A Calculus for Stochastic Interventions: Causal Effect - - PowerPoint PPT Presentation

A Calculus for Stochastic Interventions: Causal Effect Identification and Surrogate Experiments

Juan D. Correa and Elias Bareinboim

{jdcorrea, eb}@cs.columbia.edu

1

February, 2020, New York

Outline

2

Outline

• Hard/atomic interventions vs. Soft/non-atomic interventions
• Graphical representation
• Inferences rules for soft interventions (σ-calculus)
• Imperfect surrogate experiments
• Conclusions

2

Motivating example

3

Motivating example

• Consider a tutoring program in place at a certain school.

3

Motivating example

• Consider a tutoring program in place at a certain school.
• For each student, we observe the GPA at the beginning of the term, their

motivation (low, high), whether they got tutoring or not, and their GPA at the end.

3

Motivating example

• Consider a tutoring program in place at a certain school.
• For each student, we observe the GPA at the beginning of the term, their

motivation (low, high), whether they got tutoring or not, and their GPA at the end.

3

W

(previous GPA)

Z

(motivation)

• Motivation depends (among other not observed

factors) on the previous GPA.

Motivating example

• Consider a tutoring program in place at a certain school.
• For each student, we observe the GPA at the beginning of the term, their

motivation (low, high), whether they got tutoring or not, and their GPA at the end.

3

X W

(tutoring) (previous GPA)

Z

(motivation)

• Motivation depends (among other not observed

factors) on the previous GPA.

• Students get tutoring depending on their

motivation.

Motivating example

• Consider a tutoring program in place at a certain school.
• For each student, we observe the GPA at the beginning of the term, their

motivation (low, high), whether they got tutoring or not, and their GPA at the end.

3

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

• Motivation depends (among other not observed

factors) on the previous GPA.

• Students get tutoring depending on their

motivation.

• The GPA at the end of the term is a function of the

previous GPA, student’s motivation and getting tutoring or not.

Motivating example

• Consider a tutoring program in place at a certain school.
• For each student, we observe the GPA at the beginning of the term, their

motivation (low, high), whether they got tutoring or not, and their GPA at the end.

3

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Natural (current) Regime

G

• Motivation depends (among other not observed

factors) on the previous GPA.

• Students get tutoring depending on their

motivation.

• The GPA at the end of the term is a function of the

previous GPA, student’s motivation and getting tutoring or not.

4

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Natural (current) Regime

𝒣

Motivating example

4

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Natural (current) Regime

𝒣

• Using machine learning, and with enough data, a

students GPA can be predicted with small error given other features i.e., P(y | w, z, x).

Motivating example

4

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Natural (current) Regime

𝒣

• Using machine learning, and with enough data, a

students GPA can be predicted with small error given other features i.e., P(y | w, z, x).

• This distribution is a model that reflects the current/

natural regime, but we are interested in taking decisions to improve the students GPA.

Motivating example

4

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Natural (current) Regime

𝒣

• Using machine learning, and with enough data, a

students GPA can be predicted with small error given other features i.e., P(y | w, z, x).

• This distribution is a model that reflects the current/

natural regime, but we are interested in taking decisions to improve the students GPA.

• Taking decisions amount to intervening the current
• regime. Hence, we are interested in predicting

student’s GPA receiving tutoring in a hypothetical (unrealized) reality.

Motivating example

4

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Natural (current) Regime

𝒣

• Using machine learning, and with enough data, a

students GPA can be predicted with small error given other features i.e., P(y | w, z, x).

• This distribution is a model that reflects the current/

natural regime, but we are interested in taking decisions to improve the students GPA.

• Taking decisions amount to intervening the current
• regime. Hence, we are interested in predicting

student’s GPA receiving tutoring in a hypothetical (unrealized) reality.

• This is a causal inference question!

Motivating example

Some types of Interventions

5

Some types of Interventions

• Hard/atomic: σX=do(X=x) set variable X to a constant value x. 


