E o Geome y and Causal Robus ness:
A Uni ied Geome ic F amewo k om Pa ame e
Con idence Ellipsoids o Mul i-Expe imen T us
Regions
Haobo Ma1Wenlin Zhang2
1Independen Resea che
2Na ional Uni e si y o Singapo e
No embe 24, 2025
Abs ac
This pape p oposes a uni ied amewo k o ans o ming s a is ical e o s in o
“geome ic bounda ies” and sys ema ically embedding i in o causal in e ence and
expe imen al design. The co e idea is: gi en an es ima o and i s e o in pa am-
e e space, we no longe ea “con idence in e als/s anda d e o s” as auxilia y
in o ma ion, bu ele a e hem o “ us egions” in pa ame e space—geome ic
objec s wi h me ic s uc u e; all causal conclusions, obus ness judgmen s, and
expe imen al planning a e cha ac e ized h ough inclusion, in e sec ion, union, and
linea images among hese geome ic egions. Speci ically, his pape i s cons uc s
con idence ellipsoids using ypical asymp o ic no mali y and in o ma ion ma ices
unde gene al pa ame ic models, endowing pa ame e space wi h local Rieman-
nian me ic s uc u e; second, in causal in e ence, we uni y “iden i iable se s” and
“ us egions” as wo ypes o se s in he same pa ame e space, cha ac e izing
p o able causal conclusions and admissible ex apola ion di ec ions h ough hei
in e sec ions; u he , in mul i-expe imen /mul i-model scena ios, we cons uc con-
sensus egions and con lic egions ia in e sec ions, unions, and mappings o us
egions, o ming a kind o “geome ized me a-analysis”; inally, in expe imen al de-
sign and obse a ion planning, we o malize he objec i e o “sh inking us egion
olume/semi-axes” as an op imiza ion p oblem o e design a iables, p o iding se -
e al sol able ins ances unde linea models and ins umen al a iable models. The
appendix p o ides igo ous p oo s o main heo ems, including co e age p ope -
ies o con idence ellipsoids, obus ness c i e ia o causal e ec s, and equi alence
ela ions be ween design c i e ia and Fishe in o ma ion.
Keywo ds: E o Geome y; Con idence Ellipsoid; Causal In e ence; Iden i iable Se ;
Expe imen al Design; Fishe In o ma ion
MSC 2020: 62F12, 62K05, 62P25, 62R01
Con en s
1
1 In oduc ion
S a is ical in e ence is adi ionally p esen ed in he o m o “poin es ima e + con idence
in e al”, while causal in e ence ollows he basic s uc u e o “iden i ica ion assump ions
+ poin es ima e + sensi i i y analysis”. In p ac ice o decision-making and enginee ing
applica ions, esea che s o en ace h ee ypes o p oblems:
1. Gi en ini e samples, which causal conclusions a e uly suppo ed by da a,
a he han me e illusions o poin es ima es?
2. When agg ega ing di e en expe imen s, di e en models, o e en di e en da a
sou ces, how should we sys ema ically see consensus and con lic s among
a ious esul s?
3. Unde limi ed esou ces, how can we maximize “geome ic esolu ion” o
speci ic causal e ec s o pa ame e di ec ions h ough expe imen al de-
sign?
These p oblems di e in o m bu sha e a common ea u e: hey all ela e o “e o ”,
and he essence o “e o ” is no jus a a iance o a con idence in e al, bu a geome ic
objec wi h shape, di ec ion, and bounda y. The goal o his pape is o ho oughly
o malize his in ui i e “geome ic na u e”.
We adop he ollowing iewpoin :
Fo any es ima e ˆ
θo pa ame e θ∈Θ⊂Rd, i s e o na u ally induces a egion
R ⊂ Θ wi h me ic s uc u e, which can be called a “ us egion”;
Causal conclusions a e no s a emen s abou single poin ˆ
θ, bu abou he ange o
some unc ion ψ(θ) o e θ∈ R∩I, whe e Iis he iden i iable se ;
Resul s om mul iple expe imen s, models, and di e en assump ions can be uni ied
as mul iple us egions Rkin he same pa ame e space o i s p ojec ion, whose
in e sec ions, unions, and symme ic di e ences na u ally cha ac e ize pa ame e
anges ha a e “ obus ly consis en ”, “con es able”, o “signi ican ly con lic ing”;
Expe imen al design and obse a ion planning can be iewed as an op imiza ion
p oblem o “ac i ely shaping he geome ic shape o u u e us egions”, aiming o
sh ink semi-axes o us egions in speci ic di ec ions (such as some causal e ec )
o educe hei olume.
