Probabilistic and Causal Inference: The Works of Judea Pearl

Probabilistic and Causal Inference: The Works of Judea Pearl
Contents
Preface
Credits
I INTRODUCTION
	1 Biography of Judea Pearl by Stuart J. Russell
		References
	2 Turing Award Lecture
		References
	3 Interview by Martin Ford
		References
	4 An Interview with Ron Wassertein on How The Book of Why Transforms Statistics
	5 Selected Annotated Bibliography by Judea Pearl
		Search and Heuristics
		Bayesian Networks
		Causality
		Causal, Casual, and Curious
II HEURISTICS
	6 Introduction by Judea Pearl
		References
	7 Asymptotic Properties of Minimax Trees and Game-Searching Procedures
		Abstract
		7.1 The Probability of Winning a Standard h-level Game Tree with Random WIN Positions
		7.2 Game Trees with an Arbitrary Distribution of Terminal Values
		7.3 The Mean Complexity of Solving (h, d, P0)-game
		7.4 Solving, Testing, and Evaluating Game Trees
		7.5 Test and, if Necessary, Evaluate—The SCOUT Algorithm
		7.6 Analysis of SCOUT's Expected Performance
		7.7 On the Branching Factor of the ALPHA–BETA (α–β) procedure
		References
	8 The Solution for the Branching Factor of the Alpha–Beta Pruning Algorithm and its Optimality
		8.1 Introduction
			8.1.1 Informal Description of the α-β Procedure
			8.1.2 Previous Analytical Results
		8.2 Analysis
			8.2.1 An Integral Formula for Nn,d
			8.2.2 Evaluation of Rα-β
		8.3 Conclusions
		References
	9 On the Discovery and Generation of Certain Heuristics
		Abstract
		9.1 Introduction: Typical Uses of Heuristics
			9.1.1 The Traveling Salesman Problem (TSP)
			9.1.2 Some Properties of Heuristics
			9.1.3 Where do these Heuristics Come from?
		9.2 Mechanical Generation of Admissible Heuristics
		9.3 Can a Program Tell an Easy Problem When It Sees One?
		9.4 Conclusions
			9.4.1 Bibliographical and Historical Remarks
		References
III PROBABILITIES
	10 Introduction by Judea Pearl
		References
	11 Reverend Bayes on Inference Engines: A Distributed Hierarchical Approach
		Abstract
		11.1 Introduction
		11.2 Definitions and Nomenclature
		11.3 Structural Assumptions
		11.4 Combining Top and Bottom Evidences
		11.5 Propagation of Information Through the Network
		11.6 A Token Game Illustration
		11.7 Properties of the Updating Scheme
		11.8 A Summary of Proofs
		11.9 Conclusions
		References
	12 Fusion, Propagation, and Structuring in Belief Networks
		Abstract
		12.1 Introduction
			12.1.1 Belief Networks
			12.1.2 Conditional Independence and Graph Separability
			12.1.3 An Outline and Summary of Results
		12.2 Fusion and Propagation
			12.2.1 Autonomous Propagation as a Computational Paradigm
			12.2.2 Belief Propagation in Trees
				12.2.2.1 Data Fusion
				12.2.2.2 Propagation Mechanism
				12.2.2.3 Illustrating the Flow of Belief
				12.2.2.4 Properties of the Updating Scheme
			12.2.3 Propagation in Singly Connected Networks
				12.2.3.1 Fusion Equations
				12.2.3.2 Propagation Equation
			12.2.4 Summary and Extensions for Multiply Connected Networks
		12.3 Structuring Causal Trees
			12.3.1 Causality, Conditional Independence, and Tree Architecture
			12.3.2 Problem Definition and Nomenclature
			12.3.3 Star-Decomposable Triplets
			12.3.4 A Tree-Reconstruction Procedure
			12.3.5 Conclusions and Open Questions
		12.A Appendix A. Derivation of the Updating Rules for Singly Connected Networks
			12.A.1 Updating BEL
			12.A.2 Updating π
			12.A.3 Updating λ
		12.B Appendix B. Conditions for Star-decomposability
		Acknowledgments
		References
	13 GRAPHOIDS: Graph-Based Logic for Reasoning about Relevance Relations Or When Would x Tell You More about y If You Already Know z?
