io i (CMP) me hod, i p o ides a closed- o m exp ession o he
ma ginaliza ion in eg al o e he model e o hype pa ame e s, signi ican ly
Email add ess: [email p o ec ed] (Oma Kahol)
P ep in submi ed o IJUQ Ap il 5, 2025
educing he dimensionali y o he calib a ion p oblem. Such exp ession e-
lies on a se o assump ions ha a e mo e gene al and less s ingen han
he ones usually employed. The CMP me hod is applied o ou examples
o inc easing complexi y, om elemen a y o eal luid dynamics p oblems,
including o no bimodali y. Compa ed o he ue e e ence solu ion and
unlike he KOH and FMP, he CMP me hod co ec ly cap u es he shape o
he pos e io dis ibu ion, including all modes and hei weigh s. Mo eo e ,
i p o ides an accu a e es ima e o he dis ibu ion ails.
Keywo ds: Unce ain y Quan i ica ion, Model Calib a ion, Bayesian
Me hod, Model E o
1. In oduc ion
A model is a ool used by scien is s and enginee s o unde s and and p edic
he wo ld by ma hema ically desc ibing di e en phenomena. The in insic
ma hema ical de ails can be igno ed he e; howe e , i is use ul o poin ou
ha a p edic ion gene ally depends on some pa ame e s, θ. These pa am-5
e e s can be obse able quan i ies, such as dimensional cons an s, o non-
obse able a iables which in luence he p edic ion. The p ocess o lea ning
he alue o such pa ame e s, om a ailable expe imen al da a, is called
model calib a ion and belongs o he class o in e se p oblems [1].
In his pape , we adop he Bayesian pe spec i e o sol e he calib a ion p ob-10
lem. Acco ding o his pe spec i e, he pa ame e s, θ, a e andom a iables
and he calib a ion p ocess is done by upda ing hei p obabili y dis ibu ion
using a ailable expe imen al da a. The p io dis ibu ion is he p obabili y
dis ibu ion o he pa ame e s be o e he expe imen al da a is obse ed, and
he pos e io dis ibu ion is he upda ed p io dis ibu ion a e he expe i-15
men al da a is obse ed. The wo a e ela ed by Bayes’ o mula; see [1–3] o
§2 o mo e de ails.
The ma hema ical o mula ion o he model o be calib a ed migh p esen
some in e nal inconsis encies o ely on simpli ying assump ions; mo eo e ,
he nume ical ools and app oxima ions used o compu e he model p edic-20
ion in oduce e o s in he ou pu . Fo hese easons, i is easonable o
assume ha in mos cases he model is an app oxima e ep esen a ion o
he ue physical p ocess. I his is he case, he solu ion o he calib a ion
p oblem migh yield biased and o e -con iden pos e io s dis ibu ions [4,5].
2
The seminal wo k o Kennedy and O’Hagan [6] was he i s o conside 25
an addi ional model e o e m in he Bayesian calib a ion p ocedu e. The
model e o e m is a andom unc ion ha accoun s o he disc epancy be-
ween he model and he ue physical p ocess and could be exp essed in a
ini e-dimensional o m (such as polynomial expansions) o wi h an in ini e-
dimensional o m (such as Gaussian P ocesses) [1]. The model e o e m30
usually equi es an addi ional pa ame iza ion; we call i s pa ame e s hy-
pe pa ame e s and deno e hem wi h ψ o dis inguish hem om he model
pa ame e s, θ.
The model e o e m has been applied o he calib a ion o complex models
in many ields, such as ae odynamics [7], solid mechanics [8], ene gy [9] and35
clima e [10,11].
In a ull Bayesian app oach, he hype pa ame e s, ψ, a e andom a iables
and mus be lea ned join ly wi h he pa ame e s, θ. The in oduc ion o
he model e o e m he e o e inc eases he dimensionali y and complexi y
o he calib a ion p oblem, calling o simpli ied solu ions [12,13]. These40
solu ions usually a ge sampling om a lowe -dimensional dis ibu ion.
Fi s , sequen ial echniques p opose o ind a de e minis ic es ima e o he
hype pa ame e s, ψ, and hen condi ion he join pos e io dis ibu ion on
ha de e minis ic es ima e. The o iginal calib a ion amewo k p oposed by
Kennedy and O’Hagan [6,12] uses a sequen ial app oach ha is called he45
Kennedy O’Hagan (KOH) me hod.