In his amewo k, “e o geome y” is no longe jus an appendage o esul s, bu
becomes he co e s uc u e o he en i e causali y-decision p ocess. This pape will s a
om he mos basic pa ame ic models, cons uc his geome ic amewo k, and p o ide
p o able p ope ies and compu able implemen a ions unde se e al speci ic models.
2 Pa ame ic Models and Geome ic S uc u e o
T us Regions
2.1 Pa ame ic Models and Es ima o s
Le obse ed da a be X1, . . . , Xn, de ined on sample space X, assuming hei dis ibu ion
belongs o a amily o p obabili y measu es {Pθ:θ∈Θ⊂Rd}. Le θ0∈Θ be he “ ue
2
pa ame e ”, ˆ
θn=ˆ
θn(X1, . . . , Xn) some es ima o .
We assume he e exis s he usual asymp o ic linea i y and no mali y s uc u e:
√n(ˆ
θn−θ0)d
−→ N(0, I(θ0)−1),
whe e I(θ0) is he Fishe in o ma ion ma ix, posi i e de ini e and con inuous. Fu he
assume he e exis s consis en es ima e ˆ
Insuch ha ˆ
In
P
−→ I(θ0).
2.2 Local Me ic Induced by Fishe In o ma ion
A each poin θo Θ, de ine bilinea o m
gθ(u, ) := u⊤I(θ) , u, ∈Rd,
hen gis a local Riemannian me ic on Θ (unde di e en iabili y condi ions). In u-
i i ely, eigendi ec ions o I(θ) desc ibe “easy/di icul o dis inguish” pa ame e di ec-
ions: he smalle he a iance along some di ec ion, he la ge he “uni leng h” in ha
di ec ion unde in o ma ion me ic, and ice e sa.
Unde ini e sample n, using ˆ
Inwe ob ain empi ical me ic
ˆgn(u, ) := u⊤ˆ
In ,
which con e ges o gθ0in p obabili y sense.
2.3 Con idence Ellipsoid as T us Region
Fo gi en signi icance le el α∈(0,1), de ine he quan ile χ2
d,1−αo d-dimensional chi-
squa e dis ibu ion, cons uc us egion
Rn(α) := {θ∈Θ : n(θ−ˆ
θn)⊤ˆ
In(θ−ˆ
θn)≤χ2
d,1−α}.
I is an ellipsoid cen e ed a ˆ
θn, wi h shape de e mined by ˆ
I−1
n, cha ac e izing pa am-
e e unce ain y. Classical heo y gua an ees:
Theo em 2.1 (Asymp o ic Co e age).Unde he abo e egula i y condi ions, o any
ixed α∈(0,1),
Pθ0θ0∈ Rn(α)−→ 1−α, n → ∞.
This heo em is p o ed in Appendix A.1.
The e o e, Rn(α) can be iewed as a “ us egion con aining ue alue θ0wi h
p obabili y 1−α”. Unde in o ma ion me ic gθ0, semi-axis leng hs o Rn(α) a e in e sely
p opo ional o eigen alues o Fishe in o ma ion and p opo ional o 1/√n.
3 E o as Geome ic Bounda y: Ope a ions and
P ojec ions o T us Regions
This sec ion sys ema ically ans o ms “e o ” in o “geome ic bounda y” and discusses
se e al basic ope a ions: p ojec ion, linea image, and nonlinea image.
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3.1 Linea Func ion Image and Ellipsoid P ojec ion
Le he a ge o in e es be linea unc ion ψ(θ) = c⊤θ, whe e c∈Rd. On ellipsoid
Rn(α), he ange o ψis
Ψn(α) := {ψ(θ) : θ∈ Rn(α)}=ψmin,n, ψmax,n.
This in e al can be analy ically calcula ed. No e ha
ψ(θ) = c⊤θ=c⊤ˆ
θn+c⊤(θ−ˆ
θn),
wi h cons ain n(θ−ˆ
θn)⊤ˆ
In(θ−ˆ
θn)≤χ2
d,1−α. Le h=θ−ˆ
θn, he p oblem be-
comes maximizing/minimizing linea unc ion c⊤hunde ellipsoid cons ain . Classical
op imiza ion conclusion gi es
ψmax,n =c⊤ˆ
θn+sχ2
d,1−α
nc⊤ˆ
I−1
nc,
ψmin,n =c⊤ˆ
θn−sχ2
d,1−α
nc⊤ˆ
I−1
nc.
