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EQUIVALENCE OF SEM, POTENTIAL OUTCOMES AND CAUSAL GRAPHICAL MODELSAMIT SHARMAPOSTDOCTORAL RESEARCHERMICROSOFT
http://www.amitsharma.in@amt_shrma
WHAT IS CAUSALITY?
• Debatable, from the times of Aristotle and Hume.
• Practical definition:
• Interventionist causality: X causes Y if changing X leads to a change in Y, keeping everything else constant.
CAUSALITY IS MEANINGLESS WITHOUT A MODEL
• “Keeping everything else constant” requires knowing what everything else is.
• Demand increases price is valid in most economies. So seems a universal causal law.
• … except in a fully regulated economy.
• Model: Explicit specification of “everything else” that can affect causal estimate.
WITHOUT A MODEL, EVEN EXPERIMENTS DO NOT TELL YOU ANYTHING ABOUT THE FUTURE
• A/B experiments study the past.
• Provide a counterfactual answer.
• But we want to use the results for the future.
• Model: The world stays the same between:
• When the experiment was run, and
• When its results will be applied.
HOW MIGHT WE SPECIFY A MODEL?
• By qualitative knowledge about how the world works.
Encouragement Effort Outcome
HOW MIGHT WE SPECIFY A MODEL?
• By writing equations about how the world works.
• E.g. F = ma
• Encouragement (Z)• Effort (X)• Outcome (Y)
HOW MIGHT WE SPECIFY A MODEL?
• By thinking about the different worlds that changing the causal variable creates (inspired by a randomized experiment).
• Effort (X)• Outcome (Y)
• Encouragement (Z)
THREE MAJOR FRAMEWORKS FOR SPECIFYING A CAUSAL MODEL
• Causal Graphical model
• Structural Equation Model
• Potential Outcomes Framework
Encouragement Effort Outcome
ALL THREE ARE EQUIVALENT
• A theorem in one is a theorem in another (See Pearl [2009]).
• So what’s the problem?
• Different disciplines prefer one over another.
• Misconceptions abound about the frameworks.
• In general, no unified causal inference course in major universities.
A HISTORICAL TOUR OF CAUSALITY
• 1850s: John Snow uses a natural experiment to detect causal connection between water and cholera.
• 1910s: Buoyed by triumphs in physics, Bertrand Russell argues that causality is irrevelant.
• 1920s: Sewall and Philip Wright develop path diagrams and simultaneous equation modelling (SEM) for determining supply or demand from price and quantity.
• 1920s: Neyman uses potential outcomes to analyze experiments.
• 1930s: Ronald Fisher popularizes the randomized experiment.
A HISTORICAL TOUR OF CAUSALITY
• 1960s: Blalock and Duncan solve path diagrams using regression equations.
• 1960-now?: Age of regression.
• Path diagrams lose their original causal interpretation.
• SEMs, Path diagrams and regression become entangled.
• 1970s: Rubin builds on potential outcomes framework.
• Becomes popular with social scientists.
• 1980s: Pearl builds on SEM framework.
• Starting to become popular with computer scientists.
EQUIVALENCE OF GRAPHICAL MODELS AND SEM
Encouragement(Z)
Effort(X)
Outcome(Y)
P(X, Y, Z ) = P(Y|X) P(X|Z) P(Z)
EQUIVALENCE OF GRAPHICAL MODELS AND SEM
Encouragement(Z)
Effort(X)
Outcome(Y)
P(Y|do(X)) = P(Y|X) Effect =
EQUIVALENCE OF POTENTIAL OUTCOMES AND SEM
Encouragement(Z)
Effort(X)
Outcome(Y)
EQUIVALENCE OF POTENTIAL OUTCOMES AND SEM
Encouragement(Z)
Effort(X)
Outcome(Y)
Effect = Effect =
INSTRUMENTAL VARIABLES IN ALL THREE FRAMEWORKS
IV BY GRAPHICAL MODEL
Encouragement(Z)
Effort(X)
Outcome(Y)
Unobserved Confounders
(U)
Average Causal Effect =
IV BY STRUCTURAL EQUATIONS
Local average Causal effect =
IV IN POTENTIAL OUTCOMES
• Assumptions:
Local average Causal Effect =
BEST PRACTICES
• A randomized experiment, or a problem with few variables:
• Use Potential outcomes framework: Simple and practical.
• An observational study with many confounders, or any problem with many variables:
• Use graphical models to encode causal assumptions.
• If functional forms unknown, use graphical criteria or do-calculus to estimate causal effect.
• Else, a domain where functional forms are known or can be approximated• Use structural equation models to solve for causal effects, based on the
causal graph.