Non-sequen ial me hods, like he Full Maximum a Pos e io i (FMP) me hod
[13], app oxima e he ma ginal dis ibu ion o he pa ame e s by in oducing
a unc ional ela ionship be ween he hype pa ame e s, ψand he pa ame-
e s, θ, and condi ioning he join pos e io dis ibu ion on ha ela ionship.50
The ma ginal pos e io o he pa ame e s is he in eg al o he join dis i-
bu ion o e all possible alues o he hype pa ame e s, ψ. Fo his eason,
app oxima ing he ma ginal pos e io is bene icial and can po en ially cap-
u e mo e ea u es o he pos e io dis ibu ion. In [13], he au ho s show
ha in some cases, o example, bimodal dis ibu ion, he FMP me hod ou -55
pe o ms he KOH me hod as he la e some imes ails o cap u e all he
modes. In he same wo k i is poin ed ou ha , al hough able o co ec ly
cap u e all possible modes, he FMP me hod s ill ails o co ec ly ep esen
hei weigh s, inco ec ly biasing he calib a ion esul s. These ailu es a e
mos ly caused by he e y s ingen assump ions on which he FMP me hod60
elies. In pa icula , he FMP me hod assumes ha when condi ioning he
pos e io dis ibu ion o he hype pa ame e s on he pa ame e s, he esul
is a poin mass dis ibu ion. This assump ion is e y s ic and, e y a ely
i is sa is ied. The FMP me hod, he e o e, mis ep esen s he shape o he
3
hype pa ame e s’ condi ional dis ibu ion. Mo eo e , he FMP me hod does65
no wo k when he p io o he hype pa ame e s is non-uni o m (in o ma i e
p io ) as i is no p esen in i s inal exp ession.
In his pape , we e isi he FMP me hod and p opose se e al imp o emen s.
In pa icula , we p opose a new non-sequen ial me hod, he Comple e Max-
imum a Pos e io i (CMP) me hod, which p o ides a closed- o m exp ession70
o he ma ginal dis ibu ion o he model pa ame e s ha is exac in some
cases. The CMP me hod builds upon he FMP me hod and imp o es i by
elaxing some assump ions and by p o iding a mo e accu a e app oxima-
ion o he pos e io dis ibu ion. The name CMP e e s o he ac ha
he new me hod inco po a es a comple e ep esen a ion o he unce ain y75
in he model e o e m, om di e en alues o he hype pa ame e s ψ o
po en ially di e en shapes o hei dis ibu ion.
The pape is o ganized in he ollowing way. Sec ion 2p esen s a sel -
con ained desc ip ion o he calib a ion p oblem using he Bayesian ame-
wo k. Sec ion 3is dedica ed o he p esen a ion o he CMP me hod. In80
§4, we es he CMP me hod on wo elemen a y examples which, because
o hei simplici y, ha e a closed- o m solu ion. In §5, we es he CMP
me hod on wo calib a ion p oblems and compa e i o he KOH and FMP
me hods. Finally, §6concludes he discussion and ou lines possible u u e
in es iga ions.85
2. The Bayesian Calib a ion F amewo k
This sec ion b ie ly in oduces he calib a ion p oblem wi h model e o .
The basic ma hema ical o mula ion and he no a ion a e p esen ed in §2.1.
Sec ion 2.2 ocuses on he solu ion o he calib a ion p oblem discussing he
di e en me hods ha can be used o sol e he p oblem. In §2.3 we dis-90
cuss he limi a ions o he cu en me hods and ou line he mo i a ions o
in oducing he CMP me hod.
2.1. Gene al F amewo k
We conside a ma hema ical model ha desc ibes a physical sys em. The
esponse is an abs ac quan i y u∈ U ha ep esen s he sys em’s s a e95
gi en a alue o some measu able con ol a iables, x∈ X. The ma hema -
ical model, M, is ep esen ed as an ope a o ha ope a es on he esponse
and is pa ame e ized by some pa ame e s, θ∈Θ⊂Rdθ. I uis he solu ion
o he model, he ollowing equa ion holds:
M(u|x,θ) = 0 .(1)
4
The solu ion, u, is usually no di ec ly measu able as i is an abs ac ep e-100
sen a ion o he esponse o he physical sys em. One usually has access o
some obse ables,{y(i)}i=1...Nh:y(i)∈ Y(i), which a e scala physical quan-
i ies ha can be measu ed in an expe imen . To simpli y he discussion,
we ini ially deal wi h he case in which only a single scala obse able, y, is
a ailable. The ex ension o he case o mul iple obse ables is s aigh o -105
wa d and will be discussed la e . We model he obse able, y, as he sum
be ween an obse a ion ope a o , G, which ope a es on he solu ion o he
model, u, and a esidual e m, ,
y|x=G(u|x,θ) + (x|θ).(2)
The p esence o he esidual e m, , is due o he ac ha he model is an
app oxima e ep esen a ion o he ue physical sys em. The esidual e m is110
usually unknown bu depends on he model pa ame e s, θ, since a di e en
choice o θwill lead o a di e en esidual e m [13]. We call a model well-
speci ied i he e exis s a alue o he pa ame e s, θ∗, ha makes he esidual
e m, , ze o: (x|θ∗)=0. We call a model miss-speci ied i o he wise. In
his wo k, we conside he case o miss-speci ied models.115
The e m model calib a ion e e s o he p ocess o lea ning he alue o he
pa ame e s θgi en some expe imen al da a, D={(xi
obs, yi
obs)}i=1...N , whe e
xi
obs ep esen s he alue o he con ol a iables and yi
obs is he expe imen al
measu emen o he obse able, y. In his wo k, we conside he case in
which he expe imen al obse a ions, yobs, a e a ec ed by measu emen e o .120
To p oceed, we need o de ine a s a is ical model (and some assump ions)
ha explains he obse a ions. We model he obse a ions as a ec o o
N andom a iables, Y, which is he sum o he ou pu s o h ee andom
unc ions:
Y=Gθ+z+ϵ.(3)
The andom ec o Gθcon ains he ou pu o he obse a ion ope a o a 125
each con ol a iable, Gθ= (G(u|xi
obs,θ))i=1...N . Following he Bayesian
amewo k, θis a andom a iable and i is dis ibu ed acco ding o he
p io dis ibu ion, π(θ).
