The Roy Model and Pearls Do Calculus: What Do Cannot Do James - - PowerPoint PPT Presentation

Intro DAG Fix Do Exercise Roy Conclusion

The Roy Model and Pearl’s Do Calculus: What “Do” Cannot Do

James Heckman University of Chicago Rodrigo Pinto University of Chicago Econ 312, Spring 2019

James Heckman Roy Does Not

Intro DAG Fix Do Exercise Roy Conclusion

• 1. Causal Effects and the Do-calculus

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Identification of Treatment Effects of a DAG Pearl’s Do-Calculus:

1

Purpose: Identify casual effects from non-experimental data.

James Heckman Roy Does Not

Intro DAG Fix Do Exercise Roy Conclusion

Identification of Treatment Effects of a DAG Pearl’s Do-Calculus:

1

Purpose: Identify casual effects from non-experimental data.

2

Application: Bayesian network structure, i.e., Directed Acyclic Graph (DAG) that represents causal relationships.

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Identification of Treatment Effects of a DAG Pearl’s Do-Calculus:

1

Purpose: Identify casual effects from non-experimental data.

2

Application: Bayesian network structure, i.e., Directed Acyclic Graph (DAG) that represents causal relationships.

3

Tools: Three inference rules that translate graphical relations

• f a DAG into causal independence conditional relations (Pearl

1995, and Pearl 2000).

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Identification of Treatment Effects of a DAG Pearl’s Do-Calculus:

1

Purpose: Identify casual effects from non-experimental data.

2

Application: Bayesian network structure, i.e., Directed Acyclic Graph (DAG) that represents causal relationships.

3

Tools: Three inference rules that translate graphical relations

• f a DAG into causal independence conditional relations (Pearl

1995, and Pearl 2000).

4

Identification method: Iteration of do-calculus rules to generate a function that describes treatment effects statistics as a function of the observed variables only (Tian and Pearl 2002, Tian and Pearl 2003).

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Characteristics of Pearl’s Do-Calculus Completeness If some causal effect of a DAG is identifiable, then there exists a sequence of application of the Do-Calculus rules that can generate a formula that translates causal effects into an equation that only relies on observational quantities (Huang and Valtorta 2006, Shpitser and Pearl 2006). Limitation Only works for DAGs. Does not allow for additional information outside the DAG framework that could generate identification of causal distributions. Only applies to the information content of a DAG.

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• 2. Statistical Tools for DAGs

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Markovian Model A Markovian Model (Tian and Pearl, 2003) is defined by four elements: M =

• N, U, G, P(Vi|pa(Vi))
• where:

1

N = {N1, . . . , Mm} is a set of observed variables;

2

U = {U1, . . . , Un} is a set of unobserved variables;

3

G is a direct acyclic graph with nodes corresponding to the variables Vi in V = N ∪ U;

4

V comprises both observed and unobserved variables.

5

P(Vi|pa(Vi)) is the conditional probability of a variable Vi ∈ V given its parents pa(Vi) ⊂ V .

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Factorization of a Markovian Model Joint Probability Pr(V1, . . . , Vn+m) can be factorized as: Pr(V1, . . . Vn+m) =

• Vi∈V

Pr(Vi|pa(Vi)) Causal Interpretation – Structural Equations Vi = f (pa(Vi)) e.g. Y = f (X, U) Direction: Variables pa(Vi) cause Vi (e.g. X, U cause Y ) and not the contrary. Autonomy: Structural equation f is a deterministic function (y = f (x, u)) that is invariant to changes in x or u (Frisch, 1938).

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Statistical Tools of a DAG Bayesian Networks, Howard and Matheson (1981) Some Notation

1

Parents: Variables pa(Y ) ⊂ V are termed parents of Y ∈ V .

D(Y ) = ∪|V |

j=1pa−j(T), where pa−(k+1)(G) = pa−1(par−k(G)), pa−1(G) = ∪T∈Gpa−1(T)

such that G ⊂ V and pa−1(T) = {Y ∈ V ; T ∈ pa(Y )}.

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Statistical Tools of a DAG Bayesian Networks, Howard and Matheson (1981) Some Notation

1

Parents: Variables pa(Y ) ⊂ V are termed parents of Y ∈ V .

If pa(Y ) = ∅, the Y is not caused by any variable in the model.

