MCCI: Unified Theory of Mental Holes--Causality--Choice Architecture

MCCI: Uni ied Theo y o Men al Holes–Causali y–Choice
A chi ec u e
(Wi h De ini ions–C i e ia–Theo ems–P oo s–Ve i ica ion P o ocols,
Compa ible wi h WSIG / EBOC / RCA–CID)
Au ic (S-se ies / EBOC)
Ve sion: 1.7 (2025-11-05, Asia/Singapo e)
Keywo ds: Men al holes; Causal diag ams (SCM); Choice a chi ec u e (de aul / aming/o de );
Bias–noise decomposi ion; Loss a e sion; Re e ence poin ; CATE; I-p ojec ion (KL/B egman);
WSIG; EBOC; RCA–CID
MSC: 62Cxx; 62Pxx; 68Txx; 91Bxx; 94Axx
Abs ac
We cons uc a heo y o “men al holes” e i iable unde he iple no m o p obabili y–
u ili y–causali y: gi en a a ional baseline s a egy and an embedding o obse able a chi-
ec u e a iables, we de ine he o al de ia ion unc ional and i s ou -dimensional decom-
posi ion (bias, noise, causal misma ch, a chi ec u e sensi i i y), p o ide iden i ica ion c i-
e ia ia backdoo / on doo /ins umen al a iables/discon inui y/di e ence-in-di e ences,
and speci y minimal expe imen al designs. Unde I-p ojec ion and B egman geome y,
we p o e he “Py hago as–decoupling” s uc u e and de i e ealizable es ima ion–audi
pipelines (DQC). In he WSIG dic iona y, he I-p ojec ion o he a ional cons ain amily
is iewed as he “ eadou no m”, and de ia ions a e w i en as KL/B egman dis ances; in
EBOC, he pipeline is implemen ed as “window selec ion lea ” ules; in RCA–CID, e e sible
logs gua an ee in e en ion eplayabili y and ex e nal audi . We also p o ide an in-model de-
e mina ion c i e ion o “loss a e sion–lo e” ia he indica o L=η(λ−1) (conce n weigh
× ac u e coe icien ). Co e p oo s ollow Csisz´a ’s I-p ojec ion and B egman–Py hago as,
Pea l’s causal c i e ia, and mode n es ima ion heo y.
1 No a ion & Axioms / Con en ions (WSIG–EBOC–RCA Uni y)
A1 (Measu e–S a egy–Readou ): The obse a ion iple (H, w, D) induces windowed
eadou s; all s a egies and dis ibu ions on he s anda d simplex a e me ized by B egman
di e gence Dϕ(·∥·) and KL; a ional baseline gi en by I-p ojec ion on cons ain amilies [?].
A2 (Calib a ion Iden i y, WSIG ca d): Unde he uni ied calib a ion o sca e ing–
in o ma ion geome y, we adop he mo he scale φ′(E)/π =ρ el(E) = (2π)−1 Q(E), whe e
Q:= −iS†∂ESis he Wigne –Smi h g oup delay ma ix; as he measu e coo dina e connec ing
o his sys em [?].
A3 (Fini e-o de NPE discipline): All disc e e–con inuous ans o ma ions and win-
dowed in eg a ions uni o mly adop “ ini e-o de Eule –Maclau in + Poisson” h ee- e m e o
closu e, asse ing non-inc easing singula i y and pole = p ima y scale.
A4 (RCA–CID e e sibili y): Implemen a ion and audi a e uni o mly mapped o Ben-
ne e e sible compu a ion and Zeckendo -encoded logs; gua an eeing e e sible eplay o in-
e en ions and es ima ion e sions [?].
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2 Model and Baseline No m
Va iables: Con ex X, ac ion A∈ A, ou come Y, unobse ed dis u bance U;a chi ec u e
a iables C= (F, D, S) o p esen a ion aming, de aul selec ion, p esen a ion o de .
SCM: Di ec ed acyclic g aph Gand s uc u al equa ions Vi:= i(Pa(Vi), Ui).
Ra ional baseline: Unde iden i ied in e en ion dis ibu ion P(Y|do(A=a), X) and
u ili y u, he Bayes–decision op imal s a egy
π⋆(· | x)∈a g max
π
E[u(Y)|do(A∼π(· | x)), X =x].
Ac ual s a egy: π(· | x, c) may explici ly depend on c.
Di e gence: Take KL o gene al B egman di e gence Dϕ.
3 De ini ions: De ia ion Func ional and Fou -Dimensional De-
composi ion o Men al Holes
De ini ion 3.1 (To al De ia ion–Repea ed Re iew Uni ica ion).Fo each con ex X=x, ix
abaseline p esen a ion c0; le he - h e iew’s s a egy be π( )(· | x, c0). De ine
L:= EXE Dϕπ⋆(· | X)∥π( )(· | X, c0).