The andom ec o z= (z(xi
obs |ψz))i=1...N is he model o he esidual,
(x|θ), and is called model e o [6]. In his wo k, he model e o e m130
is conside ed o be a Gaussian P ocess (GP) wi h mean µψz(x)and co a i-
ance cψz(x,x′). The GP is assumed o ha e ze o mean, leading o be e
iden i iabili y [14]. The pa ame e s ψza e he hype pa ame e s ha pa-
ame e ize he model e o e m. Simila ly o he model pa ame e s, he
5
hype pa ame e s a e andom a iables and a e dis ibu ed acco ding o he135
p io dis ibu ion π(ψz).
The andom ec o ϵcon ains he measu emen e o a each con ol poin ,
ϵ= (ϵi)i=1...N . In his wo k, we assume ha he ϵia e i.i.d. cen e ed Gaussian
andom a iables wi h equal a iance σ2
eand call π(σe)i s p io dis ibu ion.
The hype pa ame e s ψ= (ψz, σe)∈Ψ⊂Rdψa e he ec o o all he140
hype pa ame e s ha pa ame e ize he model and expe imen al e o s.
The p io dis ibu ions a e upda ed using he a ailable expe imen al da a o
ob ain he pos e io dis ibu ion, p (θ,ψ|D). The pos e io dis ibu ion is
ela ed o he p io dis ibu ions by Bayes’ o mula [1]:
p(θ,ψ|D) = L(D|θ,ψ)π(θ)π(ψ)
p(D).(4)
The e m L(D|θ,ψ)is he likelihood ha measu es he p obabili y o ob-145
se ing he expe imen al da a gi en a pa icula alue o he pa ame e s and
hype pa ame e s. Following ou assump ions on Eq. 3, he likelihood is a
mul i a ia e no mal dis ibu ion:
L(D|θ,ψ) = 1
√(2π)Nde (Kψz+σ2
eIN)
exp (−1
2 T
θ(Kψz+σ2
eIN)−1 θ).
(5)
The ec o θcon ains he esiduals, θ= (yi
obs −G(u|θ,xi
obs))i=1...N , and
Kψzis he co a iance ma ix o he model e o e m e alua ed a he ob-150
se a ion poin s (Kψz)ij =cψz(xi
obs,xj
obs). The no a ion de (.)deno es he
de e minan o a ma ix and INis he iden i y ma ix o size N. In he
li e a u e, one can ind di e en exp essions o Eq. 5 ha can accoun also
o da a inconsis ency [15], mul iplica i e expe imen al e o [16], and model
e o s wi h non-ze o mean [1].155
When mul iple obse ables a e a ailable, he likelihood is he p oduc o he
likelihoods o each obse able,
L(D(1), . . . D(Nh)|θ,ψ(1), . . . ψ(Nh))=
Nh
∏
i=1 L(D(i)|θ,ψ(i)),(6)
whe e ψ(i)is he hype pa ame e s ec o ha pa ame e ize he model and
expe imen al e o e ms o he i- h obse able and D(i)is he expe imen al
da a conce ning he i- h obse able.160
Finally, he denomina o in Eq. 4is he model e idence gi en by:
6
p(D) = ∫Θ∫ΨL(D|θ,ψ)π(θ)π(ψ)dψdθ.(7)
In eg a ing he join pos e io in Eq. 4o e he hype pa ame e s yields he
pa ame e s’ ma ginal pos e io ,
p(θ|D) = ∫Ψ
p(θ,ψ|D)dψ.(8)
The co ec ed model p edic ion o an obse able a a new coo dina e, x∗,
deno ed wi h y∗
co |x∗=G(u|x∗,θ) + z(x∗|ψz), is he sum o he model165
and model e o p edic ion. I s ma ginal dis ibu ion can be compu ed using
p(y∗
co |D) = ∫Θ∫Ψz
p(y∗
co |θ,ψz,D)p(θ,ψz|D)dθdψz,(9)
whe e he dis ibu ion p (y∗
co |θ,ψz,D)is no mal wi h mean and a iance
speci ied by he Gaussian p ocess p edic i e equa ions [17]:
p(y∗
co |θ,ψz,D) =N(y∗
co |µp ed, σ2
p ed),
µp ed =G(u|x∗,θ) + k∗TK−1
ψz θ,
σ2
p ed =cψz(x∗,x∗)−k∗TK−1
ψzk∗,
(10)
whe e (k∗)i=cψz(x∗,xi
obs).
