Identifiability of Restricted Structural Equation Models Networks: Processes and Causality, Menorca Jonas Peters 1 , 2 J. Mooij 3 , D. Janzing 2 , B. Sch¨ olkopf 2 , R. Tanase 1 , P. B¨ uhlmann 1 1 Seminar for Statistics, ETH Z¨ urich, Switzerland 2 MPI for Intelligent Systems, T¨ ubingen, Germany 3 Radboud University, Nijmegen, Netherlands 3rd September 2012 Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 1 / 30
What is the Problem? Random variables: X : water temperature of Mediterranean Sea Y : # networks and causality related workshops in Cala Galdana Z : # scientists on Menorca What is the causal structure? Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 2 / 30
What is the Problem? Random variables: X : water temperature of Mediterranean Sea Y : # networks and causality related workshops in Cala Galdana Z : # scientists on Menorca What is the causal structure? Understand the (physical) process in more detail. Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 2 / 30
What is the Problem? Random variables: X : water temperature of Mediterranean Sea Y : # networks and causality related workshops in Cala Galdana Z : # scientists on Menorca What is the causal structure? Understand the (physical) process in more detail. Intervene: Organize workshop in Cala Galdana! Go swimming! Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 2 / 30
What is the Problem? Random variables: X : water temperature of Mediterranean Sea Y : # networks and causality related workshops in Cala Galdana Z : # scientists on Menorca What is the causal structure? Understand the (physical) process in more detail. Intervene: Organize workshop in Cala Galdana! Go swimming! Use observational data! Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 2 / 30
What is the Problem? ? observed iid data − → causal DAG G 0 from P ( X 1 , . . . , X 5 ) X 4 X 5 3 . 4 1 . 7 − 2 . 4 · · · X 1 X 2 X 3 − 0 . 2 7 . 0 − 1 . 2 · · · X 2 X 3 − 0 . 1 4 . 3 − 0 . 7 · · · X 4 0 . 3 5 . 8 0 . 3 · · · X 1 X 5 3 . 5 1 . 9 − 1 . 9 · · · Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 3 / 30
Relating Causal Graph and Joint Distribution X 2 X 1 X 4 X 3 1 Markov Condition: X 1 ⊥ ⊥ X 4 | { X 2 , X 3 } X 2 ⊥ ⊥ X 3 | { X 1 } ( d -separation ⇒ cond. independence) 2 Faithfulness: no more (no d -separation ⇒ no cond. independence) Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 4 / 30
Relating Causal Graph and Joint Distribution X 2 X 1 X 4 X 3 1 Markov Condition: X 1 ⊥ ⊥ X 4 | { X 2 , X 3 } X 2 ⊥ ⊥ X 3 | { X 1 } ( d -separation ⇒ cond. independence) 2 Faithfulness: no more (no d -separation ⇒ no cond. independence) Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 4 / 30
Relating Causal Graph and Joint Distribution X 2 X 1 X 4 X 3 1 Markov Condition: X 1 ⊥ ⊥ X 4 | { X 2 , X 3 } X 2 ⊥ ⊥ X 3 | { X 1 } ( d -separation ⇒ cond. independence) 2 Faithfulness: no more (no d -separation ⇒ no cond. independence) Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 4 / 30
PC Algorithm G ′′ independence Faithfuln. G ′ X 1 ⊥ ⊥ X 2 X 2 ⊥ ⊥ X 3 Markov tests X 1 ⊥ ⊥ X 4 | { X 3 } G X 1 X 1 ⊥ ⊥ X 2 | { X 3 } X 2 ⊥ ⊥ X 3 | { X 1 } X 2 X 3 P ( X 1 , ..., X 4 ) X 4 X 1 = f 1 ( N 1 ) X 2 = f 2 ( N 2 ) unique? l a i v i r X 3 = f 3 ( X 1 , N 3 ) t X 4 = f 4 ( X 2 , X 3 , N 4 ) Method: PC [Spirtes et al., 2001] N i jointly independent 1 Find all (cond.) independences from the data. 2 Select the DAG(s) that corresponds to these independences. Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 5 / 30
