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

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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

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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)

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