This amewo k, oge he wi h he abo e-s a ed assump ions, cons i u es he170
Bayesian Calib a ion F amewo k and p o ides he ools and me hodology o
calib a e mis-speci ied models. The exp ession o he pos e io dis ibu ion,
Eq. 4, is he s a ing poin o he solu ion o he calib a ion p oblem bu
one a ely has access o a closed- o m exp ession.
2.2. Sol ing he Bayesian Calib a ion P oblem175
A popula way o sol e he calib a ion p oblem is o use sampling echniques,
like Ma ko Chain Mon e Ca lo (MCMC) me hods [1,2,18]. MCMC me h-
ods cons uc a Ma ko Chain whose s eady-s a e dis ibu ion is he a ge
dis ibu ion and ex ac samples when con e gence is eached [19].
To ex ac samples om he pa ame e s’ pos e io dis ibu ion, Eq. 8, one180
can use he so-called ull Bayesian app oach. In his case, one samples om
he join pos e io dis ibu ion, p (θ,ψ|D), and hen ma ginalizes o e he
hype pa ame e s o ob ain he pa ame e s’ pos e io dis ibu ion, p (θ|D).
Examples o his echnique can be ound in [13,14,20]. The ull Bayesian
7
app oach is he mos gene al bu i is also he mos compu a ionally expen-185
si e as i s efficiency dec eases as he dimensionali y o he p oblem inc eases
[18].
Modula app oaches, on he o he hand, educe he complexi y o he p oblem
by sampling om a lowe dimensional dis ibu ion. Sequen ial app oxima-
ions ix he hype pa ame e s o a cons an alue, ¯
ψ, and sample om he190
pa ame e s’ pos e io dis ibu ion condi ioned on ha alue,
p(θ|D,¯
ψ)∝ L(D|θ,¯
ψ)π(θ).(11)
Di e en au ho s a gue o di e en choices o he de e minis ic es ima e o
he hype pa ame e s, ¯
ψ, and discuss he implica ions o such choices [21–24].
O hese, we epo he KOH me hod [6,12], which ixes he hype pa ame e s
o he maximize o he pa ame e -a e aged pos e io dis ibu ion,195
¯
ψ=ψKOH = a g max
ψ∈Ψ∫Θ
p(θ,ψ|D)dθ,
= a g max
ψ∈Ψ
p(ψ|D).
(12)
Sequen ial me hods a e po en ially cheape han he ull Bayesian app oach
bu hey migh miss impo an ea u es o he pos e io dis ibu ion. The
FMP me hod [13] is an example o a non-sequen ial me hod ha app oxi-
ma es he ma ginal dis ibu ion o he pa ame e s by in oducing a unc ional
ela ionship be ween he hype pa ame e s and he pa ame e s,200
ψMAP (θ) = a g max
ψ∈Ψ
p(ψ,θ|D).(13)
The me hod hen p oceeds by condi ioning he likelihood on he unc ional
ela ionship, ψMAP (θ), and sampling om he ollowing dis ibu ion:
pFMP (θ|D)∝ L(D|θ,ψMAP (θ))π(θ).(14)
The au ho s o he FMP me hod claim ha he me hod is mo e accu a e
han he KOH me hod and ha i app oxima es he ma ginal dis ibu ion
o he pa ame e s Eq. 8.205
No e ha ψKOH and ψMAP (θ)can be ela ed. This is done, ini ially, by using
he ma ginaliza ion o mula,
p(θ,ψ|D) = p(ψ|θ,D)p(θ|D).(15)
Subs i u ing Eq. 15 in Eq. 12 yields
8
σMAP
e(θ) = √ T
θ θ
N+a,
|de (Sθ)|=√2a+N
T
θ θ∝√1
T
θ θ
.