		Abstract
		13.1 Introduction
		13.2 Probabilistic Dependencies and their Graphical Representation
		13.3 GRAPHOIDS
		13.4 Special Graphoids and Open Problems
			13.4.1 Graph-induced Graphoids
			13.4.2 Probabilistic Graphoids
			13.4.3 Correlational Graphoids
		13.5 Conclusions
		References
	14 System Z: A Natural Ordering of Defaults with Tractable Applications to Nonmonotonic Reasoning
		Abstract
		14.1 Description
		14.2 Consequence Relations
		14.3 Illustrations
		14.4 The Maximum Entropy Approach
		14.5 Conditional Entailment
		14.6 Conclusions
		Acknowledgments
		14.I Appendix I: Uniqueness of The Minimal Ranking Function
		14.II Appendix II: Rational Monotony of Admissible Rankings
		References
IV CAUSALITY 1988–2001
	15 Introduction by Judea Pearl
		References
	16 Equivalence and Synthesis of Causal Models
		Abstract
		16.1 Introduction
		16.2 Patterns of Causal Models
		16.3 Embedded Causal Models
		16.4 Applications to the Synthesis of Causal Models
		IC-Algorithm (Inductive Causation)
		Acknowledgments
		References
	17 Probabilistic Evaluation of Counterfactual Queries
		Abstract
		17.1 Introduction
		17.2 Notation
		17.3 Party Example
		17.4 Probabilistic vs. Functional Specification
		17.5 Evaluating Counterfactual Queries
		17.6 Party Again
		17.7 Special Case: Linear-Normal Models
		17.8 Conclusion
		Acknowledgments
		References
	18 Causal Diagrams for Empirical Research (With Discussions)
		Summary
		Some key words
		18.1 Introduction
		18.2 Graphical Models and the Manipulative Account of Causation
			18.2.1 Graphs and Conditional Independence
			18.2.2 Graphs as Models of Interventions
		18.3 Controlling Confounding Bias
			18.3.1 The Back-Door Criterion
			18.3.2 The Front-Door Criteria
		18.4 A Calculus of Intervention
			18.4.1 General
			18.4.2 Preliminary Notation
			18.4.3 Inference Rules
			18.4.4 Symbolic Derivation of Causal Effects: An Example
			18.4.5 Causal Inference by Surrogate Experiments
		18.5 Graphical Tests of Identifiability
			18.5.1 General
			18.5.2 Identifying Models
			18.5.3 Nonidentifying Models
		18.6 Discussion
		Acknowledgments
		18.A Appendix
			Proof of Theorem 18.3
		References
		18.I Discussion of ‘Causal Diagrams for Empirical Research’ by J. Pearl
		18.II Discussion of ‘Causal Diagrams for Empirical Research’ by J. Pearl
		18.III Discussion of ‘Causal Diagrams for Empirical Research’ by J. Pearl
		18.IV Discussion of ‘Causal Diagrams for Empirical Research’ by J. Pearl
		18.V Discussion of ‘Causal Diagrams for Empirical Research’ by J. Pearl
		18.VI Discussion of ‘Causal Diagrams for Empirical Research’ by J. Pearl
			18.VI.A Introduction
			18.VI.B Task 1
				18.VI.B.1 General
				18.VI.B.2 A Causal Model
				18.VI.B.3 Relationship with Pearl's Work
			18.VI.C Task 2
		18.VII Discussion of ‘Causal Diagrams for Empirical Research’ by J. Pearl
			18.VII.A Successful and Unsuccessful Causal Inference: Some Examples
			18.VII.B Warranted Inferences
		18.VIII Discussion of ‘Causal Diagrams for Empirical Research’ by J. Pearl
		18.IX Discussion of ‘Causal Diagrams for Empirical Research’ by J. Pearl
			18.IX.A Introduction
			18.IX.B Ignorability and the Back-Door Criterion
		18.X Rejoinder to Discussions of ‘Causal Diagrams for Empirical Research’
			18.X.A General
			18.X.B Graphs, Structural Equations and Counterfactuals
			18.X.C The Equivalence of Counterfactual and Structural Analyses
			18.X.D Practical Versus Hypothetical Interventions
			18.X.E Intervention as Conditionalisation
			18.X.F Testing Versus using Assumptions
			18.X.G Causation Versus Dependence
			18.X.H Exemplifying Modelling Errors
			18.X.I The Myth of Dangerous Graphs
		Additional References
	19 Probabilities of Causation: Three Counterfactual Interpretations and Their Identification
		Abstract
		19.1 Introduction
		19.2 Structural Model Semantics (A Review)
