Non-parametric causal models Robin J. Evans Thomas S. Richardson - PowerPoint PPT Presentation

Non-parametric causal models Robin J. Evans Thomas S. Richardson Oxford and Univ. of Washington UAI Tutorial 12th July 2015 1 / 44

Structure Part One: Causal DAGs with latent variables Part Two: Statistical Models arising from DAGs with latents 2 / 44

Outline for Part One Intervention distributions The general identification problem Tian’s ID Algorithm Fixing: generalizing marginalizing and conditioning Non-parametric constraints aka Verma constraints 3 / 44

Intervention distributions (I) Given a causal DAG G with distribution: � p ( V ) = p ( v | pa( v )) v ∈ V we wish to compute an intervention distribution via truncated factorization: � p ( V \ X | do( X = x )) = p ( v | pa( v )) . v ∈ V \ X 4 / 44

Example L X M Y p ( X , L , M , Y ) = p ( L ) p ( X | L ) p ( M | X ) p ( Y | L , M ) 5 / 44

Example L L X M Y X M Y p ( X , L , M , Y ) = p ( L ) p ( X | L ) p ( M | X ) p ( Y | L , M ) p ( L , M , Y | do( X = ˜ x )) = p ( L ) × p ( M | ˜ x ) p ( Y | L , M ) 5 / 44

Intervention distributions (II) Given a causal DAG G with distribution: � p ( V ) = p ( v | pa( v )) v ∈ V we wish to compute an intervention distribution via truncated factorization: � p ( V \ X | do( X = x )) = p ( v | pa( v )) . v ∈ V \ X Hence if we are interested in Y ⊂ V \ X then we simply marginalize: � � p ( Y | do( X = x )) = p ( v | pa( v )) . w ∈ V \ ( X ∪ Y ) v ∈ V \ X This is the ‘g-computation’ formula of Robins (1986). 6 / 44

Intervention distributions (II) Given a causal DAG G with distribution: � p ( V ) = p ( v | pa( v )) v ∈ V we wish to compute an intervention distribution via truncated factorization: � p ( V \ X | do( X = x )) = p ( v | pa( v )) . v ∈ V \ X Hence if we are interested in Y ⊂ V \ X then we simply marginalize: � � p ( Y | do( X = x )) = p ( v | pa( v )) . w ∈ V \ ( X ∪ Y ) v ∈ V \ X This is the ‘g-computation’ formula of Robins (1986). Note: p ( Y | do( X = x )) is a sum over a product of terms p ( v | pa( v )). 6 / 44

Example L L X M Y X M Y p ( X , L , M , Y ) = p ( L ) p ( X | L ) p ( M | X ) p ( Y | L , M ) p ( L , M , Y | do( X = ˜ x )) = p ( L ) p ( M | ˜ x ) p ( Y | L , M ) � p ( Y | do( X = ˜ x )) = p ( L = l ) p ( M = m | ˜ x ) p ( Y | L = l , M = m ) l , m Note that p ( Y | do( X = ˜ x )) � = p ( Y | X = ˜ x ). 7 / 44

Example: no effect of M on Y L L X M Y X M Y p ( X , L , M , Y ) = p ( L ) p ( X | L ) p ( M | X ) p ( Y | L , M ) 8 / 44

Example: no effect of M on Y L L X M Y X M Y p ( X , L , M , Y ) = p ( L ) p ( X | L ) p ( M | X ) p ( Y | L ) 8 / 44

Example: no effect of M on Y L L X M Y X M Y p ( X , L , M , Y ) = p ( L ) p ( X | L ) p ( M | X ) p ( Y | L ) p ( L , M , Y | do( X = ˜ x )) = p ( L ) p ( M | ˜ x ) p ( Y | L ) 8 / 44

Example: no effect of M on Y L L X M Y X M Y p ( X , L , M , Y ) = p ( L ) p ( X | L ) p ( M | X ) p ( Y | L ) p ( L , M , Y | do( X = ˜ x )) = p ( L ) p ( M | ˜ x ) p ( Y | L ) � p ( Y | do( X = ˜ x )) = p ( L = l ) p ( M = m | ˜ x ) p ( Y | L = l ) l , m 8 / 44

Example: no effect of M on Y L L X M Y X M Y p ( X , L , M , Y ) = p ( L ) p ( X | L ) p ( M | X ) p ( Y | L ) p ( L , M , Y | do( X = ˜ x )) = p ( L ) p ( M | ˜ x ) p ( Y | L ) � p ( Y | do( X = ˜ x )) = p ( L = l ) p ( M = m | ˜ x ) p ( Y | L = l ) l , m � = p ( L = l ) p ( Y | L = l ) l 8 / 44