(32)
Calling =σe
√ T
θ θ
and by using Eq. 32, we can ew i e Eq. 31 as
p(σe|θ,D)∝ | de (Sθ)|e−1
2 2(1
)N+a
.(33)
We can eco e he affine ans o ma ion by no ing ha
ϕ=Sθ(σe−σMAP
e) = −√1
N+a= −ϕ0.(34)
Finally, subs i u ing Eq. 34 in Eq. 33 i yields
p(σe|θ,D)∝ | de (Sθ)|e
−1
2(ϕ+ϕ0)2(1
ϕ+ϕ0)N+a
.(35)
As i is possible o see, Eq. 35 espec s exac ly he hypo hesis o he CMP355
me hod. I can be ew i en as desc ibed by Eq. 18 wi h
(ϕ) : (−ϕ0,∞)→R+,
∝e
−1
2(ϕ+ϕ0)2(1
ϕ+ϕ0)N+a
.
(36)
0.2 0.2 0.4 0.6 0.8 1.0
1
2
3
4
5
Figu e 2: Shape unc ion.
15
Fig. 2depic s he shape unc ion o he CMP calib a ion p oblem wi h no
model e o choosing N+a= 10. In his case, he CMP app oxima ion will
yield he exac esul .
The co ec ion ac o , gi en in Eq. 32, is c ucial in his case. Omi ing360
i would lead o a pa ame e ’s pos e io dis ibu ion ha o e emphasizes
a eas whe e he esiduals a e small and unde emphasizes a eas whe e he
esiduals a e la ge. Since la ge esiduals a e ypically ound in he ails o he
dis ibu ions, he co ec ion ac o is essen ial o p e en he alse ce ain y
e ec .365
4.2. Example 2: Mix u e o Gaussians
In his subsec ion, we assess he CMP app oxima ion o he pa ame e s’
ma ginal pos e io dis ibu ion in he case whe e he pos e io dis ibu ion is
a mix u e o Gaussian dis ibu ions wi h well-sepa a ed modes. This example
was al eady ea ed by [13], which epo ed he KOH and FMP app oxima-370
ions. They ound ha he KOH me hod can cap u e only a single mode and
ha he FMP app oxima ion, while being able o cap u e all he modes, mis-
ep esen s hei weigh s. We shall he e o e pe o m simila compu a ions o
show ha he CMP me hod can cap u e bo h he modes and hei weigh s,
gi ing ise o an exac ep esen a ion o he pos e io dis ibu ion.375
We conside he case in which he join pos e io , p (γ= (θ,ψ)|D), is a
mix u e o mGaussian dis ibu ions wi h weigh s wi:
m
∑
i=1
wi= 1, means µi=
(µθ,i
µψ,i), and co a iance ma ices Ki=[Kθ,i Kθ,ψ,i
KT
θ,ψ,i Kψ,i ]. The pos e io
dis ibu ion is gi en by
p(θ,ψ|D)∝
m
∑
i=1
wi
1
√(2π)dθ+dψde (Ki)
exp (−1
2(γ−µi)TK−1
i(γ−µi)).
(37)
We assume ha he modes a e well-sepa a ed. This ansla es in o equi -380
ing ha “ he in e als: Θi={θ: (µi,θ−θ)TK−1
θ,i (µi,θ−θ)≤ dθ
95%}i=1...m
(wi h dθ
95% equal o he 95% quan ile o he χ2law wi h dθdeg ees o ee-
dom) a e disjoin ” [13]. An equi alen condi ion is supposed o hold o he
hype pa ame e s. Unde his hypo hesis, he hype pa ame e s’ condi ional
dis ibu ion can be well app oxima ed by385
16
p(ψ|θ,D)≈
m
∑
i=1 I(θ∈Θi)1
√(2π)dψde (Kψ,i|θ)
exp (−1
2(ψ−µψ,i|θ)TK−1
ψ,i|θ(ψ−µψ,i|θ)).
(38)
The condi ional means can be exp essed as µψ,i|θ=µψ,i +KT
θ,ψ,iK−1
θ,i (θ−
µθ,i)and he condi ional co a iances wi h Kψ,i|θ=Kψ,i −KT
θ,ψ,iK−1
θ,i Kθ,ψ,i.
The unc ion I(θ∈Θi)is he indica o unc ion ha is equal o 1 i θ∈Θi
and 0 o he wise. No e ha Eq. 38 holds because we assumed ha he modes
a e well-sepa a ed and hence we can ea each mode as a sepa a ed Gaussian390
dis ibu ion. Fu he mo e, condi ioning a mul i a ia e Gaussian dis ibu ion
on a subse o i s a iables esul s in a Gaussian dis ibu ion [17].