			19.2.1 Definitions: Causal Models, Actions and Counterfactuals
			19.2.2 Examples
			19.2.3 Relation to Lewis' Counterfactuals
			19.2.4 Relation to Probabilistic Causality
			19.2.5 Relation to Neyman–Rubin Model
		19.3 Necessary and Sufficient Causes: Conditions of Identification
			19.3.1 Definitions, Notations, and Basic Relationships
			19.3.2 Bounds and Basic Relationships under Exogeneity
			19.3.3 Identifiability under Monotonicity and Exogeneity
			19.3.4 Identifiability under Monotonicity and Non-Exogeneity
		19.4 Examples and Applications
			19.4.1 Example 1: Betting against a Fair Coin
			19.4.2 Example 2: The Firing Squad
			19.4.3 Example 3: The Effect of Radiation on Leukemia
			19.4.4 Example 4: Legal Responsibility from Experimental and Nonexperimental Data
		19.5 Identification in Non-Monotonic Models
		19.6 From Necessity and Sufficiency to “Actual Cause”
			19.6.1 The Role of Structural Information
			19.6.2 Singular Sufficient Causes
			19.6.3 Example: The Desert Traveler (after P. Suppes)
				19.6.3.1 Necessity and Sufficiency Ignoring Internal Structure
				19.6.3.2 Sufficiency and Necessity given Forensic Reports
				19.6.3.3 Necessity Given Forensic Reports
		19.7 Conclusion
		19.A Appendix: The Empirical Content of Counterfactuals
		References
	20 Direct and Indirect Effects
		Abstract
		20.1 Introduction
		20.2 Conceptual Analysis
			20.2.1 Direct versus Total Effects
			20.2.2 Descriptive versus Prescriptive Interpretation
			20.2.3 Policy Implications of the Descriptive Interpretation
			20.2.4 Descriptive Interpretation of Indirect Effects
		20.3 Formal Analysis
			20.3.1 Notation
			20.3.2 Controlled Direct Effects (review)
			20.3.3 Natural Direct Effects: Formulation
			20.3.4 Natural Direct Effects: Identification
			20.3.5 Natural Indirect Effects: Formulation
			20.3.6 Natural Indirect Effects: Identification
			20.3.7 General Path-specific Effects
		20.4 Conclusions
		Acknowledgments
		References
V CAUSALITY 2002–2020
	21 Introduction by Judea Pearl
		References
	22 Comment: Understanding Simpson's Paradox
		22.1 The History
		22.2 A Paradox Resolved
			22.2.1 Simpson's Surprise
			22.2.2 Which Scenarios Invite Reversals?
			22.2.3 Making the Correct Decision
		22.3 Armistead's Critique
		22.4 Conclusions
		References
	23 Graphical Models for Recovering Probabilistic and Causal Queries from Missing Data
		Abstract
		23.1 Introduction
		23.2 Missingness Graph and Recoverability
			23.2.1 Recoverability
		23.3 Recovering Probabilistic Queries by Sequential Factorization
		23.4 Recoverability in the Absence of an Admissible Sequence
		23.5 Non-recoverability Criteria for Joint and Conditional Distributions
		23.6 Recovering Causal Queries
			23.6.1 Recovering P(y|do(z)) when Y and Ry are inseparable
		23.7 Attrition
			23.7.1 Recovering Joint Distributions under Simple Attrition
			23.7.2 Recovering Causal Effects under Simple Attrition
		23.8 Related Work
		23.9 Conclusion
		Acknowledgments
		References
		23.A Appendix
			23.A.1 Missingness Process in Figure 23.1
			23.A.2 Testing Compatibility between Underlying and Manifest Distributions
			23.A.3 Proof of Theorem 23.1
			23.A.4 Recovering P(V) when Parents of R belong to Vo U Vm
			23.A.5  Proof of Theorem 23.2
			23.A.6 Example: Recoverability by Theorem 23.2
			23.A.7 Proof of Corollary 23.1
			23.A.8 Proof of Theorem 23.3
			23.A.9 Proof of Corollary 23.2
			23.A.10 Proof of Theorem 23.4
			23.A.11 Proof of Theorem 23.5
			23.A.12 Proof of Theorem 23.6
	24 Recovering from Selection Bias in Causal and Statistical Inference
		Abstract
		24.1 Introduction
			24.1.1 Related Work and Our Contributions
		24.2 Recoverability without External Data
		24.3 Recoverability with External Data
		24.4 Recoverability of Causal Effects
		24.5 Conclusions
		Acknowledgments