(Pearl’s original treatment considered mostly this intervention. )

5

Some types of Interventions

• Hard/atomic: σX=do(X=x) set variable X to a constant value x. 


(Pearl’s original treatment considered mostly this intervention. )

• Every student gets tutoring.

5

Some types of Interventions

• Hard/atomic: σX=do(X=x) set variable X to a constant value x. 


(Pearl’s original treatment considered mostly this intervention. )

• Every student gets tutoring.
• Conditional: σX=g(w) sets the variable X to output the result of a function g that depends
• n a set of observable variables W.

5

Some types of Interventions

• Hard/atomic: σX=do(X=x) set variable X to a constant value x. 


(Pearl’s original treatment considered mostly this intervention. )

• Every student gets tutoring.
• Conditional: σX=g(w) sets the variable X to output the result of a function g that depends
• n a set of observable variables W.
• Students get tutoring if and only if they have a low GPA.

5

Some types of Interventions

• Hard/atomic: σX=do(X=x) set variable X to a constant value x. 


(Pearl’s original treatment considered mostly this intervention. )

• Every student gets tutoring.
• Conditional: σX=g(w) sets the variable X to output the result of a function g that depends
• n a set of observable variables W.
• Students get tutoring if and only if they have a low GPA.
• Stochastic: σX=P*(x|w) sets the variable X to follow a given probability distribution

conditional on a set of variables W.

5

Some types of Interventions

• Hard/atomic: σX=do(X=x) set variable X to a constant value x. 


(Pearl’s original treatment considered mostly this intervention. )

• Every student gets tutoring.
• Conditional: σX=g(w) sets the variable X to output the result of a function g that depends
• n a set of observable variables W.
• Students get tutoring if and only if they have a low GPA.
• Stochastic: σX=P*(x|w) sets the variable X to follow a given probability distribution

conditional on a set of variables W.

• Students with low GPA enter a raffle for 80% of the spots, other interested students

enter for the remaining 20%.

5

Hard/Atomic Interventions

6

Hard/Atomic Interventions

• What if we make tutoring mandatory for every student?

6

Hard/Atomic Interventions

• What if we make tutoring mandatory for every student?

6

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

Hard/Atomic Interventions

• What if we make tutoring mandatory for every student?

6

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Natural (current) Regime

𝒣

Hard/Atomic Interventions

• What if we make tutoring mandatory for every student?

6

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Intervention do(X = 1)

Make tutoring mandatory for all students.

Natural (current) Regime

𝒣

Hard/Atomic Interventions

• What if we make tutoring mandatory for every student?

6

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

X=1 Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Intervention do(X = 1)

Make tutoring mandatory for all students.

Natural (current) Regime

𝒣

Hard/Atomic Interventions

• What if we make tutoring mandatory for every student?

6

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

X=1 Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Intervention do(X = 1)

Make tutoring mandatory for all students.

Natural (current) Regime Intervened (hypothesized) Regime

𝒣 𝒣X

Hard/Atomic Interventions

• What if we make tutoring mandatory for every student?

6

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

X=1 Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Intervention do(X = 1)

Make tutoring mandatory for all students.

Instead of P(y | X=1) we are reasoning about P(y | do(X=1)), or, more generally, P(y; σX=do(X=1))

Natural (current) Regime Intervened (hypothesized) Regime

𝒣 𝒣X

Soft Interventions

7

Soft Interventions

• A more realistic, or interesting, type of intervention is, for example, to consider the

effect of making tutoring mandatory for students with historically low GPA and

• nly to them, on their current GPA.

7

Soft Interventions

• A more realistic, or interesting, type of intervention is, for example, to consider the

effect of making tutoring mandatory for students with historically low GPA and

• nly to them, on their current GPA.

7

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

Soft Interventions

• A more realistic, or interesting, type of intervention is, for example, to consider the

effect of making tutoring mandatory for students with historically low GPA and

• nly to them, on their current GPA.