Acco ding o [13] and Eq. 28, he op imal hype pa ame e s, ψMAP (θ), and
he co ec ion ac o , |de (Sθ)|, a e well app oxima ed by he ollowing
piecewise cons an unc ions395
ψMAP (θ)≈
m
∑
i=1
µψ,i|θI(θ∈Θi),
|de (Sθ)| ≈
m
∑
i=1
1
√de (Kψ,i|θ)I(θ∈Θi).
(39)
2 1 0 1 2
2.0
1.5
1.0
0.5
0.0
0.5
1.0
Figu e 3: Example o a mix u e o Gaussian dis ibu ion. The ed line ep esen s he MAP
alue o he hype pa ame e s, ψMAP (θ), and he blue line ep esen s he KOH alue.
Fig. 3shows an example o a mix u e o Gaussian dis ibu ion wi h wo
modes cen e ed in (−1,0) and (1,−1) espec i ely. The co a iance ma ices
a e diagonal wi h elemen s (0.1,0.01) and (0.1,0.1) espec i ely. The ed line
17
ep esen s he MAP alue o he hype pa ame e s, ψMAP (θ), and he blue
line ep esen s he KOH alue whose exp ession is epo ed in [13]. Using400
he de ini ion o he CMP me hod, Eq. 21, wi h Eq. 39 and he hypo hesis
o well-sepa a ed modes, we can compu e he CMP app oxima ion o he
pa ame e s’ ma ginal pos e io dis ibu ion,
ˆ
pCMP (θ|D)∝
m
∑
i=1
wi√de (Kψ,i|θ)
√(2π)dθ+dψde (Ki)
exp (−1
2(θ−µθ,i
µψ,i|θ−µψ,i)T
K−1
i(θ−µθ,i
µψ,i|θ−µψ,i)).
(40)
We can exploi he block s uc u e o he co a iance ma ix, Ki, o exp ess i s
de e minan as de (Ki) = de (Kθ,i)de (Kψ,i|θ)[25]. Also, he a gumen 405
o he exponen ial simpli ies o −1
2(θ−µθ,i)TK−1
θ,i (θ−µθ,i), see [13] o
he de ails. In oducing hese simpli ica ions in Eq. 40 and no malizing he
dis ibu ion yields
ˆ
pCMP (θ|D)≈
m
∑
i=1
wi
1
√(2π)dθde (Kθ,i)
exp (−1
2(θ−µθ,i)TK−1
θ,i (θ−µθ,i))=p(θ|D).
(41)
The exp ession in Eq. 41 is he exac ep esen a ion o he pa ame e s’
ma ginal pos e io dis ibu ion [13].410
2 1 1 2
0.2
0.4
0.6
0.8
1.0
1.2
FMP
KOH
CMP
Figu e 4: Pa ame e s’ ma ginal pos e io dis ibu ion.
18
Fig. 4shows he esul ing pa ame e s’ ma ginal pos e io dis ibu ion o he
mix u e o Gaussian dis ibu ion epo ed in Fig. 3. The CMP app oxima-
ion is exac and can co ec ly cap u e he modes and hei weigh s. This is
in con as wi h he KOH me hod which can cap u e only a single mode and
he FMP me hod which can cap u e all he modes bu mis ep esen s hei 415
weigh s. The eade is e e ed o [13] o he de i a ion o he KOH and
FMP app oxima ions.
I is impo an o no e ha he CMP app oxima ion is exac due o he
well-sepa a ed na u e o he modes. When he modes a e no well-sepa a ed,
he CMP me hod o e s an app oxima e ma ginal pos e io dis ibu ion o 420
he pa ame e s ha may no be exac . None heless, we con end ha he
CMP me hod emains p e e able, as i is likely o yield a mo e accu a e
app oxima ion compa ed o he KOH and FMP me hods.
5. Examples
The i s example in §5.1 add esses he calib a ion o an inadequa e model,425
inspi ed by [13], ea u ing a bimodal pos e io . The esul s demons a e
ha he CMP me hod can cap u e bo h modes and hei ela i e weigh s,
unlike he FMP and KOH me hods. The second example, in §5.2, in ol es
he calib a ion o a p oblem ha depends on h ee pa ame e s and h ee
hype pa ame e s, esul ing in an unimodal pos e io dis ibu ion. The no el430
me hod accu a ely cap u es he a iabili y o his mode, leading o mo e
obus p edic ions and co ec ing he alse ce i ude e ec cha ac e is ic o
modula me hods.