		References
	25 External Validity: From Do-Calculus to Transportability Across Populations
		Abstract
		Key words and phrases
		25.1 Introduction: Threats vs. Assumptions
		25.2 Preliminaries: The Logical Foundations of Causal Inference
			25.2.1 Causal Models as Inference Engines
			25.2.2 Assumptions in Nonparametric Models
			25.2.3 Representing Interventions, Counterfactuals and Causal Effects
			25.2.4 Identification, d-Separation and Causal Calculus
			25.2.5 The Rules of do-Calculus
		25.3 Inference Across Populations: Motivating Examples
		25.4 Formalizing Transportability
			25.4.1 Selection Diagrams and Selection Variables
			25.4.2 Transportability: Definitions and Examples
		25.5 Transportability of Causal Effects—A Graphical Criterion
		25.6 Conclusions
		25.A Appendix
		Acknowledgments
		References
	26 Detecting Latent Heterogeneity
		Abstract
		Keywords
		26.1 Introduction
		26.2 Covariate-Induced Heterogeneity
			26.2.1 Assessing Covariate-Induced Heterogeneity
			26.2.2 Special Cases
		26.3 Latent Heterogeneity between the Treated and Untreated
			26.3.1 Two Types of Confounding
			26.3.2 Separating Fixed-Effect from Variable-Effect Bias
		26.4 Three Ways of Detecting Heterogeneity
			26.4.1 Detecting Heterogeneity in Randomized Trials
			26.4.2 Detecting Heterogeneity Through Adjustment
			26.4.3 Detecting Heterogeneity Through Mediating Instruments
		26.5 Example: Heterogeneity in Recruitment
		26.6 Conclusions
		Acknowledgments
		Declaration of Conflicting Interests
		Funding
		References
		Author Biography
		26.A Appendix A (An Extreme Case of Latent Heterogeneity)
		26.B Appendix B (Assessing Heterogeneity in Structural Equation Models)
			26.B.1 The Structural Origin of Counterfactuals
			26.B.2 Illustration
VI CONTRIBUTED ARTICLES
	27 On Pearl's Hierarchy and the Foundations of Causal Inference
		Abstract
		27.1 Introduction
			27.1.1 Roadmap of the Chapter
			27.1.2 Notation
		27.2 Structural Causal Models and the Causal Hierarchy
			27.2.1 Pearl Hierarchy, Layer 1—Seeing
			27.2.2 Pearl Hierarchy, Layer 2—Doing
			27.2.3 Pearl Hierarchy, Layer 3—Imagining Counterfactual Worlds
		27.3 Pearl Hierarchy—A Logical Perspective
		27.4 Pearl Hierarchy—A Graphical Perspective
			27.4.1 Causal Inference via L2-constraints—Markovian Causal Bayesian Networks
			27.4.2 Causal Inference via L2-constraints—Semi-Markovian Causal Bayes Networks
				27.4.2.1 Revisiting Locality in Semi-Markovian Models
				27.4.2.2 CBNs with Latent Variables—Putting All the Pieces Together
				27.4.2.3 Cross-layer Inferences through CBNs with Latent Variables
		27.5 Conclusions
		Acknowledgments
		References
	28 The Tale Wags the DAG
		Abstract
		28.1 Introduction
		28.2 The Ladder of Causation
		28.3 Ground Level: Syntax
		28.4 Rung 1: Seeing
			28.4.1 Qualitative Structure
			28.4.2 Quantitative Structure
			28.4.3 Empirical Assessment
			28.4.4 Functional DAGs
			28.4.5 Downsizing and Upsizing
			28.4.6 Empirical Assessment
		28.5 Rung 2: Doing
			28.5.1 Intervention DAGs
			28.5.2 Augmented DAGs
			28.5.3 Empirical Assessment
			28.5.4 Downsizing and Upsizing
			28.5.5 Functional Intervention DAGs
		28.6 Rung 3: Imagining
		28.7 Conclusion
		References
	29 Instrumental Variables with Treatment-induced Selection: Exact Bias Results
		29.1 Introduction
		29.2 Causal Graphs
		29.3 Instrumental Variables
		29.4 Selection Bias in IV: Qualitative Analysis
		29.5 Selection Bias in IV: Quantitative Analysis
			29.5.1 Selection as a Function of Treatment Alone
			29.5.2 Selection as a Function of a Mediator
			29.5.3 Selection on Treatment and the Unobserved Confounder
		29.6 Conclusion
		29.A Appendix
			29.A.1 Proof of Truncation Bias Expressions
			29.A.2 Proof of Adjustment as Point Truncation (Proposition 29.3)
		References
	30 Causal Models and Cognitive Development
		References
