[PDF] - Causal Discovery Richard Scheines Peter Spirtes, Clark Glymour, PDF Document

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Causal Discovery

Richard Scheines Peter Spirtes, Clark Glymour, and many others

Dept. of Philosophy & Machine Learning

Carnegie Mellon

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Day Care Aggressivenes John Mary A lot None A lot A little Graphical Models --11/29/06 4

Disease

[Heart Disease, Reflux Disease, other]

Shortness of Breath

[Yes, No]

Chest Pain

[Yes, No]

!

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" ,#% Disease

[Heart Disease, Reflux Disease, other]

Shortness of Breath

[Yes, No]

Chest Pain

[Yes, No]

Updating

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Causal Inference

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&9

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4 !2!!

∃$≠ $ 2345 $≠ 2345 $

%%-42⇔ 42 42 4<33<2!! ∃$≠ $ 2345$≠ 2345$ / %%-4<33<2⇔ 2<33<4

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= =5>?

& 4→ 2%- 4 !2

Exposure

Rash

Exposure

Infection Rash

Chicken Pox

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% . %%%

Exposure

Infection Symptoms

Exposure

Infection Symptoms Omitted Common Causes Omitted Causes 2 Omitted Causes 1

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255

55" 8% " 8% " 8% " 8% ; 33 ( 8 ; !28

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Sweaters On Room Temperature

Pre-experimental System Post

255 5>((

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255

Sweaters On Room Temperature

Pre-experimental System Post

5

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Interventions & Causal Graphs

Model an ideal intervention by adding an “intervention” variable

utside the original system as a direct cause of its target.

Education

Income Taxes

Pre-intervention graph Intervene on Income “Soft” Intervention

Education

Income Taxes

I

“Hard” Intervention

Education

Income Taxes

I

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5,5@

55 2A5,35,5BB C5,35,5BD 2A535,5, C535,5,D 2A5,355, C5,355E, 2A5355E, C5355,

S m o k in g [0,1 ] Lu n g C ancer [0 ,1] Y ello w F in gers [0,1 ]

P(S,YF, L) = P(S) P(YF | S) P(LC | S) The Joint Distribution Factors According to the Causal Graph, i.e., for all X in V P(V) = ΠP(X|Immediate Causes of(X))

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.>B

&9

Education

Longevity Income

Statistical Model Causal Graph

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.>B

z &9-

F&9! + - +5!+1+ !&!!

z -

G. &%

Education Longevity Income

Causal Graph

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.>B

&9- &5ε %5 β1 & + ε% C5 β2 & + εC

ε1ε%1ε%H",1Σ

− Σ # I

Education Longevity Income

Causal Graph

Education εIncome εLongevity β1 β2 Longevity Income

SEM Graph (path diagram)

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+2

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Semantics of Causation

Choice 1: Define X Y, or X Y in terms of intervention, i.e., (hypothetical) treatment) Choice 2: Causal systems over V Probabilistic Independence Relations in P(V)

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X is a cause of Y iff

∃x1 ≠ x2 P(Y | X set= x1) ≠ P(Y | X set= x2)

Causation is asymmetric: X Y ⇔ X Y X and Y are associated (X _||_ Y) iff ∃x1 ≠ x2 P(Y | X = x1) ≠ P(Y | X = x2) Association is symmetric: X _||_ Y ⇔ Y _||_ X

Choice 1: Define Causation from Manipulation

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4 !21!! ∃1$≠ $2345 $1 5 ≠ 2345 $1 5

* 5 # >412? X

Y

Choice 1: Define Direct Causation from Intervention

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Choice 2: Causal Markov Axiom

If G is a causal graph, and P a probability distribution over the variables in G, then in P: every variable V is independent of its non- effects, conditional on its immediate causes.

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'*-

%% %;!! ! % ; %% %; !!

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%% %;!! !

% ; E || S | I E = Exposure to Chicken Pox I = Infected S = Symptoms S I E

Markov Cond.

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&!! %%
YF || LC | S

Sm oking (S) Y ellow Fingers (Y F) Lung C ancer (LC)

Markov Cond.

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. .

X

3 | X 2

X

1

X

2

X

3

X

1 C ausal M arkov A xiom (D

separation)

Independence

A cyclic C ausal G raph

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A.

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A 2 52

X3 T X2 X1 X3 T X2 X1 I

P(X3 | X2) ≠ P(X3 | X2, X1) X3 _||_ X1 | X2 P(X3 | X2 set= ) = P(X3 | X2 set=, X1) X3 _||_ X1 | X2 set=

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C. D3.

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. .

Background Knowledge

X2 before X3
no unmeasured common causes

X3 | X2 X

1 Independence

Data

Statistical Inference

X2 X

3

X

1

Equivalence Class of Causal Graphs

X2 X3 X1 X2 X3 X1

Discovery Algorithm Causal Markov Axiom (D-separation)

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'( A>B

7* - ) I & &9 ) ! %%! 9

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9:

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X2

X1 X2 X1 X2 X1 X4 X3 X2 X1

Possible Edges Example

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4>2

X2

X1 X2 X1 X1 → X2 in some members of the equivalence class, and X2 → X1 in

thers.

X1 → X2 (X1 is a cause of X2) in every member of the equivalence class. X2 X1 X1 and X2 are not adjacent in any member of the equivalence class

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X2

X4 X3 X1 X2 X4 X3 Represents Pattern X1 X2 X4 X3 X1

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::

X2 X1 X2 X1 X2 X1 X2 There is a latent common cause of X1 and X2 No set d-separates X2 and X1 X1 is a cause of X2 X2 is not an ancestor of X1 X1 X2 X1 X1 and X2 are not adjacent

What PAG edges mean.

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::

X 2

X 3 X 1 X 2 X 3 Represents PAG X 1 X 2 X 3 X 1 X 2 X 3 T 1 X 1 X 2 X 3 X 1 etc. T 1 T 1 T2 Graphical Models --11/29/06 42

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E'2

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ρ412 5, !!4 <33<23F

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E'2

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:- β 5,⇐ ∃⊆ F14 <33<23 ∃⊆ F14 <33<23 β 5, H∃⊆ F14 <33<23 don’t know *S

;

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'2>)3

X2 Y X1 True Model

b2 = 0 b1≠ 0

X1 _||_ Y | X2 X2 _||_ Y | X1

Don’t know

H∃⊆ >4?X1 _||_ Y | S ∃⊆ >4?X2 _||_ Y | {X1}

β2 = 0

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'2>)3

X2 Y X3 X1 T1 True Model T2 b1≠ 0

H∃⊆ >414?14 <33<23 4 <33<23>414? 4 <33<23>414?

b2≠ 0 b3≠ 0

4 <33<23>414?

DK

∃⊆ >414?14 <33<23>4?

β2 = 0 DK

H∃⊆ >414?14 <33<23

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'2>)3

X2 Y X3 X1 T1 True Model T2

X2 Y X3 X1 PAG

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