D(Y ) = ∪|V |

j=1pa−j(T), where pa−(k+1)(G) = pa−1(par−k(G)), pa−1(G) = ∪T∈Gpa−1(T)

such that G ⊂ V and pa−1(T) = {Y ∈ V ; T ∈ pa(Y )}.

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Intro DAG Fix Do Exercise Roy Conclusion

Statistical Tools of a DAG Bayesian Networks, Howard and Matheson (1981) Some Notation

1

Parents: Variables pa(Y ) ⊂ V are termed parents of Y ∈ V .

If pa(Y ) = ∅, the Y is not caused by any variable in the model. In such cases, Y is termed external variable.

D(Y ) = ∪|V |

j=1pa−j(T), where pa−(k+1)(G) = pa−1(par−k(G)), pa−1(G) = ∪T∈Gpa−1(T)

such that G ⊂ V and pa−1(T) = {Y ∈ V ; T ∈ pa(Y )}.

James Heckman Roy Does Not

Intro DAG Fix Do Exercise Roy Conclusion

Statistical Tools of a DAG Bayesian Networks, Howard and Matheson (1981) Some Notation

1

Parents: Variables pa(Y ) ⊂ V are termed parents of Y ∈ V .

If pa(Y ) = ∅, the Y is not caused by any variable in the model. In such cases, Y is termed external variable.

2

Descendants: Variables D(Y ) ⊂ V that are caused by Y (directly or indirectly) are termed descendants of Y ∈ V .

D(Y ) = ∪|V |

j=1pa−j(T), where pa−(k+1)(G) = pa−1(par−k(G)), pa−1(G) = ∪T∈Gpa−1(T)

such that G ⊂ V and pa−1(T) = {Y ∈ V ; T ∈ pa(Y )}.

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• 3. Fixing versus Conditioning

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Pearl’s Definition of Causal Effects, the Do-operator

The Do-operator is based on the Truncated Factorization of the probability factor of the fixed variable is deleted: Let X ⊂ V : Then Pr(V (x) = v) = Pr(V1 = v1, . . . , Vm+n = vm+n, |do(X) = x) and: Pr(V (x) = v) =

Vi∈V \X P(Vi = vi|pa(Vi))

if v is consistent with x; if v is inconsistent with x.

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Example of the Do-operator

X Z Y

Variables: Y , X, Z Factorization: Pr(Y , X, Z) = Pr(Y |Z, X) Pr(X|Z) Pr(Z) = Pr(Y |X) Pr(X|Z) Pr(Z) Do-operator: Pr(Z, Y |do(X) = x) = Pr(Y |X = x) Pr(Z) Conditional operator: Pr(Y , Z|X = x) = Pr(Y |Z, X = x) Pr(X|Z, X = x) Pr(Z|X = x) = Pr(Y |X = x) Pr(Z|X = x) Do-operator targets variables, not causal links.

Example of the Do-operator

X Y U V

Variables: Y , X, U, V Factorization: Pr(V , U, X, Y ) = Pr(Y |U, X) Pr(X|V ) Pr(U|V ) Pr(V ) Do-operator: Pr(V , U, Y |do(X) = x) = Pr(Y |U, X = x) Pr(U|V ) Pr(V ) Conditional operator: Pr(V , U, Y |X = x) = Pr(Y |U, V , X = x) Pr(U|V , X = x) Pr(V |X = x) = Pr(Y |U, X = x) Pr(U|V ) Pr(V |X = x)

Intro DAG Fix Do Exercise Roy Conclusion

Empirical and Hypothetical Models Heckman and Pinto (2013)

Empirical Model Hypothetical Model

X Y U V

X Y U V X ~

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Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation;

Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters;

Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters;

1

Causal parameters are defined under the hypothetical model;

Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters;

1

Causal parameters are defined under the hypothetical model;

2

Observed data is generated through empirical model;

Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters;

1

Causal parameters are defined under the hypothetical model;

2

Observed data is generated through empirical model;

Distinguish definition from identification;

Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters;

1

Causal parameters are defined under the hypothetical model;

2

Observed data is generated through empirical model;

Distinguish definition from identification;

1

Identification requires to connect the hypothetical and empirical models such that

Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters;

1

Causal parameters are defined under the hypothetical model;

2

Observed data is generated through empirical model;

Distinguish definition from identification;

1

Identification requires to connect the hypothetical and empirical models such that