	31 The Causal Foundations of Applied Probability and Statistics
		Abstract
		31.1 Introduction: Scientific Inference is a Branch of Causality Theory
		31.2 Causality is Central Even for Purely Descriptive Goals
		31.3 The Strength of Probabilistic Independence Demands Physical Independence
		31.4 The Superconducting Supercollider of Selection
		31.5 Data and Algorithms are Causes of Reported Results
		31.6 Getting Causality into Statistics by Putting Statistics into Causal Terms from the Start
		31.7 Causation in 20th-century Statistics
		31.8 Causal Analysis versus Traditional Statistical Analysis
		31.9 Relating Causality to Traditional Statistical Philosophies and “Objective” Statistics
		31.10 Discussion
		31.11 Conclusion
		31.A Appendix
			31.A.1 A Counting Measure for the Logical Content of a Finite Exchangeability Assumption
		Acknowledgments
		References
	32 Pearl on Actual Causation
		Abstract
		32.1 Introduction
		32.2 Actual Causation
		32.3 Causal Models and But-for Causation
		32.4 Pre-emption and Lewis
		32.5 Intransitivity and Overdetermination
		32.6 Pearl's Definitions of Actual Causation
		32.7 Pearl's Achievement
		References
	33 Causal Diagram and Social Science Research
		33.1 Graphical Causal Models and Social Science Research
		33.2 Two Applications of Graphical Causal Models
			33.2.1 Causal Inference with Panel Data
			33.2.2 Causal Inference with Interference between Units
		33.3 The Future of Causal Research in the Social Sciences
		References
	34 Causal Graphs for Missing Data: A Gentle Introduction
		34.1 Introduction
		34.2 Missingness Graphs
			34.2.1 Graphical Representation of Missingness Categories
		34.3 Recoverability
			34.3.1 Recoverability in MAR and MCAR Problems
				34.3.1.1 Recoverability of Joint Distribution in MCAR and MAR Models
				34.3.1.2 Recoverability as a Guide for Estimation
			34.3.2 Recoverability in MNAR Problems
				34.3.2.1 Recovering P(X, Y) Given the m-graph G in Figure 34.2(a)
				34.3.2.2 Recovering P(X, Y) Given the m-graph in Figure 34.2(b)
				34.3.2.3 Recovering P(X, Y) Given the m-graph in Figure 34.2(c)
				34.3.2.4 Recovering P(X, Y) Given the m-graph in Figure 34.2(d)
				34.3.2.5 Recovering P(X) Given the m-graph in Figure 34.2(e)
			34.3.3 Non-recoverability
		34.4 Testability
		References
	35 A Note of Appreciation
		35.1 A Magic Square
		35.2 A Magic Shield of David
	36 Causal Models for Dynamical Systems
		Abstract
		36.1 Introduction
			36.1.1 Structural Causal Models with Measurement Noise
			36.1.2 Structural Causal Models with Driving Noise
			36.1.3 Interventions
			36.1.4 Time-dependent Data
		36.2 Chemical Reaction Networks and ODEs
		36.3 Causal Kinetic Models
			36.3.1 Causal Kinetic Models with Measurement Noise
			36.3.2 Causal Kinetic Models with Driving Noise
			36.3.3 Interventions
			36.3.4 Other Causal Models for Dynamical Systems and Related Work
		36.4 Challenges in Causal Inference for ODE-based Systems
		36.5 From Invariance to Causality and Generalizability
		36.6 Conclusions
		Acknowledgments
		References
	37 Probabilistic Programming Languages: Independent Choices and Deterministic Systems
		37.1 Probabilistic Models and Deterministic Systems
		37.2 Possible Worlds Semantics
		37.3 Inference
		37.4 Learning
		37.5 Other Issues
		37.6 Causal Models
		37.7 Some Pivotal References
		37.8 Conclusion
		References
	38 An Interventionist Approach to Mediation Analysis
		38.1 Introduction
		38.2 Approaches to Mediation Based on Counterfactuals Defined in Terms of the Mediator: The CDE and PDE
			38.2.1 Two Hypothetical River Blindness Treatment Studies
			38.2.2 The PDE and CDE in the River Blindness Studies
			38.2.3 Identification of the PDE via the Mediation Formula under the NPSEM-IE for Figure 38.3(a)
			38.2.4 Partial Identification of the PDE Under the FFRCISTG for Figure 38.3(a)
