Causal inference finds applications in a variety of fields, ranging from the social to the biomedical sciences. Over the years, different fields have approached the problem separately and have developed varying methods and terminologies. On the surface, these look very different and proponents of these different languages of causal inference seem divided on their validity. However, at their heart, the key frameworks---structural equation modeling, causal graphical models, and potential outcomes---are all valid tools to solve the same problem and for most common applications, provide equivalent results. In this talk, I will review these three frameworks and provide high-level comparisons between them, hoping to resolve some of the debates around these different frameworks.

1. EQUIVALENCE OF SEM,
POTENTIAL OUTCOMES AND
CAUSAL GRAPHICAL MODELS
AMIT SHARMA
POSTDOCTORAL RESEARCHER
MICROSOFT
http://www.amitsharma.in
@amt_shrma

2. 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.

3. 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.

4. 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.

5. HOW MIGHT WE SPECIFY A MODEL?
• By qualitative knowledge about how the world works.
Encouragement Effort Outcome

6. HOW MIGHT WE SPECIFY A MODEL?
• By writing equations about how the world works.
• E.g. F = ma
• Encouragement (Z)
• Effort (X)
• Outcome (Y)
𝑦 ∶= 𝛽𝑥 + 𝜖 𝑦
𝑥 ∶= 𝛾𝑧 + 𝜖 𝑥

7. 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)
𝑌𝑥1 𝑜𝑟 𝑌𝑥2 … .
• Encouragement (Z)
𝑌𝑥1𝑧1 𝑜𝑟 𝑌𝑥2𝑧1 𝑜𝑟 𝑌𝑥1𝑧2 𝑜𝑟 𝑌𝑥2𝑧2 … .

8. THREE MAJOR FRAMEWORKS FOR
SPECIFYING A CAUSAL MODEL
• Causal Graphical model
• Structural Equation Model
𝑦 ∶= 𝛽𝑥 + 𝜖 𝑦
𝑥 ∶= 𝛾𝑧 + 𝜖 𝑥
• Potential Outcomes Framework
𝑌𝑥1𝑧1 𝑜𝑟 𝑌𝑥2𝑧1 𝑜𝑟 𝑌𝑥1𝑧2 𝑜𝑟 𝑌𝑥2𝑧2 … .
Encouragement Effort Outcome

9. 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.

10. 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.

11. 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.

12. EQUIVALENCE OF GRAPHICAL
MODELS AND SEM
Encouragement
(Z)
Effort
(X)
Outcome
(Y)
P(X, Y, Z ) = P(Y|X) P(X|Z) P(Z)
𝑦 ∶= 𝑓 𝑥, 𝜖 𝑦
𝑥 ∶= 𝑓(𝑧, 𝜖 𝑥)

13. EQUIVALENCE OF GRAPHICAL
MODELS AND SEM
Encouragement
(Z)
Effort
(X)
Outcome
(Y)
P(Y|do(X)) = P(Y|X)
𝑦 ∶= 𝑓 𝑥, 𝜖 𝑦
Effect =
𝑑𝑦
𝑑𝑥

14. EQUIVALENCE OF POTENTIAL
OUTCOMES AND SEM
Encouragement
(Z)
Effort
(X)
Outcome
(Y)
𝑦 ∶= 𝑓 𝑥, 𝜖 𝑦
𝑥 ∶= 𝑓(𝑧, 𝜖 𝑥)
𝑌𝑥1, 𝑌𝑥2, …
𝑋𝑧1, 𝑋𝑧2, …
𝑌𝑥1𝑧1, 𝑌𝑥1𝑧2, …

15. EQUIVALENCE OF POTENTIAL
OUTCOMES AND SEM
Encouragement
(Z)
Effort
(X)
Outcome
(Y)
Effect = 𝑌𝑥2 − 𝑌𝑥1
𝑦 ∶= 𝑓 𝑥, 𝜖 𝑦
Effect =
𝑑𝑦
𝑑𝑥

16. INSTRUMENTAL
VARIABLES IN
ALL THREE
FRAMEWORKS

17. IV BY GRAPHICAL MODEL
Encouragement
(Z)
Effort
(X)
Outcome
(Y)
Unobserved
Confounders
(U)
𝑃 𝑦 𝑑𝑜 𝑥 =
𝑢
𝑃 𝑦 𝑥, 𝑢 𝑃(𝑢)
Average Causal Effect = 𝑃 𝑦2 𝑑𝑜 𝑥2 − 𝑃 𝑦2 𝑑𝑜 𝑥1
=
𝑢
𝑃 𝑦2 𝑥2, 𝑢 − 𝑃(𝑦2|𝑥1, 𝑢) 𝑃(𝑢)

18. IV BY STRUCTURAL EQUATIONS
𝑦 ∶= 𝛽𝑥 + 𝜖 𝑦
𝑥 ∶= 𝛾𝑧 + 𝜖 𝑥
Local average Causal effect =
𝐶𝑜𝑣(𝑌, 𝑋)
𝐶𝑜𝑣(𝑋,𝑍)

19. IV IN POTENTIAL OUTCOMES
• Assumptions:
• 𝑌𝑥1𝑧1 = 𝑌𝑥1𝑧2
Local average Causal Effect =
𝐸 𝑌 𝑋2 – 𝐸 𝑌 𝑋1
𝐸 𝑋 𝑍2 −𝐸 𝑋 𝑍1

20. 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.

21. THANK YOU!
Amit Sharma
http://www.amitsharma.in
@amt_shrma