2

allows us to evaluate causal parameters defined of the hypothetical model using data generated by the empirical model

Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters;

1

Causal parameters are defined under the hypothetical model;

2

Observed data is generated through empirical model;

Distinguish definition from identification;

1

Identification requires to connect the hypothetical and empirical models such that

2

allows us to evaluate causal parameters defined of the hypothetical model using data generated by the empirical model

Versatility: Targets causal links, not variables

Benefits of a Hypothetical Model Formalizes Haavelmo insight of Hypothetical variation; Clarifies the definition of causal parameters;

1

Causal parameters are defined under the hypothetical model;

2

Observed data is generated through empirical model;

Distinguish definition from identification;

1

Identification requires to connect the hypothetical and empirical models such that

2

allows us to evaluate causal parameters defined of the hypothetical model using data generated by the empirical model

Versatility: Targets causal links, not variables Completeness : Automatically generates the Do-calculus (Pinto (2013))

Benefits of a Hypothetical Model Most Important Fixing in the empirical model is translated to statistical conditioning in the hypothetical model: EE(Y (x))

• Causal Operation Empirical Model

= EH(Y | ˜ X = x)

• Statistical Operation Hypothetical Model

do-Operator and Statistical Conditioning Let ˜ X be the hypothetical variable in GH associated with variable X in the empirical model GE, such that ChH( ˜ X) = ChE(X), then: PH(TE \ {X}| ˜ X = x) = PE(TE \ {X}|do(X) = x).

Connecting Hypothetical and Empirical Models

1

L-1: Let W , Z be any disjoint set of variables in TE \ DH( ˜ X) then: PH(W |Z) = PH(W |Z, ˜ X) = PE(W |Z)∀{W , Z} ⊂ TE\DH( ˜ X).

2

T-1: Let W , Z be any disjoint set of variables in TE then: PH(W |Z, X = x, ˜ X = x) = PE(W |Z, X = x) ∀ {W , Z} ⊂ TE.

3

C-1: Let ˜ X be uniformly distributed in the support of X and let W , Z be any disjoint set of variables in TE then: PH(W |Z, X = ˜ X) = PE(W |Z) ∀ {W , Z} ⊂ TE.

4

Matching: Let Z, W be any disjoint set of variables in TE such that, in the hypothetical model, X ⊥ ⊥ W |(Z, ˜ X), then PH(W |Z, ˜ X = x) = PE(W |Z, X = x),

Hypothetical and Empirical Front-door Models

Empirical Model Hypothetical Model

X M U Y

X M U Y X ~ ~

Pa(U) = ∅, Pa(U) = Pa( ˜ X) = ∅, Pa(X) = {U} Pa(X) = {U} Pa(M) = {X} Pa(M) = { ˜ X} Pa(Y ) = {M, U} Pa(Y ) = {M, U}

L-2: In the Front-Door hypothetical model:

1

Y ⊥ ⊥ ˜ X|M,

2

X ⊥ ⊥ M, and

3

Y ⊥ ⊥ ˜ X|(M, X)

Using the Hypothetical Model Framework (Front-door)

PH(Y | ˜ X = x) =

• m∈supp(M)

PH(Y |M = m, ˜ X = x) PH(M = m| ˜ X = x) =

• m∈supp(M)

PH(Y |M = m) PH(M = m| ˜ X = x) =

• m∈supp(M)
• x′∈supp(X)

PH(Y |X = x′, M = m) PH(X = x′|M = m)

• PH(M = m| ˜

X = x) =

• m∈supp(M)
• x′∈supp(X)

PH(Y |X = x′, M = m) PH(X = x′)

• PH(M = m| ˜

X = x) =

• m∈supp(M)
• x′∈supp(X)

PH(Y |X = x′, ˜ X = x′, M = m) PH(X = x′)

• PH(M = m| ˜

X = x) =

• m∈supp(M)
• x′∈supp(X)

PE(Y |M, X = x′)

• by T-1

PE(X = x′)

• by L-1
• PE(M = m|X = x)
• by Matching

. The second equality from (1) Y ⊥ ⊥ ˜ X|M of L-2. The fourth equality from (2) X ⊥ ⊥ M of L-2. The fifth equality from (3) Y ⊥ ⊥ ˜ X|(M, X) of L-2.