The e o e, con idence in e al o linea a ge is na u ally gi en by geome ic ela-
ionship be ween ellipsoid and di ec ion ec o c.
P oposi ion 3.1 (Op imal Bounds o Linea Ta ge ).Fo any c∈Rd, he minimum
and maximum alues o ψ(θ) = c⊤θon Rn(α)a e as shown abo e, and his in e al has
co e age p obabili y 1−αin asymp o ic sense.
P oo is in Appendix A.2.
3.2 Local Linea App oxima ion o Nonlinea Func ions
I ψ: Θ →Rkis di e en iable, hen nea ˆ
θnwe can make i s -o de app oxima ion
ψ(θ)≈ψ(ˆ
θn) + Dψ(ˆ
θn)(θ−ˆ
θn),
whe e Jacobian ma ix Dψ(ˆ
θn)∈Rk×dhas i- h ow as ∇ψi(ˆ
θn)⊤. Then ψ(Rn(α))
unde i s -o de app oxima ion is an ellipsoid in Rk:
Sn(α) := ny∈Rk:n(y−ψ(ˆ
θn))⊤Dψ(ˆ
θn)ˆ
I−1
nDψ(ˆ
θn)⊤−1(y−ψ(ˆ
θn)) ≤χ2
k,1−αo,
whe e exis ence o in e se ma ix equi es ow ec o s o Dψ(ˆ
θn) o be linea ly in-
dependen unde in o ma ion me ic. This esul is essen ially a es a emen o Del a
me hod in geome ic language.
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3.3 Suppo Func ion Fo m o Mul i-Pa ame e and Mul i-
Ta ge
In mo e gene al cases, we can use suppo unc ions o cha ac e ize a bi a y con ex
a ge se s. On con ex ellipsoid Rn(α), i s suppo unc ion is
hRn(α)(u) := sup
θ∈Rn(α)
u⊤θ=u⊤ˆ
θn+sχ2
d,1−α
nu⊤ˆ
I−1
nu.
The e o e, any a ge se de ined by collec ion o linea unc ionals {u:u∈ U} can
ha e i s bounda y di ec ly calcula ed h ough suppo unc ion. An impo an applica ion
in causal in e ence scena ios is: when we ca e abou a amily o linea causal e ec s
(such as he e ogeneous e ec s ac oss mul iple g oups), we can uni o mly p o ide hei
wo s /bes cases on us egions h ough suppo unc ions.
4 Iden i iable Se s and T us Regions in Causal In-
e ence
4.1 Causal Models and Iden i iable Se s
In causal in e ence, pa ame e θusually has s uc u al in e p e a ion, such as a e age
ea men e ec in po en ial ou come models, pa h coe icien s in s uc u al equa ion
models, local a e age ea men e ec in ins umen al a iable models, e c. In cases o
non-iden i ica ion o pa ial iden i ica ion, wha da a and assump ions can de e mine is
only an iden i iable se
I:= {θ∈Θ : θis compa ible wi h obse ed dis ibu ion and causal assump ions}.
Fo example, when iola ing ce ain exclusion es ic ions o wi h selec ion bias, iden-
i iable se s a e o en con ex se s, semi-algeb aic se s, o gene al closed se s, a he han
single poin s.
4.2 Da a-D i en Es ima ion o Iden i iable Se s
Unde ini e samples, we ypically app oxima e I h ough es ima ed inequali ies. Fo
example, i causal cons ain s can be exp essed as pa ame e cons ain s
gj(θ)≤0, j = 1, . . . , m,
while we can only es ima e empi ical e sion ˆgj,n(θ) o gj(θ), common p ac ice is using
“ elaxed inequali ies”
ˆgj,n(θ)≤bj,n,
whe e bj,n is uppe bound (such as h eshold a e mul iple es ing co ec ion), hus
ob aining da a-d i en iden i iable se es ima e
ˆ
In:= {θ∈Θ : ˆgj,n(θ)≤bj,n, j = 1, . . . , m}.
5
Unde sui able condi ions i can be p o ed ha ˆ
Incon e ges o Iin app op ia e
sense.