(A chi ec u e sensi i i y is sepa a ely measu ed by AS and i s egula iza ion e m RAS; see
Theo em ??.)
De ini ion 3.2 (Same-Case Repe i ion and Fou Componen s).Fo he same case x, epea
e iews A( )∼π( )(· | x, c). He e π( )(· | x, c)deno es he ac ion dis ibu ion o he - h e iew
(o e iewe ); i s B egman cen oid
¯πϕ(· | x, c) := (∇ϕ)−1E [∇ϕ(π( )(· | x, c))].
De ine
Bias(x) := Dϕπ⋆(· | x)∥¯πϕ(· | x, c),
Noise(x) := E Dϕ¯πϕ(· | x, c)∥π( )(· | x, c),
CM(x) := E[u(Y)|A∼¯πϕ(· | x, c), X =x]
−E[u(Y)|do(A∼¯πϕ(· | x, c)), X =x]2≥0,
AS(x) := sup
c,c′
Dϕπ(· | x, c)∥π(· | x, c′).
De ini ion 3.3 (S eng h Indica o ).Gi en weigh s ω≻0, de ine
De ec := EXhωbBias(X) + ωnNoise(X) + ωcCM(X) + ωaAS(X)i.
No e: The ou e ms he e co espond one- o-one wi h B,N,C,RAS in Sec ion ??, whe e C=
EX[CM(X)] and RAS is he penal y unc ional o AS.
4 Causal Embedding and Iden i ica ion C i e ia
A chi ec u e embedding: Inco po a e Cas pa en o co-pa en o Ain o G:C→A→
Y; allow C o al e in o ma ion p esen a ion and obse a ion channels bu no he s uc u al
equa ions o po en ial ou comes Y(a).
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Backdoo c i e ion: I he e exis s Z⊂Xblocking all backdoo pa hs om A o Y, hen
P(y|do(a)) = PzP(y|a, z)P(z) [?].
F on doo /IV/RD/DiD: Fo unobse ed con ounding, use on doo a iables, quali ied
ins umen s ( ele ance, exclusion, mono onici y), eg ession discon inui y, and mode n mul i-
pe iod DiD (including s agge ed ea men iming and con inuous in ensi y) espec i ely [?,?].
5 Th ee Co e Theo ems and P oo s
Theo em 5.1 (B egman–Py hago as Dual Decomposi ion + Regula iza ion).Fo each x, ak-
ing expec a ion o e yields
E hDϕπ⋆∥π( )i=Dϕπ⋆∥¯πϕ+E hDϕ¯πϕ∥π( )i.
Taking expec a ion o e X, by De ini ion ?? we ob ain
L=EXDϕ(π⋆∥¯πϕ)
| {z }
B
+EXE Dϕ(¯πϕ∥π( ))
| {z }
N
.
In oducing egula iza ion o penalize causal misma ch and a chi ec u e sensi i i y, de ine
Laug := L+EX[CM(X)]
| {z }
C
+ ΨAS
|{z}
RAS
⇒ Laug =B+N+C+RAS,
whe e C,RAS ≥0.
P oo . The B egman h ee-poin iden i y Dϕ(x1∥x3) = Dϕ(x1∥x2)+Dϕ(x2∥x3)+⟨x1−x2,∇ϕ(x3)−
∇ϕ(x2)⟩, aking x1=π⋆, x2= ¯πϕ, x3=π( )and condi ional expec a ion o e , using ¯πϕ=
(∇ϕ)−1E[∇ϕ(π( ))] o make he c oss e m 0 (B egman cen oid i s -o de condi ion), yields
he i s iden i y and baseline equali y; CM(X) is de ined as a nonnega i e squa ed di e ence
by De ini ion ??, ΨAS is he penal y unc ional o AS; inco po a ing bo h as egula iza ion
e ms gi es he augmen ed Laug [?].
Theo em 5.2 (A chi ec u e Equi alence and A chi ec u e E ec ).I wo p esen a ions c, c′
only a ec in o ma ion channels wi hou al e ing he s uc u e o Y(a), hen
AS(x) = 0 ⇐⇒ π(· | x, c) = π(· | x, c′)almos su ely.
I AS(x)>0, he e exis s an a chi ec u e e ec induced by pu e p esen a ion di e ence
P(a|x, c)=P(a|x, c′).
P oo . By posi i e de ini eness o di e gence and he de ini ion, immedia e.
Theo em 5.3 (In-Model De e mina ion o “Loss A e sion–Lo e”).Le s∈ {0,1}, e e ence
poin s∗= 1, o he ’s wel a e weigh η≥0, ac u e loss coe icien λ > 1,
U(x, y, s) = u(x) + η u(y) + (s−s∗), (z) = (α z, z ≥0,
−λ β(−z), z < 0.
whe e β(·)>0, β(0) = 0. Ope a ionalize “lo e” as: WTP o educe sepa a ion p obabili y om
ε↓0 o 0exceeds he baseline implied solely by isk a e sion o u. Then unde he p emise
λ > 1,
Lo e ⇐⇒ η > 0, L := η(λ−1) >0.
