Structure Learning Using Causation Rules

Structure Learning Using Causation Rules

Raanan Yehezkel

PAML Lab. Journal Club

March 13, 2003

Main References

• Pearl, J., Verma, T., A Theory of Inferred Causation, Proceedings of the Second International Conference of Representation and Reasoning, San Francisco. 1991.

• Spirtes, P., Glymour, C., Scheines, R., Causation Prediction and Search, second edition, 2000, MIT Press.

Taken from Judea Pearl web-site

Simpson’s “Paradox”

The sure thing principle (Savage, 1954)

Let a, b be two alternative acts of any sort, and let G be any event.

If you would definitely prefer b to a, either knowing that the event G obtained, or knowing that the event G did not obtain, then you definitely prefer b to a.

Taken from Judea Pearl web-site

New treatment is preferred for male group (G).

New treatment is preferred for female group (G’).

=> New treatment is preferred.

Simpson’s “Paradox”Local Success Rate

G = male patients G’ = female patients

Old 5% (50/1000) 50% (5000/10000)

New 10% (1000/10000) 92% (95/100)

Global Success Rateall patients

Old 46% (5050/11000)

New 11% (1095/10100)


• Intuitive way of thinking:

G T

S

P(S,G,T)=P(G) P(T) · P(S|G,T)

P(S=1 | G,T=new) = 0.51

P(S=1 | G,T=old) = 0.27


• The faithful DAG: G T

S

P(S,G,T)=P(G) · P(T | G) · P(S | G,T)

P(S=1 | G,T=new) = 0.11

P(S=1 | G,T=old) = 0.46

Assumptions:

• Directed Acyclic Graph, Bayesian Networks.

• All variables are observable.

• No errors in Conditional Independence test results.

Identifying cause and effect relations

• Statistical data.• Statistical data and temporal information.

Identifying cause and effect relations

• Potential Cause• Genuine Cause• Spurious Association

Intransitive Triplet

• I(C1,C2)

• ~I(C1,E)

• ~I(C2,E)

C1 C2

E

H1 H2

C1 C2

E

H1 H2

C1 C2

E

Potential Cause

X has a potential causal influence on Y if:

• X and Y are dependent in every context.

• ~I(Z,Y|Scontext)

• I(X,Z|Scontext)

X

Y

Z

Genuine Cause

X has a genuine causal influence on Y if:

• Z is a potential cause of X.


• I(Z,Y|X,Scontext)

Z XPotential

Y

Given context S

Given X and context S

Z XPotential

Y

Spurious Association

X and Y are spuriously associated if:

1. ~I(X,Y| Scontext)

2. ~I(Z1,X|Scontext)

3. ~I(Z2,Y|Scontext)

4. I(Z1,Y|Scontext)

5. I(Z2,X|Scontext)

Z1

X Y

From conditions 1,2,4

From conditions 1,3,5Z2

X Y

Genuine Cause with temporal information

X has a genuine causal influence on Y if:

• Z and Scontext precedes X.


• I(Z,Y|X,Scontext)

Z

Y

Given context S

Given X and context SZ

X

Y

Spurious Association with temporal information

X and Y are spuriously associated if:

1. ~I(X,Y|S)

2. X precedes Y.

3. I(Z,Y|Scontext)

4. ~I(Z,X|Scontext)

Z

Y

From conditions 1,2

X

From conditions 1,3,4

X

Y

Algorithms

• Inductive Causation (IC).

• PC.

• Other.

Pearl and Verma, 1991

• For each pair of non-adjacent nodes (X,Y) with a common neighbor C, if C is not in SXY then add arrowheads to C: X C Y.

• For each pair (X,Y) find the set of nodes SXY such that I(X,Y|SXY). If SXY is empty, place an undirected link between X and Y.

• For each pair (X,Y) find the set of nodes SXY such that I(X,Y|SXY). If SXY is empty, place an undirected link between X and Y.

• For each pair of non-adjacent nodes (X,Y) with a common neighbor C, if C is not in SXY then add arrowheads to C: X C Y.

Inductive Causation (IC)


• Recursively:

1. If X-Y and there is a strictly directed path from X to Y then add an arrowhead at Y.

2. If X and Y aren’t adjacent but XC and there is Y-C then direct the link CY.

• Recursively:



• Mark uni-directed links XY if there is some link with an arrow head at X.

Inductive Causation (IC)

• Mark uni-directed links XY if there is some link with an arrow head at X.

Example (IC)

X1 X2

X3 X4 X5

True graph

Example (IC)

X1 X2

X3 X4 X5

For each pair (X,Y) find the set of nodes SXY such that I(X,Y|SXY). If SXY is empty, place an undirected link between X and Y.