Intro DAG Fix Do Exercise Roy Conclusion

• 4. The Do-Calculus (Pearl, 1995)

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Identifying Causal Effects for Markovian Models Tools needed: Rules: A set of graphical/statistical rules that convert expressions of causal inference into probability equations. Complete algorithm: an algorithm based on these rules that, for any causal effect in question, we can generate an expression involving observed conditional probabilities or report that the causal effect is not identifiable.

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Some Notation DAG Notation Let X, Y , Z be arbitrary disjoint sets of variables (nodes) in a causal graph G. GX : DAG that modifies G by deleting the arrows pointing to X. GX : DAG that modifies G by deleting arrows emerging from X. GX,Z : DAG that modifies G by deleting arrows pointing to X and emerging from Z. Probability Notation Probability of Y when X is fixed at x and Z is observed. Pr(Y |do(X) = x, Z) = Pr(Y , Z|do(X) = x) Pr(Z|do(X) = x)

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Example of a DAG G

X Z U Y

This figure represents causal relations between four variables. Arrows represent direct causal relations. Circles represent unobserved variables. Squares represent observed variables The “If” Question DAG defines causal relations among variables. It answers the question if a variable causes or is caused by any other variable.

Example of DAG Notation GX = GZ GZ X Z U Y

X Z U Y

GX,Z GX,Z

X Z U Y X Z U Y

Three Rules of Do-Calculus (Pearl, 2000) Let G be a DAG then for any disjoint sets of variables X, Y , Z, W :

Rule 1: Insertion/deletion of observations Y ⊥ ⊥ Z|(X, W ) under GX ⇒ Pr(Y |do(X), Z, W ) = Pr(Y |do(X), W ) Rule 2: Action/observation exchange Y ⊥ ⊥ Z|(X, W ) under GX,Z ⇒ Pr(Y |do(X), do(Z), W ) = Pr(Y |do(X), Z, W ) Rule 3: Insertion/deletion of actions Y ⊥ ⊥ Z|(X, W ) under GX,Z(W ) ⇒ Pr(Y |do(X), do(Z), W ) = Pr(Y |do(X), W ) where Z(W ) is the set of Z-nodes that are not ancestors of any W -node in GX.

Intro DAG Fix Do Exercise Roy Conclusion

Understanding Rules of Do-Calculus Let G be a DAG then for any disjoint sets of variables X, Y , Z, W : Rule 1: Insertion/deletion of observations If Y ⊥ ⊥ Z|(X, W )

• Statistical Relation

under GX

• Graphic Criterion

then Pr(Y |do(X), Z, W ) = Pr(Y |do(X), W )

• Equivalent Probability Expression

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• 5. Do-Calculus Exercise

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Using the Do-Calculus : Task 1 – Compute Pr(Z|do(X)) X ⊥ ⊥ Z in GX, by Rule 2, Pr(Z|do(X)) = Pr(Z|X). G GX

X Z U Y X Z U Y

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Using the Do-Calculus : Task 2 – Compute Pr(Y |do(Z))

Z ⊥ ⊥ X in GZ, by Rule 3, Pr(X|do(Z)) = Pr(X) Z ⊥ ⊥ Y |X in GZ, by Rule 2, Pr(Y |X, do(Z)) = Pr(Y |X, Z) ∴ Pr(Y |do(Z)) =

• X

Pr(Y |X, do(Z)) Pr(X|do(Z)) =

• X

Pr(Y |X, Z) Pr(X) G GZ GZ X Z U Y

X Z U Y X Z U Y

Using the Do-Calculus : Task 3 – Compute Pr(Y |Z, do(X))

Y ⊥ ⊥ Z|X in GX,Z, by Rule 2, Pr(Y |Z, do(X)) = Pr(Y |do(Z), do(X)) Y ⊥ ⊥ X|Z in GX,Z, by Rule 3, Pr(Y |do(X), do(Z)) = Pr(Y |do(Z)) ∴ Pr(Y |Z, do(X)) = Pr(Y |do(Z), do(X)) = Pr(Y |do(Z)) G GX,Z GX,Z X Z U Y

X Z U Y X Z U Y

Using the Do-Calculus : Task 4 – Compute Pr(Y |do(X)) ∴ Pr(Y |do(X)) =

• Z

Pr(Y |Z, do(X)) Pr(Z|do(X)) =

• Z

Pr(Y |do(Z), do(X))