5.1. Example 1: Bimodal Pos e io
In his example, he ue unc ion is y(x) = x. The expe imen al da a435
consis s o 11 syn he ically gene a ed obse a ions, equally spaced wi hin
he in e al [0,1], based on he ue model wi h he addi ion o whi e noise,
ϵ∼ N(0,0.012).
The compu e model, which is an inadequa e ep esen a ion o he eal p o-
cess, depends on a single pa ame e called θ. The s a is ical model is de ined440
in Eq. 3, whe e he measu emen e o is modeled as a whi e noise p ocess
ϵ|σe∼ N(0, σ2
e)and he model e o is ep esen ed by a Gaussian p ocess
wi h ze o mean and a squa ed exponen ial ke nel as he co a iance unc ion:
Compu e model: (x, θ) = (1 −θ)(x+ 0.15) + xsin(2θx)
Model e o co a iance: cψz(x, x′) = σ2exp −1
2(x−x′
l)2(42)
19
The p io o θis conside ed uni o m, while a se o in e se gamma unc ions,
IG(x, α, β), is used o he hype pa ame e s. The scale and shape pa ame e s445
o hese dis ibu ions a e he same as he ones used in [13].
The pos e io dis ibu ions compu ed using he KOH, FMP, and CMP me h-
ods all ha e a dimension o 1, and hey a e e alua ed using he analy ical
o mula. In con as , he ull Bayesian pos e io , wi h a dimension o 4,
was sampled using an MCMC sample . These samples we e hen used o450
cons uc a ke nel densi y es ima ion (KDE) o he ue pos e io dis ibu-
ion, employing a ec angula ke nel on he same g id as ha used o he
quad a u e o he KOH, FMP, and CMP pos e io s.
Figu e 5: Slice o he un-no malized pos e io dis ibu ion wi h σe= 0.01 and l= 0.1
(MAP alue o he hype pa ame e s in ed and he KOH alue in blue). The ligh ed
backg ound is p opo ional o he magni ude o he in e se o he co ec ion ac o .
Fig. 5shows a slice o he esul ing un-no malized pos e io dis ibu ion,
ob ained by ixing wo hype pa ame e s: he measu emen noise σe= 0.01455
and he co ela ion leng h l= 0.1. The ed line shows he MAP alue
o he hype pa ame e s, wi h a ia ions ep esen a i e o he scale o he
in e se o he co ec ion ac o |de (Sθ)|, and he blue line shows he
KOH alue. This igu e illus a es how di e en modula app oaches simpli y
he esul ing dis ibu ion by e alua ing i along a single line in he case o 460
he KOH me hod, o along a cu e o he CMP and FMP me hods. The
20
pos e io has wo modes co esponding o θ≈ −0.1,0.9wi h he second mode
being less signi ican . The KOH me hod misses his second mode because i s
e alua ion line does no pass h ough i . While o he sequen ial app oaches
migh p o ide be e app oxima ions, hey a e no gua an eed o cap u e all465
explana ions accu a ely.
As expec ed, no including he co ec ion ac o unde es ima es he e ec o
he ails, leading o a alse ce i ude e ec . This can be in e ed om Fig. 5
by obse ing he ela i e scale o he co ec ion ac o , |de (Sθ)|, which
inc eases a he ails.470
Figu e 6: Pos e io dis ibu ion compu ed using di e en app oxima ion echniques. The
his og am co esponds o samples om he ull-Bayesian pos e io .
Fig. 6p esen s he esul s o he calib a ion. I shows ha he pos e io
has wo modes a θ≈ −0.1and θ≈0.9, wi h he second mode being mo e
signi ican . The KOH app oxima ion comple ely misses he second mode and
exhibi s he alse ce i ude e ec p e iously men ioned. The FMP me hod
cap u es bo h modes bu signi ican ly o e es ima es he weigh o he i s 475
mode and also displays a simila alse ce i ude e ec . In con as , he CMP
app oxima ion p o ides an excellen app oxima ion o he ue pos e io .
21
(a) Hype pa ame e s (MAP in ed and KOH
in blue)
(b) Co ec ion ac o
Figu e 7: Hype pa ame e s and co ec ion ac o used by he modula app oaches.
Fig. 7 epo s he alue o he hype pa ame e s and he Hessian used by he
CMP me hod. The hype pa ame e s, see Fig. 7a, used by he KOH me hod
co espond o he a e age alue, compu ed wi h espec o he inal pos e io 480
dis ibu ion (see Eq. 16), o he ones used by he FMP/CMP me hods. Thei
alue will be he e o e in luenced by he p esence o a s ong peak a ound
θ≈ −0.1. The magni ude o he co ec ion ac o , depic ed in Fig. 7b, is
signi ican . Howe e , i is impo an o ocus on ela i e a ia ions a he
han he absolu e scale, as he CMP me hod’s app oxima ion is alid up o485
a mul iplica i e cons an . I is ne e heless in e es ing o obse e ha no
including he Hessian would lead o a signi ican o e es ima ion o he second
mode.
