1 INTRODUCTION TO PROBABILITIES, GRAPHS, AND CAUSAL MODELS

1.1 Introduction to Probability Theory

1.1.1 Why probabilities

1.1.2 Basic concepts in probability theory

1.1.3 Combining predictive and diagnostic supports

1.1.4 Random Variables and Expectations

1.1.5 Conditional independence and graphoids

1.2 Graphs and Probabilities

1.2.1 Graphical Notation and Terminology

1.2.2 Bayesian Networks

1.2.3 The -separation criterion

1.2.4 Inference with Bayesian networks

1.3 Causal Bayesian Networks

1.3.1 Causal networks as oracles for interventions

1.3.2 Causal relationships and their stability

1.4 Functional Causal Models

1.4.1 Structural Equations

1.4.2 Probabilistic predictions in causal models

1.4.3 Interventions and causal effects in functional models

1.4.4 Counterfactuals in functional models

2 A THEORY OF INFERRED CAUSATION

2.1 Introduction

2.2 The Causal Modeling Framework

2.3 Model Preference (Occam's razor)

2.4 Stable Distributions

2.5 Recovering DAG Structures

2.6 Recovering Latent Structures

2.7 Local Criteria for Causal Relations

2.8 Non-Temporal Causation and Statistical Time

2.9 Conclusions

2.9.1 On minimality, Markov, and stability

3 CAUSAL DIAGRAMS AND THE IDENTIFICATION OF CAUSAL EFFECTS

3.1 Introduction

3.2 Intervention in Markovian Models

3.2.1 Graphs as models of interventions

3.2.2 Interventions as variables

3.2.3 Computing the effect of interventions

An example: Process control

3.2.4 Identification in causal models

3.3 Controlling Confounding Bias

3.3.1 The back-door criterion

3.3.2 The front-door criterion

3.3.3 Example: Smoking and the genotype theory

3.4 A Calculus of Intervention

3.4.1 Preliminary notation

3.4.2 Inference rules

3.4.3 Symbolic derivation of causal effects: An example

3.4.4 Causal inference by surrogate experiments

3.5 Graphical Tests of Identifiability

3.5.1 Identifying models

3.5.2 Nonidentifying models

3.6 Discussion

3.6.1 Qualifications and extensions

3.6.2 Diagrams as a mathematical language

3.6.3 Translation from Graphs to Potential Outcomes

4 ACTIONS, PLANS, AND DIRECT EFFECTS

4.1 Introduction

4.1.1 Actions and probabilities

4.1.2 Actions in decision analysis

4.1.3 Actions and counterfactuals

4.1.4 Actions and mechanisms

4.2 Learning to Act by Observing Other Actors

4.3 Conditional Actions and Stochastic Policies

4.4 When is the Effect of an Action Identifiable?

4.4.1 Four graphical conditions

4.4.2 Remarks on efficiency

4.4.3 Deriving a closed-form expression for control queries

4.4.4 Summary

4.5 The Identification of Plans

4.5.1 Motivation

4.5.2 Plan identification: Notation and assumptions

4.5.3 Plan identification: A general criterion

4.5.4 Plan identification: A procedure

4.6 Direct Effects and their Identification

4.6.1 Direct vs. Total Effects:

4.6.2 Direct effects, definition and identification

4.6.3 Example: Sex Discrimination in College Admission

4.6.4 Average direct effects

5 CAUSALITY AND STRUCTURAL MODELS IN THE SOCIAL SCIENCES

Outline

5.1 Introduction

5.1.1 Causality in search of a language

5.1.2 SEM: How its meaning got obscured

5.1.3 Graphs as a mathematical language: An example

5.2 Graphs and Model Testing

5.2.1 The testable implications of structural models

Preliminary notation

-separation and partial correlations

5.2.2 Testing the testable

5.2.3 Model equivalence

Generating equivalent models

The significance of equivalent models

5.3 Graphs and Identifiability

5.3.1 Parameter identification in linear models

5.3.2 Comparison to nonparametric identification

Parametric vs. nonparametric models: An example

5.3.3 Causal effects: The interventional interpretation of structural equation models

5.4 Some Conceptual Underpinnings

5.4.1 What do structural parameters really mean?

Structural equations: Operational definition

The structural parameters: Operational definition

The mystical error term: Operational definition

The mystical error term: Conceptual interpretation

5.4.2 Interventional interpretation of effect decomposition

5.4.3 Exogeneity, superexogeneity and other frills

The mystical error term revisited

5.5 Conclusion

6 SIMPSON'S PARADOX, CONFOUNDING, AND COLLAPSIBILITY225

6.1 Simpson's Paradox: An Anatomy

6.1.1 A tale of a non-paradox

6.1.2 A tale of statistical agony