7

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Natural (current) Regime

𝒣

Soft Interventions

• A more realistic, or interesting, type of intervention is, for example, to consider the

effect of making tutoring mandatory for students with historically low GPA and

• nly to them, on their current GPA.

7

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) Intervention σX = 1[W = 1]

Assign tutoring only to students with low GPA.

Natural (current) Regime

𝒣

Soft Interventions

• A more realistic, or interesting, type of intervention is, for example, to consider the

effect of making tutoring mandatory for students with historically low GPA and

• nly to them, on their current GPA.

7

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) σX Intervention σX = 1[W = 1]

Assign tutoring only to students with low GPA.

Natural (current) Regime

𝒣

Soft Interventions

• A more realistic, or interesting, type of intervention is, for example, to consider the

effect of making tutoring mandatory for students with historically low GPA and

• nly to them, on their current GPA.

7

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) σX Intervention σX = 1[W = 1]

Assign tutoring only to students with low GPA.

Natural (current) Regime Intervened (hypothesized) Regime

𝒣σX 𝒣

Soft Interventions

• A more realistic, or interesting, type of intervention is, for example, to consider the

effect of making tutoring mandatory for students with historically low GPA and

• nly to them, on their current GPA.

7

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation) σX Intervention σX = 1[W = 1]

Assign tutoring only to students with low GPA.

P(y ; )

σX

Natural (current) Regime Intervened (hypothesized) Regime

𝒣σX 𝒣

σ-calculus (simplified)

8

σ-calculus (simplified)

• Insertion/deletion of observations:

8

P(y ∣ w, t; σX) = P(y ∣ w; σX) (Y ⊥ T ∣ W) 𝒣σX

if in

σ-calculus (simplified)

• Insertion/deletion of observations:
• Change of regimes under observation:

8

P(y ∣ w, t; σX) = P(y ∣ w; σX) (Y ⊥ T ∣ W) 𝒣σX

if in

P(y ∣ x, w; σX) = P(y ∣ x, w) (Y ⊥ Z ∣ W) 𝒣σXX 𝒣X

if in and

σ-calculus (simplified)

• Insertion/deletion of observations:
• Change of regimes under observation:
• Change of regimes without observations:

8

P(y ∣ w, t; σX) = P(y ∣ w; σX) (Y ⊥ T ∣ W) 𝒣σX

if in

P(y ∣ x, w; σX) = P(y ∣ x, w) (Y ⊥ Z ∣ W) 𝒣σXX 𝒣X

if in and

P(y ∣ w; σX) = P(y ∣ w) (Y ⊥ Z ∣ W) 𝒣σXX(W) 𝒣X(W)

if in and

Using σ-calculus

9

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

𝒣

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

σX

𝒣σX

P(y; σX) = ∑

w,z

P(y|x, w, z; σX)P(x ∣ w, z; σX)P(w, z; σX)

Using σ-calculus

9

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

𝒣

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

σX

𝒣σX

P(y; σX) = ∑

w,z

P(y|x, w, z; σX)P(x ∣ w, z; σX)P(w, z; σX) = ∑

w,z

P(y|w, z)P(x ∣ w; σX)P(w, z; σX) P(x ∣ w; σX)

Rule 1 (X ⊥ Z ∣ W)

𝒣σX

in

Using σ-calculus

9

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

𝒣

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

σX

𝒣σX

P(y; σX) = ∑

w,z

P(y|x, w, z; σX)P(x ∣ w, z; σX)P(w, z; σX) = ∑

w,z

P(y|w, z)P(x ∣ w; σX)P(w, z; σX)

Rule 1 (X ⊥ Z ∣ W)

𝒣σX

in

Using σ-calculus

9

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

𝒣

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

σX

𝒣σX

P(y; σX) = ∑

w,z

P(y|x, w, z; σX)P(x ∣ w, z; σX)P(w, z; σX) = ∑

w,z

P(y|w, z)P(x ∣ w; σX)P(w, z; σX)