			38.2.5 An Example in Which an FFRCISTG Model Holds, but an NPSEM-IE Does Not
			38.2.6 Testable Versus Untestable Assumptions and Identifiability
		38.3 Interventionist Theory of Mediation
			38.3.1 Interventional Interpretation of the PDE Under an Expanded Graph
			38.3.2 Direct and Indirect Effects via the Expanded Graph
			38.3.3 Expanded Graphs for a Single Treatment
			38.3.4 On the Substantive Relationship between Different Gex  Graphs and Gex
			38.3.5 Generalizations
			38.3.6 Identification of Cross-world Nested Counterfactuals of DAG G under an FFRCISTG Model for its Expanded Graph Gex
		38.4 Path-Specific Counterfactuals
			38.4.1 Conditional Path-specific Distributions
		38.5 Conclusion
		Acknowledgments
		38.A Appendix
			38.A.1 Proof of PDE Bounds under the FFRCISTG Model
			38.A.2 Proof that the PDE is Not Identified in the River Blindness Study
			38.A.3 Detecting Confounding via Interventions on A and S
			38.A.4 Proof of Proposition 38.3
		References
	39 Causality for Machine Learning
		Abstract
		39.1 Introduction
		39.2 The Mechanization of Information Processing
		39.3 From Statistical to Causal Models
			39.3.1 Methods Driven by Independent and Identically Distributed Data
			39.3.2 Structural Causal Models
		39.4 Levels of Causal Modeling
		39.5 Independent Causal Mechanisms
		39.6 Cause–Effect Discovery
		39.7 Half-sibling Regression and Exoplanet Detection
		39.8 Invariance, Robustness, and Semi-supervised Learning
			39.8.1 Semi-supervised Learning
			39.8.2 Adversarial Vulnerability
			39.8.3 Multi-task Learning
			39.8.4 Reinforcement Learning
		39.9 Causal Representation Learning
			39.9.1 Learning Transferable Mechanisms
			39.9.2 Learning Disentangled Representations
			39.9.3 Learning Interventional World Models and Reasoning
		39.10 Personal Notes and Conclusion
		Acknowledgments
		References
	40 Why Did They Do That?
		Abstract
		40.1 Introduction
		40.2 Some Examples
		40.3 Back to the Garden of Eden
		40.4 Decision Theory and Decision Analysis
		40.5 Back Again in the Garden of Eden
		40.6 Conclusion: God's Decision
		References
	41 Multivariate Counterfactual Systems and Causal Graphical Models
		41.1 Introduction
		41.2 Graphs, Non-parametric Structural Equation Models, and the g-/do Operator
			41.2.1 Graphical Models
			41.2.2 Causal Models Associated with DAGs
				41.2.2.1 Non-parametric Structural Equations with Independent Errors
				41.2.2.2 A Less Restrictive Model: Non-parametric Structural Equations with Single-World (FFRCISTG) Independences
			41.2.3 Single-World Intervention Graphs
			41.2.4 Factorization Associated with the SWIG Global Markov Property
			41.2.5 SWIG Representation of the Defining FFRCISTG Assumptions
		41.3 The Potential Outcomes Calculus and Identification
		41.4 Identification in Hidden Variable Causal Models
			41.4.1 Latent Projection ADMGs
			41.4.2 The Identified Splitting Operation in a SWIG
			41.4.3 The Extended ID Algorithm
			41.4.4 Identification of Conditional Interventional Distributions
			41.4.5 Representing Context-specific Independence using SWIGs
		41.5 Conclusion
		Acknowledgments
		41.A Appendix
			41.A.1 Incompleteness of d-Separation in Twin Networks due to Deterministic Relations
			41.A.2 Weaker Causal Models to Which the po-Calculus Also Applies
			41.A.3 Completeness Proofs
		References
	42 Causal Bayes Nets as Psychological Theory
		Abstract
		42.1 The Human Conception of Causality
		42.2 Core Properties
		42.3 The Broader Perspective: The Community of Knowledge
		42.4 Collective Causal Models
		42.5 Conclusion
		Acknowledgments
		References
	43 Causation: Objective or Subjective?
		Abstract
		43.1 Causation: A Bunch of Attitudes
		43.2 The Model Relativity of Causation
		43.3 Laws
		43.4 Probability
		Acknowledgments
		References
	Editors' Biographies
	Index