Example (IC)

X1 X2

X3 X4 X5

For each pair of non-adjacent nodes (X,Y) with a common neighbor C, if C is not in SXY then add arrowheads to C:

X C Y

Example (IC)

X1 X2

X3 X4 X5

Recursively:



Example (IC)

X1 X2

X3 X4 X5

Mark uni-directed links XY if there is some link with an arrow head at X.

Spirtes and Glymour, 1993

1. Form a complete undirected graph C on vertex set V.

1. Form a complete undirected graph C on vertex set V.

PC


2. n = 0;

3. Repeat

Repeat

• Select an ordered pair X and Y such that:

|Adj(C,X)\{Y}| n, and a subset S such that:

S Adj(C,X)\{Y}, |S| = n

• if: I(X,Y|S) = true, then delete edge(X,Y)

Until all possible sets were tested. n = n + 1.

Until: X,Y, |Adj(C,X)\{Y}| < n.

2. n = 0;

3. Repeat

Repeat

• Select an ordered pair X and Y such that:

|Adj(C,X)\{Y}| n, and a subset S such that:

S Adj(C,X)\{Y}, |S| = n

• if: I(X,Y|S) = true, then delete edge(X,Y)

Until all possible sets were tested. n = n + 1.

Until: X,Y, |Adj(C,X)\{Y}| < n.

PC


4. For each triple of vertices X, Y, Z,

such that edge(X,Z) and edge(Y,Z),

orient X Z Y, if and only if:

Z SXY

PC4. For each triple of vertices X, Y, Z,

such that edge(X,Z) and edge(Y,Z),

orient X Z Y, if and only if:

Z SXY


Mark uni-directed links XY if there is some link with an arrow head at X.

Recursively:



Use Inductive Causation (IC)

Spirtes, Glymour and Scheines. 2000.

Example (PC)

True graph

X5X2

X4

X1

X3

Example (PC)

Form a complete undirected graph C on vertex set V.

X5X2

X4

X1

X3

Example (PC)

n = 0; |SXY| = n

Independencies:

None

X5X2

X4

X1

X3

Example (PC)

n = 1; |SXY| = n

Independencies:

I(X1,X3|X2)

X5X2

X4

X1

X3

I(X1,X4|X2) I(X1,X5|X2) I(X3,X4|X2)

Example (PC)

n = 2; |SXY| = n

Independencies:

I(X2,X5|X3,X4)

X5X2

X4

X1

X3

Example (PC)For each triple of vertices X, Y, Z, such that edge(X,Z) and edge(Y,Z),

orient X Z Y, if and only if: Z SXY

X5X2

X4

X1

X3

D-Separation set:

S3,4={X2}S1,3 = {X2}

• PC* - tests conditional independence between X,Y given a subset S, where

S { [(Adj(X) Adj(Y)] path(X,Y) }• CI test prioritization according to:

for a given variable X, first test those variables Y that are least dependent on X, conditional on those subsets of variables that are most dependent on X.

• PC* - tests conditional independence between X,Y given a subset S, where

S { [(Adj(X) Adj(Y)] path(X,Y) }• CI test prioritization according to:

for a given variable X, first test those variables Y that are least dependent on X, conditional on those subsets of variables that are most dependent on X.

Possible PC improvements(2)

Markov Equivalence

• (Verma and Pearl, 1990). Two casual models are equivalent if and only if their dags have the same links and same set of uncoupled head-to-head nodes (colliders).

Z

X Y

P=P(X)·P(Y)·P(Z|X,Y)

Z

X Y

Z

X Y

P=P(Z)·P(X|Z)·P(Y|Z) = P(Y)·P(X|Z)·P(Z|Y)

• Algorithms such as PC and IC produce a partially directed graphs, which represent a family of Markov equivalent graphs.

• The remaining undirected arcs can be oriented arbitrarily (under DAG restrictions), in order to construct a classifier.

• The main flaw of the IC and PC algorithms, is that they might be unstable in a noisy environment. An error in one CI test for an arc, might lead to an error in other arcs. And one erroneous orientation might lead to other erroneous orientations.

Summery

• Algorithms such as PC and IC produce a partially directed graphs, which represent a family of Markov equivalent graphs.

• The remaining undirected arcs can be oriented arbitrarily (under DAG restrictions), in order to construct a classifier.

• The main flaw of the IC and PC algorithms, is that they might be unstable in a noisy environment. An error in one CI test for an arc, might lead to an error in other arcs. And one erroneous orientation might lead to other erroneous orientations.

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Structure Learning Using Causation Rules