P oo . A i s -o de app oxima ion,
WTP ∼εη·∆u+ (λ−1) ·β(1),
whe e ∆u ep esen s he ma ginal di e ence in o he ’s wel a e be ween s= 1 and s= 0; i
η= 0, his e m anishes; i λ= 1, he e is no loss a e sion co ec ion o sepa a ion. Bo h
being posi i e yields posi i e WTP excess.
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6 Iden i ica ion and Es ima ion (DQC: Documen –Coun e –Causalize–
Audi )
D1 Documen : Case ile con ains (X, C, A,objec i e,cons ain s).
D2 Coun e - aming: Apply wo o mo e C o he same case (gain/loss aming, de aul
swi ching, o de shu ling), compu e
c
AS(x) = max
c,c′Dϕˆπ(· | x, c),ˆπ(· | x, c′),
lag as “a chi ec u e sensi i e” i abo e h eshold.
D3 Causaliza ion: D aw DAG and iden i y ia backdoo / on doo /IV/RD/DiD c i e ia;
p io i ize small-scale andomiza ion o andomizable cases. Es ima e ATE = E[Y(1) −Y(0)],
CATE(x) = E[Y(1) −Y(0) |X=x]. Fo obse a ional da a, use IPW/DR/TMLE and causal
o es s; pe o m Γ-sensi i i y analysis o unobse ed con ounding [?].
D4 Audi (Noise audi ): Same-case mul i-e alua ion es ima es Noise and agg ega es
De ec = ωbb
B+ωnb
N+ωcb
C+ωac
AS.
Dis inguish “le el noise/pa e n noise/occasion noise” in epo s and p o ide “decision hygiene”
p o ocols (independen judgmen , agg ega ion, mul i-sou ce e idence) [?].
7 Iden i ica ion C i e ia and Minimal Expe imen al Design (Quick
Re e ence)
Backdoo : Selec Zblocking all pa hs wi h a ows in o A, use PzP(y|a, z)P(z) [?].
F on doo : When comple e media o Mexis s and A→Mhas no backdoo , M→Yis
backdoo -adjus able, P(y|do(a)) is iden i iable [?].
Ins umen al Va iables (IV): Z ele an o A, independen o Y(a), a ec s Yonly
h ough A; unde mono onici y iden i ies LATE [?].
Reg ession Discon inui y (RD): Con inui y assump ion a h eshold gua an ees local
a e age causal e ec iden i ica ion [?].
Mul i-pe iod DiD: Unde s agge ed ea men and he e ogeneous e ec s, use Callaway–
San ’Anna / Sun–Ab aham amilies and ex ensions o con inuous ea men in ensi y [?].
8 Es ima o s and E o Discipline (Non-Asymp o ic Implemen-
a ion)
IPW / DR: U ilize double obus ness o p opensi y sco e and ou come eg ession; epo
small-sample co ec ions and imming obus ness [?].
TMLE: Two-s ep subs i u ion es ima ion espec ing e iciency in luence unc ion o a ge
unc ional, easy o in eg a e wi h ML; p o ide in luence unc ion s anda d e o s [?].
Causal o es s / Gene alized andom o es s: Es ima e CATE and unce ain y, handle
clus e e o s [?].
Sensi i i y analysis: Rosenbaum Γ bounds, ma ginal sensi i i y model and i s sha pe
a ian s [?].
NPE e o budge : Fo all disc e e–con inuous ans o ma ions, epo h ee pa s: alias-
ing, bounda y laye (Be noulli), and ail wi h o al bounds.
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9 Isomo phic Connec ion wi h WSIG / EBOC / RCA–CID
WSIG (I-p ojec ion = Bo n eadou ): The I-p ojec ion q⋆= a g minq∈Q KL(p∥q) on
a ional cons ain amily Qis he “no m eadou ”; o al de ia ion L= KL(q⋆∥pπ) is he
eadou –s a egy ela i e de ia ion; B egman–Py hago as gi es he addi i e “bias + noise”
s uc u e [?].
EBOC (s a ic block): Case iles and andomized designs a e window selec ion ules on
s a ic block measu es, no al e ing global measu e; ime is iewed as lea eading o blocks, wi h
o de induced by selec ion ules.
RCA–CID ( e e sible log): Embed DQC pipeline in e e sible cellula au oma a; all
in e en ion–es ima ion e sions eco ded in CID logs encoded in Zeckendo no mal o m, and
Benne e e sible embedding gua an ees eplayabili y and ex e nal audi [?].