4.3 In e sec ion o Iden i iable Se and T us Region
This pape p oposes: Causal conclusions should be based on Rn(α)∩ˆ
In a he
han me ely on ˆ
θn. Fo gi en causal unc ion ψ: Θ →Rk, we ca e abou
Cn(α) := {ψ(θ) : θ∈ Rn(α)∩ˆ
In}.
I Cn(α) has consis en sign in some di ec ion o componen , o is es ic ed o some
desi ed in e al, we can say his causal conclusion is “geome ically obus ” a signi icance
le el α.
De ini ion 4.1 (Geome ic Robus ness).Le ψ: Θ →Rbe a scala causal a ge ,
Rn(α)a1−αle el us egion, ˆ
Insample app oxima ion o iden i iable se . I he e
exis s in e al [L, U]⊂Rsuch ha
Cn(α) = {ψ(θ) : θ∈ Rn(α)∩ˆ
In} ⊂ [L, U],
hen say “a le el α, causal conclusion ψ(θ)∈[L, U] is geome ically obus ”.
In pa icula , when L > 0 (o U < 0), obus judgmen can be made abou e ec
di ec ion.
4.4 A Typical C i e ion: Linea Causal E ec
Le ψ(θ) = c⊤θbe linea causal e ec (such as some linea combina ion in mul i-pa ame e
model co esponding o a e age ea men e ec ), and iden i iable se can be ep esen ed
as linea inequali ies
Aθ ≤b,
hen Rn(α)∩ˆ
Inis in e sec ion o ellipsoid and polyhed on, a con ex se . Ex eme
alues o causal e ec can be gi en by ollowing con ex op imiza ion p oblem:
ψ∗
max,n := sup{c⊤θ:θ∈ Rn(α), Aθ ≤b},
ψ∗
min,n := in {c⊤θ:θ∈ Rn(α), Aθ ≤b}.
In many applica ions, his p oblem can be e icien ly sol ed h ough quad a ic p o-
g amming o semide ini e p og amming. Clea ly,
Cn(α) = ψ∗
min,n, ψ∗
max,n.
Theo em 4.2 (Geome ic Robus ness C i e ion o Linea Causal E ec ).Unde abo e
se ing, i o some δ > 0,
ψ∗
min,n ≥δ > 0,
hen a signi icance le el α, can conclude causal conclusion “e ec is posi i e and a
leas δ”, and his conclusion holds o all θ∈ Rn(α)∩ˆ
In.
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P oo is in Appendix A.3.
This c i e ion ele a es “poin es ima e signi icance” o “signi icance o e all us ed
candida e pa ame e s”, na u ally excluding spu ious signi icance ha may a ise om
elying only on poin es ima es while igno ing pa ame e co ela ions.
5 Mul iple Expe imen s and Models: In e sec ion,
Union, and Con lic S uc u e o T us Regions
In eali y, we o en need o syn hesize esul s om di e en expe imen s, di e en da a
sou ces, o e en di e en models. This pape ad oca es: The na u al objec s o
mul i-expe imen agg ega ion a e no “se e al poin es ima es”, bu “se e al
us egions”.
5.1 In e sec ion and Consensus o Mul iple T us Regions
Suppose he e a e Kexpe imen s/da a sou ces/models gi ing us egions R(k)
nk(αk) on
same pa ame e space Θ, k= 1, . . . , K. De ine o e all consensus egion
Rcons :=
K
k=1 R(k)
nk(αk).
I he causal unc ion o in e es is ψ: Θ →Rd, hen i s image on consensus egion is
Ccons := {ψ(θ) : θ∈ Rcons}.
I Rcons is non-emp y and “small”, i indica es high consis ency among di e en ex-
pe imen s; con e sely, i Rcons is emp y, can clea ly say “ he e exis s undamen al con lic
among hese expe imen s/models”, a he han aguely elying on “some es ima ion di -
e ences”.
5.2 Union and Admissible Se s
On he o he hand, de ine admissible egion as
Rpe m :=
K
[
k=1 R(k)
nk(αk),
which cha ac e izes he pa ame e se “suppo ed by a leas one expe imen ”. In
some decision p oblems (such as ole a ing pa ial expe imen ailu e o model misspeci-
ica ion), we may only equi e conclusions o hold on Rpe m.
