comments on by judea pearl (ucla). notation 1990’s artificial intelligence hoover

COMMENTS ON

By Judea Pearl (UCLA)

notation

1990’s

Artificial Intelligence

Hoover

Hoover slideHoover slide

From Hoover (2004) “Lost Causes”

notation

1990’s

Artificial Intelligence

additional?^

more?

^

Already unified

additional?^

additional?^

Already permitted

Commendable!

(Statistical)

P. 215

additional?^

additional?^

Already permitted

Commendable!

(Statistical)

WHITE & CHALAKPICTURE OF UNIFICATION

1950 - 2005

SEMNeyman-RubinDAGs

Settable System2006

MY PICTURE OF UNIFICATION1920 - 1990

Informal SEMNeyman-RubinInformal Diagrams

Formal SEM1990 - 2000 2006 (W&C)

Complete Neyman-Rubin

Graphs

DAGs

Multi-agent

extension

CAUSAL ANALYSIS WITHOUT TEARS

TRADITIONAL STATISTICALINFERENCE PARADIGM

Data

Inference

Q(P)(Aspects of P)

PJoint

Distribution

e.g.,Infer whether customers who bought product Awould also buy product B.Q = P(B|A)

THE CAUSAL INFERENCEPARADIGM

Data

Inference

Q(M)(Aspects of M)

Data Generating

Model

Some Q(M) cannot be inferred from P.e.g.,Infer whether customers who bought product Awould still buy A if we were to double the price.

JointDistribution

FROM STATISTICAL TO CAUSAL ANALYSIS:1. THE DIFFERENCES

Datajoint

distribution

inferencesfrom passiveobservations

Probability and statistics deal with static relations

ProbabilityStatistics

CausalModel

Data

Causalassumptions

1. Effects of interventions

2. Causes of effects

3. Explanations

Causal analysis deals with changes (dynamics)

Experiments

Z

YX

INPUT OUTPUT

FAMILIAR CAUSAL MODELORACLE FOR MANIPILATION

WHY CAUSALITY NEEDS SPECIAL MATHEMATICS

Y = 2XX = 1

X = 1 Y = 2

Process information

Had X been 3, Y would be 6.If we raise X to 3, Y would be 6.Must “wipe out” X = 1.

Static information

SEM Equations are Non-algebraic:

CAUSAL MODELS ANDCAUSAL DIAGRAMS

Definition: A causal model is a 3-tupleM = V,U,F

with a mutilation operator do(x): M Mx where:

(i) V = {V1…,Vn} endogenous variables,(ii) U = {U1,…,Um} background variables(iii) F = set of n functions, fi : V \ Vi U Vi

vi = fi(pai,ui) PAi V \ Vi Ui U•

CAUSAL MODELS ANDCAUSAL DIAGRAMS




vi = fi(pai,ui) PAi V \ Vi Ui U

U1 U2I W

Q P PAQ 222

111uwdqbp

uidpbq

++=++=




vi = fi(pai,ui) PAi V \ Vi Ui U(iv) Mx= U,V,Fx, X V, x X

where Fx = {fi: Vi X } {X = x}(Replace all functions fi corresponding to X with the

constant functions X=x)•

CAUSAL MODELS ANDMUTILATION





vi = fi(pai,ui) PAi V \ Vi Ui U

U1 U2I W

Q P 222

111uwdqbp

uidpbq

++=++=

(iv)

(attributes)





vi = fi(pai,ui) PAi V \ Vi Ui U(iv)

U1 U2I W

Q P P = p0

0

222

111

pp

uwdqbp

uidpbq

=++=

++=

Mp

(attributes)




vi = fi(pai,ui) PAi V \ Vi Ui U(iv) Mx= U,V,Fx, X V, x X

where Fx = {fi: Vi X } {X = x}(Replace all functions fi corresponding to X with the

constant functions X=x)Definition (Probabilistic Causal Model): M, P(u)P(u) is a probability assignment to the variables in U.

PROBABILISTIC CAUSAL MODELS

CAUSAL MODELS AND COUNTERFACTUALS

Definition: The sentence: “Y would be y (in situation u), had X been x,” denoted Yx(u) = y, means:

The solution for Y in a mutilated model Mx, (i.e., the equations for X replaced by X = x) and U=u, is equal to y.

)(),()(,)(:

uPzZyYPzuZyuYu

wxwx

∑=====

Joint probabilities of counterfactuals:•

GRAPHICAL – COUNTERFACTUALS SYMBIOSIS

Every causal model implies constraints on counterfactuals

e.g.,

XZY

uYuY

yx

xzx

|

)()(,

⊥⊥

=

consistent, and readable from the graph.

Every theorem in SEM is a theorem in N-R, and conversely.

GRAPHICAL TEST OF IDENTIFICATION

The causal effect of X on Y,

is identifiable in G if there is a set Z ofvariables such that Z d-separates X from Y in Gx.

Z6

Z3

Z2

Z5

Z1

X Y

Z4

Z6

Z3

Z2

Z5

Z1

X Y

Z4

Z

Moreover, P(y | do(x)) = P(y | x,z) P(z)(“adjusting” for Z) z

Gx G

))(())(do|( yuYPxyP x ==

RULES OF CAUSAL CALCULUSRULES OF CAUSAL CALCULUS

Rule 1: Ignoring observations P(y | do{x}, z, w) = P(y | do{x}, w)

Rule 2: Action/observation exchange P(y | do{x}, do{z}, w) = P(y | do{x},z,w)

Rule 3: Ignoring actions P(y | do{x}, do{z}, w) = P(y | do{x}, w)

XG Z|X,WY )( ⊥⊥ if

Z(W)XGZ|X,WY )⊥⊥( if

ZXGZ|X,WY )( if ⊥⊥

RECENT RESULTS ON IDENTIFICATION

Theorem (Tian 2002):

We can identify P(v | do{x}) (x a singleton)

if and only if there is no child Z of X connected

to X by a bi-directed path.

X

Z Z

Z

k

1

• do-calculus is complete• A complete graphical criterion available

for identifying causal effects of any set on any set

• References: Shpitser and Pearl 2006 (AAAI, UAI)

RECENT RESULTS ON IDENTIFICATION (Cont.)

CONCLUSIONS

Structural-model semantics enriched with

logic + graphs leads to formal interpretation

and practical assessments of wide variety

of (if not all) causal and counterfactual

relationships.

causal effects, responsibility,

direct and indirect effects

Multi-agent systems?

e.g.,

MULTI-AGENT GRAPHS

Agent 1 Agent 2

Y1 Z1

X1

ux1

uy1uz1

Y2 Z2

X2

ux2

uy2uz2

WHITE & CHALAKPICTURE OF UNIFICATION

1950 - 2005

SEMNeyman-RubinDAGs

Settable System2006

MY PICTURE OF UNIFICATION1920 - 1990

Informal SEMNeyman-RubinInformal Diagrams

Formal SEM1990 - 2000 2006 (W&C)

Complete Neyman-Rubin

Graphs

DAGs

Multi-agent

extension

comments on by judea pearl (ucla). notation 1990’s artificial intelligence hoover

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