Calib a ion alignmen : In scena ios equi ing con e gence wi h ene gy spec um calib a-
ion, ci e φ′/π =ρ el = (2π)−1 Qas uni e sal coo dina e; g oup delay–bandwid h esou ce
cons ain s become global budge o DQC [?].
10 Expe imen al Bluep in and Rep oducibili y Checklis
A/B (de aul e ec ): Randomize D∈ {op -in,op -ou }; es ∆ATE and c
AS.
Dual- aming e iew: Same no i ica ion p esen ed in gain/loss e sions; es ima e CATE
wi h TMLE [?].
Noise audi : Same-case mul i-e alua ion; dis inguish le el/occasion/pa e n noise and e-
po pos - educ ion magni ude and s abili y [?].
“Lo e” indica o : Cons uc insu ance- ype choice wi h small-p obabili y sepa a ion on
olun a y sample; es ima e b
L= ˆη(ˆ
λ−1) and link wi h sa is ac ion/ ecip oci y seconda y end-
poin s.
Go e nance and ai ness: Repo CATE, c
AS o key subg oups; se “a chi ec u e ai -
ness” h esholds and no i ica ion no ms.
11 Fu he P ope ies and Co olla ies
Co olla y 11.1 (Backdoo Adjus men ⇒Causal Misma ch Te m Vanishes).I he e exis s Z
sa is ying he backdoo c i e ion, and when compu ing CM ull adjus men is pe o med on Z,
hen C= 0 [?].
Co olla y 11.2 (KL Special Case Cen oid).When Dϕ= KL and he i s a gumen is on he
simplex, ¯πϕis a geome ic-mean- ype cen oid, ensu ing he c oss e m in Theo em ?? anishes
[?].
Co olla y 11.3 (Su iciency o Decision Hygiene).Independen judgmen and de-echo-chambe
agg ega ion on he B egman pla o m a e equi alen o minimizing E [Dϕ(¯πϕ∥π( ))], hus di ec ly
educing N[?].
Co olla y 11.4 (G oup Delay Budge ).In sys ems calib a ed wi h Q, o al complexi y o
windowed e alua ion is cons ained by g oup delay–bandwid h p oduc uppe bound, se ing as
esou ce budge o DQC [?].
12 P oo De ails (Selec ed)
(I) B egman–Py hago as: Bane jee e al.’s gene al ea men o B egman h ee-poin iden-
i y and clus e ing cen oid, combined wi h Csisz´a I-p ojec ion geome y, gi es he i s -o de
condi ion ¯πϕ= (∇ϕ)−1E[∇ϕ(π( ))], hence c oss e m is 0 [?].
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(II) Causal iden i ica ion: Pea l’s backdoo / on doo ; Ang is –Imbens–Rubin IV and
LATE; Hahn–Todd– an de Klaauw RD; Callaway–San ’Anna (and subsequen ex ensions)
mul i-pe iod and con inuous ea men DiD [?].
(III) Es ima ion heo y: Bang–Robins DR; an de Laan–Rubin TMLE; A hey–Wage
causal and gene alized andom o es s; Rosenbaum and ecen sensi i i y e iews [?].
(IV) WSIG calib a ion: Wigne –Smi h g oup delay and Bi man–K e˘ın o mula p o ide
equi alen coo dina es o calib a ion–phase–spec um, used as measu e coo dina e con e ging
wi h his heo y [?].
(V) RCA–CID e e sibili y: Benne ’s logical e e sibili y and Zeckendo heo em gua -
an ee e e sible eplay and unique ac o iza ion o logs, enabling ex e nal audi o in e en ion–
es ima ion e sions [?].
13 Implemen a ion Bluep in (Enginee ing Minimal Se )
1. Diag amming and c i e ia: Each online decision low i s d aws DAG and ma ks
backdoo se s/a ailable ins umen s/possible h esholds and empo al s agge ing.
2. Online DQC: Case ile empla e + dual- aming ques ionnai e + small-scale andomiza-
ion; au oma ed IPW/DR/TMLE/causal o es s; accompanied by Rosenbaum Γ epo
[?].
3. Audi and go e nance: Repo CATE, c
AS and
De ec (including b
B,b
N,b
C,c
AS) o key
subg oups; se “a chi ec u e ai ness” h esholds and e iew equency.
4. RCA–CID: Use Zeckendo -log o ca y e sions; decla e e e sible eplay in e ace and
audi API.
One-Sen ence Summa y
“Men al holes” a e decomposable de ia ions o s a egy ela i e o a ional baseline; ia causal
c i e ia and I-p ojec ion, hey a e ope a ionalized in o measu able indica o s; DQC con e s
doub in o ins i u ionalized imp o emen and gua an ees audi abili y and po abili y wi hin
he uni ied language o WSIG / EBOC / RCA–CID.
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