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 February, 2020, New York 1

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. • Motivation depends (among other not observed (previous GPA) W factors) on the previous GPA. Z (motivation) 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. • Motivation depends (among other not observed (previous GPA) W factors) on the previous GPA. • Students get tutoring depending on their Z (motivation) motivation. X (tutoring) 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. • Motivation depends (among other not observed (previous GPA) W factors) on the previous GPA. • Students get tutoring depending on their Z (motivation) motivation. • The GPA at the end of the term is a function of the X Y previous GPA, student’s motivation and getting (GPA) (tutoring) tutoring or not. 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. • Motivation depends (among other not observed (previous GPA) W factors) on the previous GPA. • Students get tutoring depending on their Z (motivation) motivation. • The GPA at the end of the term is a function of the X Y previous GPA, student’s motivation and getting (GPA) (tutoring) tutoring or not. G Natural (current) Regime 3

Motivating example (previous GPA) W Z (motivation) X Y (GPA) (tutoring) Natural (current) Regime 𝒣 4

Motivating example • Using machine learning, and with enough data, a students GPA can be predicted with small error (previous GPA) W given other features i.e., P(y | w, z, x) . Z (motivation) X Y (GPA) (tutoring) Natural (current) Regime 𝒣 4

Motivating example • Using machine learning, and with enough data, a students GPA can be predicted with small error (previous GPA) W given other features i.e., P(y | w, z, x) . • This distribution is a model that reflects the current/ Z (motivation) natural regime, but we are interested in taking decisions to improve the students GPA. X Y (GPA) (tutoring) Natural (current) Regime 𝒣 4

Motivating example • Using machine learning, and with enough data, a students GPA can be predicted with small error (previous GPA) W given other features i.e., P(y | w, z, x) . • This distribution is a model that reflects the current/ Z (motivation) natural regime, but we are interested in taking decisions to improve the students GPA. X Y • Taking decisions amount to intervening the current (GPA) (tutoring) regime. Hence, we are interested in predicting Natural (current) Regime 𝒣 student’s GPA receiving tutoring in a hypothetical (unrealized) reality. 4

Motivating example • Using machine learning, and with enough data, a students GPA can be predicted with small error (previous GPA) W given other features i.e., P(y | w, z, x) . • This distribution is a model that reflects the current/ Z (motivation) natural regime, but we are interested in taking decisions to improve the students GPA. X Y • Taking decisions amount to intervening the current (GPA) (tutoring) regime. Hence, we are interested in predicting Natural (current) Regime 𝒣 student’s GPA receiving tutoring in a hypothetical (unrealized) reality. • This is a causal inference question! 4

Some types of Interventions 5

Some types of Interventions • Hard/atomic : σ X =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 : σ X =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 : σ X =do(X=x) set variable X to a constant value x . 
 (Pearl’s original treatment considered mostly this intervention. ) • Every student gets tutoring. • Conditional : σ X =g( w ) sets the variable X to output the result of a function g that depends on a set of observable variables W . 5

Some types of Interventions • Hard/atomic : σ X =do(X=x) set variable X to a constant value x . 
 (Pearl’s original treatment considered mostly this intervention. ) • Every student gets tutoring. • Conditional : σ X =g( w ) sets the variable X to output the result of a function g that depends on 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 : σ X =do(X=x) set variable X to a constant value x . 
 (Pearl’s original treatment considered mostly this intervention. ) • Every student gets tutoring. • Conditional : σ X =g( w ) sets the variable X to output the result of a function g that depends on a set of observable variables W . • Students get tutoring if and only if they have a low GPA. • Stochastic : σ X =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 : σ X =do(X=x) set variable X to a constant value x . 
 (Pearl’s original treatment considered mostly this intervention. ) • Every student gets tutoring. • Conditional : σ X =g( w ) sets the variable X to output the result of a function g that depends on a set of observable variables W . • Students get tutoring if and only if they have a low GPA. • Stochastic : σ X =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 ra ffl e 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? (previous GPA) W Z (motivation) X Y (GPA) (tutoring) 6

Hard/Atomic Interventions • What if we make tutoring mandatory for every student? (previous GPA) W Z (motivation) X Y (GPA) (tutoring) Natural (current) Regime 𝒣 6

Hard/Atomic Interventions • What if we make tutoring mandatory for every student? (previous GPA) W Intervention do ( X = 1) Z (motivation) Make tutoring mandatory for all students. X Y (GPA) (tutoring) Natural (current) Regime 𝒣 6

Hard/Atomic Interventions • What if we make tutoring mandatory for every student? (previous GPA) (previous GPA) W W Intervention do ( X = 1) Z Z (motivation) Make tutoring (motivation) mandatory for all students. X Y X=1 Y (GPA) (GPA) (tutoring) (tutoring) Natural (current) Regime 𝒣 6

Hard/Atomic Interventions • What if we make tutoring mandatory for every student? (previous GPA) (previous GPA) W W Intervention do ( X = 1) Z Z (motivation) Make tutoring (motivation) mandatory for all students. X Y X=1 Y (GPA) (GPA) (tutoring) (tutoring) 𝒣 X Natural (current) Regime Intervened (hypothesized) Regime 𝒣 6

Hard/Atomic Interventions • What if we make tutoring mandatory for every student? (previous GPA) (previous GPA) W W Intervention do ( X = 1) Z Z (motivation) Make tutoring (motivation) mandatory for all students. X Y X=1 Y (GPA) (GPA) (tutoring) (tutoring) 𝒣 X Natural (current) Regime Intervened (hypothesized) Regime 𝒣 Instead of P(y | X=1) we are reasoning about P(y | do(X=1)), or, more generally, P(y; σ X =do(X=1)) 6

Soft Interventions 7

Soft Interventions • A more realistic, or interesting, type of intervention is, for example, to consider the e ff ect of making tutoring mandatory for students with historically low GPA and only to them, on their current GPA. 7

Soft Interventions • A more realistic, or interesting, type of intervention is, for example, to consider the e ff ect of making tutoring mandatory for students with historically low GPA and only to them, on their current GPA. (previous GPA) W Z (motivation) X Y (GPA) (tutoring) 7