Example: no effect of M on Y L L X M Y X M Y p ( X , L , M , Y ) = p ( L ) p ( X | L ) p ( M | X ) p ( Y | L ) p ( L , M , Y | do( X = ˜ x )) = p ( L ) p ( M | ˜ x ) p ( Y | L ) � p ( Y | do( X = ˜ x )) = p ( L = l ) p ( M = m | ˜ x ) p ( Y | L = l ) l , m � = p ( L = l ) p ( Y | L = l ) l = p ( Y ) � = P ( Y | ˜ x ) since X �⊥ ⊥ Y . ‘Correlation is not Causation’. 8 / 44

Example with M unobserved L L X M Y X M Y � p ( Y | do( X = ˜ x )) = p ( L = l ) p ( M = m | ˜ x ) p ( Y | L = l , M = m ) l , m 9 / 44

Example with M unobserved L L X M Y X M Y � p ( Y | do( X = ˜ x )) = p ( L = l ) p ( M = m | ˜ x ) p ( Y | L = l , M = m ) l , m � = p ( L = l ) p ( M = m | ˜ x , L = l ) p ( Y | L = l , M = m , X = ˜ x ) l , m � = p ( L = l ) p ( Y , M = m | L = l , X = ˜ x ) l , m � = p ( L = l ) p ( Y | L = l , X = ˜ x ) . l ⇒ can find p ( Y | do( X = ˜ x )) even if M not observed. This is an example of the ‘back door formula’. 9 / 44

Example with L unobserved L L X M Y X M Y p ( Y | do( X = ˜ x )) 10 / 44

Example with L unobserved L L X M Y X M Y p ( Y | do( X = ˜ x )) � = p ( M = m | do( X = ˜ x )) p ( Y | do( M = m )) m 10 / 44

But with both L and M unobserved.... L X M Y ...we are out of luck! 11 / 44

But with both L and M unobserved.... L X M Y ...we are out of luck! Given P ( X , Y ), absent further assumptions we cannot distinguish: L X Y X M Y 11 / 44

General Identification Question Given: a latent DAG G ( O ∪ H ), where O are observed, H are hidden, and disjoint subsets X , Y ⊆ O . Q: Is p ( Y | do( X )) identified given p ( O )? 12 / 44

General Identification Question Given: a latent DAG G ( O ∪ H ), where O are observed, H are hidden, and disjoint subsets X , Y ⊆ O . Q: Is p ( Y | do( X )) identified given p ( O )? A: Provide either an identifying formula that is a function of p ( O ) or report that p ( Y | do( X )) is not identified. 12 / 44

Latent Projection Can preserve conditional independences and causal coherence with latents using paths. DAG G on vertices V = O ˙ ∪ H , define latent projection as follows: (Verma and Pearl, 1992) 13 / 44

Latent Projection Can preserve conditional independences and causal coherence with latents using paths. DAG G on vertices V = O ˙ ∪ H , define latent projection as follows: (Verma and Pearl, 1992) Whenever there is a path of the form y x · · · h 1 h k add y x 13 / 44

Latent Projection Can preserve conditional independences and causal coherence with latents using paths. DAG G on vertices V = O ˙ ∪ H , define latent projection as follows: (Verma and Pearl, 1992) Whenever there is a path of the form y x · · · h 1 h k add y x Whenever there is a path of the form y x · · · h 1 h k add y x 13 / 44

Latent Projection Can preserve conditional independences and causal coherence with latents using paths. DAG G on vertices V = O ˙ ∪ H , define latent projection as follows: (Verma and Pearl, 1992) Whenever there is a path of the form y x · · · h 1 h k add y x Whenever there is a path of the form y x · · · h 1 h k add y x Then remove all latent variables H from the graph. 13 / 44

ADMGs x z x z − → u project w t t y y 14 / 44

ADMGs x z x z − → u project w t t y y Latent projection leads to an acyclic directed mixed graph (ADMG) 14 / 44

ADMGs x z x z − → u project w t t y y Latent projection leads to an acyclic directed mixed graph (ADMG) Can read off independences with d/m-separation. The projection preserves the causal structure; Verma and Pearl (1992). 14 / 44

Non-parametric causal models Robin J. Evans Thomas S. Richardson - PowerPoint PPT Presentation

Non-parametric causal models Robin J. Evans Thomas S. Richardson Oxford and Univ. of Washington UAI Tutorial 12th July 2015 1 / 44 Structure Part One: Causal DAGs with latent variables Part Two: Statistical Models arising from DAGs with

Semi-parametric and response setup non-parametric approaches to Parametric models

Parametric vs Nonparametric Models Parametric models assume some finite set of parameters .