Rule 1 (X ⊥ Z ∣ W)

𝒣σX

in

= ∑

w,z

P(y|x, w, z)P(x ∣ w, z; σX)P(w, z; σX) P(y|x, w, z)

Rule 2 (Y ⊥ X ∣ W, Z)

𝒣σXX 𝒣X

in and

Using σ-calculus

9

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

𝒣

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

σX

𝒣σX

P(y; σX) = ∑

w,z

P(y|x, w, z; σX)P(x ∣ w, z; σX)P(w, z; σX) = ∑

w,z

P(y|w, z)P(x ∣ w; σX)P(w, z; σX)

Rule 1 (X ⊥ Z ∣ W)

𝒣σX

in

= ∑

w,z

P(y|x, w, z)P(x ∣ w, z; σX)P(w, z; σX)

Rule 2 (Y ⊥ X ∣ W, Z)

𝒣σXX 𝒣X

in and

Using σ-calculus

9

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

𝒣

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

σX

𝒣σX

P(y; σX) = ∑

w,z

P(y|x, w, z; σX)P(x ∣ w, z; σX)P(w, z; σX) = ∑

w,z

P(y|w, z)P(x ∣ w; σX)P(w, z; σX)

Rule 1 (X ⊥ Z ∣ W)

𝒣σX

in

= ∑

w,z

P(y|x, w, z)P(x ∣ w, z; σX)P(w, z; σX)

Rule 2 (Y ⊥ X ∣ W, Z)

𝒣σXX 𝒣X

in and

= ∑

w,z

P(y|x, w, z)P(x ∣ w; σX)P(w, z) P(w, z)

Rule 3 (W, Z ⊥ X)

𝒣σXX 𝒣X

in and

Using σ-calculus

9

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

𝒣

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

σX

𝒣σX

P(y; σX) = ∑

w,z

P(y|x, w, z; σX)P(x ∣ w, z; σX)P(w, z; σX) = ∑

w,z

P(y|w, z)P(x ∣ w; σX)P(w, z; σX)

Rule 1 (X ⊥ Z ∣ W)

𝒣σX

in

= ∑

w,z

P(y|x, w, z)P(x ∣ w, z; σX)P(w, z; σX)

Rule 2 (Y ⊥ X ∣ W, Z)

𝒣σXX 𝒣X

in and

= ∑

w,z

P(y|x, w, z)P(x ∣ w; σX)P(w, z)

Rule 3 (W, Z ⊥ X)

𝒣σXX 𝒣X

in and

Using σ-calculus

9

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

𝒣

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

σX

𝒣σX

P(y; σX) = ∑

w,z

P(y|x, w, z; σX)P(x ∣ w, z; σX)P(w, z; σX) = ∑

w,z

P(y|w, z)P(x ∣ w; σX)P(w, z; σX)

Rule 1 (X ⊥ Z ∣ W)

𝒣σX

in

= ∑

w,z

P(y|x, w, z)P(x ∣ w, z; σX)P(w, z; σX)

Rule 2 (Y ⊥ X ∣ W, Z)

𝒣σXX 𝒣X

in and

= ∑

w,z

P(y|x, w, z)P(x ∣ w; σX)P(w, z)

Rule 3 (W, Z ⊥ X)

𝒣σXX 𝒣X

in and Defined by σX

Using σ-calculus

9

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

𝒣

X Y W

(tutoring) (GPA) (previous GPA)

Z

(motivation)

σX

𝒣σX

P(y; σX) = ∑

w,z

P(y|x, w, z; σX)P(x ∣ w, z; σX)P(w, z; σX) = ∑

w,z

P(y|w, z)P(x ∣ w; σX)P(w, z; σX)

Rule 1 (X ⊥ Z ∣ W)