• Task 3

Pr(Z|do(X)) =

• Z

Pr(Y |do(Z))

• Task 3

Pr(Z|do(X)) =

• Z

X ′

Pr(Y |X ′, Z) Pr(X ′)

• Task 2

Pr(Z|X)

• Task 1

Intro DAG Fix Do Exercise Roy Conclusion

• 6. Do-Calculus and The Roy Model

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Generalized Roy Model The Generalized Roy Model stems from six variables:

1

V: Unobserved confounding variable V not caused by any variable;

2

X: observed pre-treatment variables X caused by V ;

3

Z: instrumental variable Z caused by X;

4

T: treatment choice T that caused by Z, V and X;

5

U: unobserved variable U caused by T, V and X;

6

Y: outcome of interest Y caused by T, U and X.

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Generalized Roy Model

Z T Y U V X

This figure represents causal relations of the Generalized Roy Model. Arrows represent direct causal relations. Circles represent unobserved variables. Squares represent observed variables

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Key Aspects of the Generalized Roy Model

1

T is caused by Z, V ;

2

U mediates the effects of V on Y (that is V causes U);

3

T and U cause Y and

4

Z (instrument) not caused by V , U and does not directly cause Y , U. We are left to examine the cases whether:

1

V causes X (or vice-versa),

2

X causes Z (or vice-versa),

3

X causes T,

4

X causes U,

5

T causes U, and

6

X causes Y . The combinations of all these causal relations generate 144 possible models (Pinto, 2013).

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Intro DAG Fix Do Exercise Roy Conclusion

Key Aspects of the Generalized Roy Model (Pinto, 2013)

Z T Y U V X

Dashed lines denote causal relations that may not exist or, if they exist, the causal direction can go either way. Dashed arrows denote causal relations that may not exist, but, if they exist, the causal direction must comply the arrow direction.

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Marginalizing the Generalized Roy Model We examine the identification of causal effects of the Generalized Roy Model using a simplified model w.l.o.g. Suppress variables X and U. This simplification is usually called marginalization in the DAG literature (Koster (2002), Lauritzen (1996), Wermuth (2011)).

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Marginalizing the Generalized Roy Model G = GZ

X Z U Y

This figure represents causal relations of the Marginalized Roy

• Model. Arrows represent direct causal relations. Circles represent

unobserved variables. Squares represent observed variables Note: Z is exogenous, thus conditioning on Z is equivalent to fixing Z.

Examining the Marginalized Roy Model – 1/4 Y ⊥ ⊥ Z in GX, by Rule 1 Pr(Y |do(X), Z) = Pr(Y |do(X)) Y ⊥ ⊥ Z, in GX,Z, by Rule 3 Pr(Y |do(X), Z) = Pr(Y |do(X)) Y ⊥ ⊥ Z|X in GX,Z, by Rule 2 Pr(Y |do(X), do(Z)) = Pr(Y |do(X), Z) GX = GX,Z = GX,Z

X Z U Y

Examining the Marginalized Roy Model – 2/4 Under GX, Y ⊥ ⊥ X, thus Rule 2 does not apply. Under GX,Z, Y ⊥ ⊥ X|Z, thus Rule 2 does not apply. GX = GX,Z

X Z U Y

Examining the Marginalized Roy Model – 3/4 GZ ⇒ Y ⊥ ⊥ Z, thus by Rule 2 Pr(Y |do(Z)) = Pr(Y |Z). GZ

X Z U Y

Examining the Marginalized Roy Model – 4 of 4 Modifications Under GX,Z, Y ⊥ ⊥ (X, Z), thus Rule 2 does not apply. GX,Z

X Z U Y

Intro DAG Fix Do Exercise Roy Conclusion

• 6. Conclusion

The Do-Calculus applied to the Marginalized Roy Model generates:

1

Pr(Y |do(X), do(Z)) = Pr(Y |do(X), Z) = Pr(Y |do(X)),

2

Pr(Y |do(Z)) = Pr(Y |Z) These relations only corroborate the exogeneity of the instrumental variable Z and are not sufficient to identify Pr(Y |do(X)). Identification of the Roy Model To identify the Roy Model, we make assumption on how Z impacts X, i.e. monotonicity/separability. These assumptions cannot be represented in a DAG. These assumptions are associated with properties of how Z causes X and not only if Z causes X.

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