5.2. Example 2: Highe Dimensional Unimodal Dis ibu ion
In his sec ion, we will calib a e a model ha ela es he d ag coefficien ,490
Cd, o a smoo h sphe e mo ing h ough a luid o i s Reynolds numbe , Re.
The i s comp ehensi e cha ac e iza ion o his ela ionship was p o ided
by Lapple and Shephe d in 1940 [26], and is commonly e e ed o as he
s anda d d ag cu e. The cu en li e a u e o e s a wide ange o abula ed
expe imen al da a and models, mos o which a e de i ed om empi ical495
co ela ions. Fo a ecen e iew o hese esou ces, he in e es ed eade is
e e ed o [27].
Fo simplici y, his wo k will u ilize only he expe imen al da a om [26],
which pe ains o Reynolds numbe s below 200,000, p io o he ansi ion
om lamina o ully u bulen low. The model o be calib a ed is based on500
he modi ied S okes law (Eq. 8 in [27]) and depends on h ee pa ame e s:
22
Cd=A
ReB+C . (43)
Fo each es ed me hod, he pos e io dis ibu ion was sampled using an
MCMC algo i hm. The pa ame e space has a dimensionali y o 3, and we
included a model e o e m wi h ze o mean and a squa ed exponen ial ke nel
as he co a iance unc ion, along wi h an expe imen al e o e m. This505
esul s in a o al o 3 hype pa ame e s, b inging he o e all dimensionali y
o 6.
The p io s o he model pa ame e s a e uni o m, while a se o in e se-
gamma unc ions, de ailed in Tab. 1, is used o he hype pa ame e s.
Va iable α β mean mode
σe4 0.15 0.05 0.03
σ3 0.2 0.1 0.05
l3 4 2 1
Table 1: Pa ame e s o he in e se-gamma p io s o he hype pa ame e s and he co e-
sponding mean and mode o he dis ibu ion.
In e se gamma p io s a e a popula choice o hype pa ame e p io as hey510
a e conjuga e p io s o scale pa ame e s [1] and pu ze o p obabili y mass
on unwan ed e en s, such as ze o, in ini e o nega i e alues. The mean and
mode o he in e se gamma dis ibu ion we e chosen in o de o sepa a e
he model and expe imen al e o s’ ke nel s eng hs and o ensu e ha he
co ela ion leng h is la ge han he leng h scale o he expe imen al da a.515
We choose o pe o m he calib a ion o he log Cd e sus log Re o accoun
o he di e en o de s o magni ude achie ed by Eq. 43.
A KDE o he pos e io dis ibu ion is cons uc ed using 10,000 independen
samples. Independence is ensu ed by inc easing he subsampling a io o
make i compa ible wi h he co ela ion leng h.520
Fo he FMP me hod, he MCMC chains o he pos e io dis ibu ion we e
unable o each con e gence, in he sense o [19]. This happens because
he p io o he hype pa ame e s is no p esen in he FMP app oxima ion,
making he esul ing pos e io dis ibu ion w ong and ha d o sample om
(see §2.3 o a de ailed explana ion). In [13] he au ho s use only uni o m525
dis ibu ions o he p io o he hype pa ame e s hence his p oblem does
no appea .
23
(a) A (b) B
(c) C
No Model E o
Full Bayes
CMP
KOH
Figu e 8: Compa ison be ween di e en app oxima ions o he ma ginal pos e io .
A B C
Pa ame e µ σ µ σ µ σ
BAYES 31.2 6.87 0.89 0.054 0.41 0.07
CMP 30.9 5.65 0.89 0.050 0.41 0.06
KOH 29.7 2.09 0.89 0.029 0.41 0.03
Table 2: Mean and a iance o he pa ame e s.
Fig. 8 epo s he app oxima ion o he ma ginal pos e io dis ibu ion o
he pa ame e s compu ed using he h ee di e en me hods along wi h he
one compu ed wi hou he model e o e m. The la e was included o530
jus i y he inclusion o he model e o since he esul ing dis ibu ion is oo
o e con iden and explains all e o s wi h measu emen noise. The mean and
s anda d de ia ions o he ma ginal pos e io s a e also epo ed in Tab. 2.
Bo h he able and he KDE plo show ha he KOH me hod unde es ima es
he a iance o he dis ibu ion by mo e han 50 %. The CMP es ima ion o 535
he a iance, on he o he hand, is much close o he ull Bayesian one. This
24
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