𝒣σX

in

= ∑

w,z

P(y|x, w, z)P(x ∣ w, z; σX)P(w, z; σX)

Rule 2 (Y ⊥ X ∣ W, Z)

𝒣σXX 𝒣X

in and

= ∑

w,z

P(y|x, w, z)P(x ∣ w; σX)P(w, z)

Rule 3 (W, Z ⊥ X)

𝒣σXX 𝒣X

in and Defined by σX Estimable from current regime

Surrogate Experiments

10

Surrogate Experiments

• It’s not uncommon that the effect of a certain

intervention is not identifiable (not uniquely computable) from observational data alone whenever unobserved confounders are present.

10

Identifiable?

Input: { P(v) } Query: { P(y; σX) } No

𝒣

Surrogate Experiments

• It’s not uncommon that the effect of a certain

intervention is not identifiable (not uniquely computable) from observational data alone whenever unobserved confounders are present.

• Experiments over a set of surrogate variables Z

may be more accessible to manipulation than the target effect σX, e.g., randomizing diet vs randomizing cholesterol.

10

Identifiable?

Input: { P(v) } Query: { P(y; σX) } No

𝒣

Surrogate Experiments

• It’s not uncommon that the effect of a certain

intervention is not identifiable (not uniquely computable) from observational data alone whenever unobserved confounders are present.

• Experiments over a set of surrogate variables Z

may be more accessible to manipulation than the target effect σX, e.g., randomizing diet vs randomizing cholesterol.

• Those surrogate experiments can be leveraged to

identify the effect of the interventions of interest.

10

Identifiable?

Input: { P(v) } Query: { P(y; σX) } No

𝒣

Identifiable?

Input: { P(v), P(v;σZ1), P(v;σZ2), … } Query: { P(y; σX) } Yes

𝒣

Surrogate Experiments

11

W Y Z R X

Surrogate Experiments

• Input: {P(v), P(v | σZ=P*(Z|X))}

11

W Y Z R X

Surrogate Experiments

• Input: {P(v), P(v | σZ=P*(Z|X))}
• Query: P(y | σX=P*(X|R))

11

W Y Z R X

Surrogate Experiments

• Input: {P(v), P(v | σZ=P*(Z|X))}
• Query: P(y | σX=P*(X|R))

11

W Y Z R X

W Y Z R X

σX

W Y Z R X

σZ

Surrogate Experiments

• Input: {P(v), P(v | σZ=P*(Z|X))}
• Query: P(y | σX=P*(X|R))
• Not identifiable from P(v) alone, but

11

W Y Z R X

W Y Z R X

σX

W Y Z R X

σZ

Surrogate Experiments

• Input: {P(v), P(v | σZ=P*(Z|X))}
• Query: P(y | σX=P*(X|R))
• Not identifiable from P(v) alone, but

11

W Y Z R X

P(y; σX)

W Y Z R X

σX

W Y Z R X

σZ

Surrogate Experiments

• Input: {P(v), P(v | σZ=P*(Z|X))}
• Query: P(y | σX=P*(X|R))
• Not identifiable from P(v) alone, but

11

W Y Z R X

P(y; σX)

W Y Z R X

σX

W Y Z R X

σZ

= ∑

r,w,x,z

P(r)P(x ∣ r; σX)P(z ∣ r, x, w)P(w ∣ r)∑

x′

P(y ∣ r, x′ , z; σZ)P(x′ ∣ r)

Surrogate Experiments

• Input: {P(v), P(v | σZ=P*(Z|X))}
• Query: P(y | σX=P*(X|R))
• Not identifiable from P(v) alone, but

11

W Y Z R X

P(y; σX)

W Y Z R X

σX

W Y Z R X

σZ

From surrogate
 experiment

= ∑

r,w,x,z

P(r)P(x ∣ r; σX)P(z ∣ r, x, w)P(w ∣ r)∑

x′

P(y ∣ r, x′ , z; σZ)P(x′ ∣ r)