6.1.3 Causality vs. enchangeability

6.1.4 A paradox resolved (or what kind of machine is man)

6.2 Why there is no statistical test for confounding, why many think there is, and why they are almost right

6.2.1 Introduction

6.2.2 Causal and associational definitions

6.3 How the Statistical Criterion Fails

6.3.1 Failing sufficiency due to marginality

6.3.2 Failing sufficiency due to closed-world assumptions

6.3.3 Failing necessity due to barren proxies

6.3.4 Failing necessity due to incidental cancelations

6.4 Stable vs. Incidental Unbiasedness

6.4.1 Motivation

6.4.2 Formal definitions

6.5 Operational Test for Stable no Confounding

6.5.1 Confounding and collapsibility

6.5.2 Confounding vs. confounders

Proof of necessity

6.5.3 Exchangeability vs. structural analysis of confounding

6.6 Conclusions

6.7 Acknowledgment

7 THE LOGIC OF COUNTERFACTUALS AND STRUCTURAL MODELS

7.1 Structural Model Semantics

7.1.1 Definitions: Causal models, actions and counterfactuals

7.1.2 Evaluating counterfactuals: Deterministic analysis

Example-1, The firing squad

Evaluating standard sentences

Evaluating action sentences

Evaluating counterfactuals

7.1.3 Evaluating counterfactuals: Probabilistic analysis

7.1.4 The twin-networks method

7.2 Applications and Interpretation of Structural Models

7.2.1 Policy analysis in linear econometric models: An example

7.2.2 The empirical content of counterfactuals

7.2.3 Causal explanations, utterances, and their interpretation

7.2.4 From mechanisms to actions to causation

Action, Mechanisms and surgeries

Laws vs.facts

Mechanisms and causal relationships306

7.2.5 Simon's Causal ordering

7.3 Axiomatic Characterization

7.3.1 The axioms of structural counterfactuals

7.3.2 Causal effects from counterfactual logic: An example

7.3.3 Axioms of causal relevance

7.4 Structural and Similarity-based Counterfactuals

7.4.1 Relations to Lewis' counterfactuals

Causality from counterfactuals

Structure vs. similarity

Axiomatic comparism

7.4.2 Imaging vs.conditioning

7.4.3 Relation to Neyman-Rubin framework

A language in search of a model

Graphical vs. counterfactual analysis

7.4.4 Exogeneity revisited with counterfactuals or On Errors, graphs and counterfactuals

Exogeneity: Graphical criterion

Instrumental variables: Three definitions

7.5 Structural vs. Probabilistic Causality

7.5.1 The reliance on temporal ordering

7.5.2 The perils of circularity

7.5.3 The closed-world assumption

7.5.4 Singular vs.general causes

8 COMPUTING AND APPLYING COUNTERFACTUALS

8.1 Introduction

8.2 Problem Statement

8.3 Summary of Results

8.4 Example 1: The Effects of Cholestyramine

8.5 Example 2: The Paradoxical Episode of Dr.Pearson

8.5.1 The Story

8.5.2 The Story Unfolds

8.5.3 The Story Explained

8.6 Can the Model Be Validated?

8.7 Concluding Remarks

8.8 Introduction

8.9 The Graphical Model

8.10 The Potential-Response Model

8.11 Learning the Causal Effect

8.12 Gibbs Sampling

8.13 Experimental Results

8.13.1 Simulated Data Example: Identifiable Causal Effect

8.13.2 Real Data Example: Effect of Cholestyramine on Reduced Cholesterol

8.13.3 Real Data Example: Effect of Vitamin A Supplements on Child Mortality

8.14 A Counterfactual Query

8.15 Conclusion

9 PROBABILITIES OF CAUSATION: THREE COUNTERFACTUAL INTERPRETATIONS AND THEIR IDENTIFICATION

9.1 Introduction

9.2 Necessary and Sufficient Causes: Conditions of Identification

9.2.1 Definitions, notation, and basic relationships

9.2.2 Bounds and basic relationships under exogeneity

9.2.3 Identifiability under monotonicity and exogeneity

9.2.4 Identifiability under monotonicity and non-exogeneity

9.3 Examples and Applications

9.3.1 Example-1: Betting against a fair coin

9.3.2 Example-2: The firing squad

9.3.3 Example-3: The effect of radiation on leukemia

9.3.4 Example-4: Legal responsibility from experimental and non-experimental data

9.4 Identification in Non-monotonic Models

9.5 From Necessity and Sufficiency to ``Actual Cause''

9.5.1 The Role of Structural Information

9.5.2 Singular sufficient causes

9.5.3 Example: The Desert Traveler (after P.Suppes)

Necessity and Sufficiency Ignoring Internal Structure

Sufficiency and Necessity given Forensic Reports

Necessity given Forensic reports

9.6 Conclusion

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