5.3 Con lic Region and Unce ain y Decomposi ion
De ine con lic egion as symme ic di e ence
Rcon lic := K
[
k=1 R(k)
nk! K
k=1 R(k)
nk!,
7
whe e i some poin θis only suppo ed by pa ial expe imen s (while excluded by
o he s), hen belongs o his egion. By isualizing pa ame e space as pa i ion o
“consensus–con lic –uncons ained”, can in ui i ely iden i y which pa ame e di ec ions’
conclusions a e mos sensi i e o expe imen selec ion.
6 Expe imen al Design and Obse a ion Planning:
T us Region as Objec i e Func ion
In abo e amewo k, he essence o expe imen al design is: Shaping he shape and
size o u u e us egions h ough choosing expe imen al schemes o ob-
se a ion s a egies. This sec ion p o ides o malized cha ac e iza ion unde classical
linea models and gene al Fishe in o ma ion backg ound.
6.1 Fishe In o ma ion and Region Volume
In egula models, wi h sample size n, in o ma ion ma ix can usually be exp essed as
In(θ) = nI1(θ),
whe e I1(θ) is in o ma ion om single obse a ion. Fo gi en design pa ame e ξ
(such as dis ibu ion o samples o e di e en ea men /co a ia e con igu a ions), single
obse a ion in o ma ion can be w i en as I1(θ;ξ). The e o e,
In(θ;ξ) = nI1(θ;ξ).
Volume o ellipsoidal us egion is p opo ional o de In(θ0;ξ)−1/2. Mo e p ecisely,
when Rn(α;ξ) is ellipsoid based on In(θ0;ξ), i s Lebesgue olume is
VolRn(α;ξ)=Cd,α de In(θ0;ξ)−1/2,
whe e cons an Cd,α only depends on dimension dand α. The e o e, minimizing egion
olume is equi alen o maximizing de In(θ0;ξ).
De ini ion 6.1 (E o Geome ic Cha ac e iza ion o D-Op imal Design).I design ξ∗
sa is ies
de In(θ0;ξ∗) = sup
ξ
de In(θ0;ξ),
hen call ξ∗D-op imal design. Geome ically, i makes us egion olume minimal
unde gi en n, hus mos compac o e all.
Classical D-op imali y conclusions a e es a ed in e o geome ic language in Ap-
pendix A.4.
6.2 Di ec ional Resolu ion: A-Op imal and c-Op imal
I ocus is on speci ic linea causal e ec ψ(θ) = c⊤θ, hen i s asymp o ic a iance is
Va ˆ
ψ≈1
nc⊤I1(θ0;ξ)−1c.
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F om e o geome y pe spec i e, his is exac ly he squa ed leng h o p incipal semi-
axis o us ellipsoid in di ec ion c(igno ing cons an s). The e o e, minimizing his
a iance is equi alen o maximizing esolu ion in di ec ion c.
De ini ion 6.2 (c-Op imal Design).I design ξ∗sa is ies
c⊤I1(θ0;ξ∗)−1c= in
ξc⊤I1(θ0;ξ)−1c,
hen call ξ∗c-op imal design.
F om geome ic pe spec i e: c-op imal design does no pu sue minimal o e all ellip-
soid olume, bu speci ically comp esses semi-axis in di ec ion c, i.e., ocuses on enhancing
geome ic esolu ion o his causal e ec .
7 Examples o E o Geome y in Typical Models
This sec ion b ie ly demons a es speci ic o ms o e o geome y amewo k unde se e al
common models o eade s o gain in ui i e imp ession.
7.1 Con idence Ellipsoid and E ec In e al in Linea Reg es-
sion Models
Conside linea eg ession model
Yi=x⊤
iβ+εi, εi∼ N(0, σ2),
whe e xi∈Rpa e known co a ia es, β∈Rpa e eg ession coe icien s. Le Xbe
design ma ix, OLS es ima e is
ˆ
β= (X⊤X)−1X⊤Y.
Classical esul gi es
ˆ
β∼ N β, σ2(X⊤X)−1.
The e o e, in o ma ion ma ix is I(β) = σ−2X⊤X, us ellipsoid is
R(α) = {β: (β−ˆ
β)⊤X⊤X(β−ˆ
β)≤σ2χ2
p,1−α}.
Fo any linea p edic ion ψ(β) = x⊤
newβ, i s in e al on R(α) is
x⊤
new ˆ
β±qχ2
p,1−ασ2x⊤
new(X⊤X)−1xnew,
which is comple ely consis en wi h classical linea eg ession con idence in e al, bu
in his amewo k is in e p e ed as “geome ic p ojec ion o us ellipsoid in di ec ion
xnew”.
9