Surrogate Experiments

• Input: {P(v), P(v | σZ=P*(Z|X))}
• Query: P(y | σX=P*(X|R))
• Not identifiable from P(v) alone, but

11

W Y Z R X

P(y; σX)

W Y Z R X

σX

W Y Z R X

σZ

Natural regime From surrogate
 experiment

= ∑

r,w,x,z

P(r)P(x ∣ r; σX)P(z ∣ r, x, w)P(w ∣ r)∑

x′

P(y ∣ r, x′ , z; σZ)P(x′ ∣ r)

Surrogate Experiments

• Input: {P(v), P(v | σZ=P*(Z|X))}
• Query: P(y | σX=P*(X|R))
• Not identifiable from P(v) alone, but

11

W Y Z R X

P(y; σX)

W Y Z R X

σX

W Y Z R X

σZ

Natural regime From surrogate
 experiment

= ∑

r,w,x,z

P(r)P(x ∣ r; σX)P(z ∣ r, x, w)P(w ∣ r)∑

x′

P(y ∣ r, x′ , z; σZ)P(x′ ∣ r)

Defined by
 intervention

Summary of the Results

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Summary of the Results

We introduce a set of inference rules called σ-calculus, which generalizes Pearl’s do-calculus, to reason about the effect of general types of

• interventions. Further, we provide a syntactical method for deriving and

verifying claims about such interventions given a causal graph.

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1

Summary of the Results

We introduce a set of inference rules called σ-calculus, which generalizes Pearl’s do-calculus, to reason about the effect of general types of

• interventions. Further, we provide a syntactical method for deriving and

verifying claims about such interventions given a causal graph. We develop an efficient procedure to determine the identifiability of the (conditional) effect of non-atomic interventions from a combination of

• bservational and experimental data given a causal diagram.

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1 2

Proposed Strategy

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Proposed Strategy

Encode qualitative assumptions natural and intervened domain graphically.

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1

Proposed Strategy

Encode qualitative assumptions natural and intervened domain graphically.

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Diagrams annotated with
 nodes. σX

1

Proposed Strategy

Encode qualitative assumptions natural and intervened domain graphically. Find the mechanisms composing the effect of intervention.

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Diagrams annotated with
 nodes. σX

1 2

Proposed Strategy

Encode qualitative assumptions natural and intervened domain graphically. Find the mechanisms composing the effect of intervention. Derive the needed mechanisms from the given distributions.

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Diagrams annotated with
 nodes. σX

1 2 3

Proposed Strategy

Encode qualitative assumptions natural and intervened domain graphically. Find the mechanisms composing the effect of intervention. Derive the needed mechanisms from the given distributions. Construct an estimator from the available data.

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Diagrams annotated with
 nodes. σX

1 2 3 4

Proposed Strategy

Encode qualitative assumptions natural and intervened domain graphically. Find the mechanisms composing the effect of intervention. Derive the needed mechanisms from the given distributions. Construct an estimator from the available data.

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Diagrams annotated with
 nodes. σX Use σ-calculus or equivalent algorithmic procedure.

1 2 3 4

Conclusions

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Conclusions

• σ-calculus allows one to discover and verify, from a causal graph, logical

statements about general interventions suitable to capture real-world situations.

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Conclusions

• σ-calculus allows one to discover and verify, from a causal graph, logical

statements about general interventions suitable to capture real-world situations.

• These rules can be used to identify the effect of interventions from a combination
• f observational and experimental data.

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Conclusions

• σ-calculus allows one to discover and verify, from a causal graph, logical

statements about general interventions suitable to capture real-world situations.

• These rules can be used to identify the effect of interventions from a combination
• f observational and experimental data.
• Our algorithm searches for a reduction of the effect of interest to the set of
• bserved distributions (observational and experimental); if found, it returns a

corresponding mapping expression.

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Thank you!

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