On estimation of functional causal models: Post - nonlinear causal model as an

Causal dynamics within the food-versus-fuel nexus: parametric and nonparametric causality testing

TOPIC 3: COUNTERFACTUALS & CAUSAL MODELS Dan Lassiter Paris VII Stanford Linguistics

Models for Probability Distributions and Density Functions 1 General Concepts Parametric:

h ( n ) w ( n ) x ( n ) Linear nonparametric vs. parametric models Many researchers use

Causality in Econometrics and Statistics: Structural Models are Causal Models Some Formal

Chapter 3 Parametric Models and Methods Models: Weibull ( includes the exponential model)

High-dimensional causal inference, graphical modeling and structural equation models Peter B

Two Optimal Strategies for Active Learning of Causal Models from Interventions Alain Hauser

Towards AI systems that can build coherent causal models of what they read! 0 State of

Tutorial on Parametric Timed Automata for RT Scheduling Examples of scheduling models Giuseppe

Causal analysis within the framework of structural autoregressive models Alessio Moneta Scuola

More is Less? Non-parametric Language Models and Efficiency Graham Neubig Carnegie Mellon

Non-Parametric Models Review of last class: Decision Tree Learning dealing with the

Rina Dechter Causal Inference in Statistics, A primer, J. Pearl, M Glymur and N. Jewell slides12a

The PLS approach to Generalised Linear Models and Causal Path Modeling: Algorithms and

Automatic Generation of Minimal and Reduced Models for Structured Parametric Dynamical Systems

The Relationship between Heuristic, Causal and Statistical Models of Knowledge, and Big Data D.

Foundations of Causal Discovery Frederick Eberhardt KDD Causality Workshop 2016 Causal Discovery

Combining parametric and nonparametric models for off-policy evaluation Omer Gottesman 1 , Yao Liu

Learning Neural Causal Models From Unknown Interventions Summary The relationship between

Gumbel-Max Structural Causal Models Michael Oberst David Sontag MIT MIT @MichaelOberst

Non-parametric causal models Robin J. Evans Thomas S. Richardson - PowerPoint PPT Presentation

Non-parametric causal models Robin J. Evans Thomas S. Richardson Oxford and Univ. of Washington UAI Tutorial 12th July 2015 1 / 44 Structure Part One: Causal DAGs with latent variables Part Two: Statistical Models arising from DAGs with

Semi-parametric and response setup non-parametric approaches to Parametric models

Parametric vs Nonparametric Models Parametric models assume some finite set of parameters .

On estimation of functional causal models: Post - nonlinear causal model as an

Causal dynamics within the food-versus-fuel nexus: parametric and nonparametric causality testing

TOPIC 3: COUNTERFACTUALS &amp; CAUSAL MODELS Dan Lassiter Paris VII Stanford Linguistics

Models for Probability Distributions and Density Functions 1 General Concepts Parametric:

h ( n ) w ( n ) x ( n ) Linear nonparametric vs. parametric models Many researchers use

Causality in Econometrics and Statistics: Structural Models are Causal Models Some Formal

Chapter 3 Parametric Models and Methods Models: Weibull ( includes the exponential model)

High-dimensional causal inference, graphical modeling and structural equation models Peter B

Two Optimal Strategies for Active Learning of Causal Models from Interventions Alain Hauser

Towards AI systems that can build coherent causal models of what they read! 0 State of

Tutorial on Parametric Timed Automata for RT Scheduling Examples of scheduling models Giuseppe

Causal analysis within the framework of structural autoregressive models Alessio Moneta Scuola

More is Less? Non-parametric Language Models and Efficiency Graham Neubig Carnegie Mellon

Non-Parametric Models Review of last class: Decision Tree Learning dealing with the

Rina Dechter Causal Inference in Statistics, A primer, J. Pearl, M Glymur and N. Jewell slides12a

The PLS approach to Generalised Linear Models and Causal Path Modeling: Algorithms and

Automatic Generation of Minimal and Reduced Models for Structured Parametric Dynamical Systems

The Relationship between Heuristic, Causal and Statistical Models of Knowledge, and Big Data D.

Foundations of Causal Discovery Frederick Eberhardt KDD Causality Workshop 2016 Causal Discovery

Combining parametric and nonparametric models for off-policy evaluation Omer Gottesman 1 , Yao Liu

Learning Neural Causal Models From Unknown Interventions Summary The relationship between

Gumbel-Max Structural Causal Models Michael Oberst David Sontag MIT MIT @MichaelOberst

TOPIC 3: COUNTERFACTUALS & CAUSAL MODELS Dan Lassiter Paris VII Stanford Linguistics