PC Algorithm G ′′ independence Faithfuln. G ′ X 1 ⊥ ⊥ X 2 X 2 ⊥ ⊥ X 3 Markov tests X 1 ⊥ ⊥ X 4 | { X 3 } G X 1 X 1 ⊥ ⊥ X 2 | { X 3 } X 2 ⊥ ⊥ X 3 | { X 1 } X 2 X 3 P ( X 1 , ..., X 4 ) X 4 X 1 = f 1 ( N 1 ) X 2 = f 2 ( N 2 ) unique? l a i v i r X 3 = f 3 ( X 1 , X 2 , N 3 ) t X 4 = f 4 ( X 2 , X 3 , N 4 ) Method: PC [Spirtes et al., 2001] N i jointly independent 1 Find all (cond.) independences from the data. 2 Select the DAG(s) that corresponds to these independences. Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 5 / 30
PC Algorithm G ′′ independence Faithfuln. G ′ X 1 ⊥ ⊥ X 2 X 2 ⊥ ⊥ X 3 Markov tests X 1 ⊥ ⊥ X 4 | { X 3 } G X 1 X 1 ⊥ ⊥ X 2 | { X 3 } X 2 ⊥ ⊥ X 3 | { X 1 } X 2 X 3 P ( X 1 , ..., X 4 ) X 4 X 1 = f 1 ( N 1 ) X 2 = f 2 ( N 2 ) unique? l a i v i r X 3 = f 3 ( X 1 , X 2 , N 3 ) t X 4 = f 4 ( X 2 , X 3 , N 4 ) Method: PC [Spirtes et al., 2001] N i jointly independent 1 Find all (cond.) independences from the data. Be smart. 2 Select the DAG(s) that corresponds to these independences. Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 5 / 30
Relating Causal Graph and Joint Distribution The PC algorithm makes very few assumptions. Can we gain something by making more/different assumptions? Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 6 / 30
Relating Causal Graph and Joint Distribution PC assumptions: assumption Strong strong Faithfulness Faithfulness assumption weak Markov Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 7 / 30
Relating Causal Graph and Joint Distribution New assumptions: assumption Strong strong Faithfulness Restricted SEM Faithfulness assumption weak Causal SEM Markov Minimality Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 8 / 30
Relating Causal Graph and Joint Distribution New assumptions: assumption Strong strong Faithfulness Restricted SEM Faithfulness assumption weak Causal SEM Markov Minimality Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 9 / 30
Causal Minimality Causal Minimality is a weak form of faithfulness: Definition Let G 0 be the true causal graph. If P ( X 1 , . . . , X p ) is not Markov to any proper subgraph of G 0 , causal minimality is satisfied. � “Each arrow does something.” Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 10 / 30
Violation of Causal Minimality X 2 X 1 X 4 X 3 1 Markov Condition: X 2 ⊥ ⊥ X 3 | { X 1 } X 1 ⊥ ⊥ X 4 | { X 2 , X 3 } ( d -separation ⇒ cond. independence) 2 Faithfulness: no more (no d -separation ⇒ no cond. independence) Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 11 / 30
Violation of Causal Minimality X 2 X 1 X 4 X 3 1 Markov Condition: X 2 ⊥ ⊥ X 3 | { X 1 } X 1 ⊥ ⊥ X 4 | { X 2 , X 3 } ( d -separation ⇒ cond. independence) X 4 ⊥ ⊥ X 3 | { X 2 } 2 Faithfulness: no more (no d -separation ⇒ no cond. independence) Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 11 / 30
Violation of Faithfulness X 2 X 1 X 4 X 3 1 Markov Condition: X 2 ⊥ ⊥ X 3 | { X 1 } X 1 ⊥ ⊥ X 4 | { X 2 , X 3 } ( d -separation ⇒ cond. independence) X 1 ⊥ ⊥ X 4 2 Faithfulness: no more (no d -separation ⇒ no cond. independence) Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 11 / 30
Relating Causal Graph and Joint Distribution New assumptions: assumption Strong strong Faithfulness Restricted SEM Faithfulness assumption weak Causal SEM Markov Minimality Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 12 / 30
Structural Equation Models The joint distribution P ( X 1 , . . . , X p ) satisfies a Structural Equation Model (SEM) with graph G 0 if X i = f i ( PA i , N i ) 1 ≤ i ≤ p with PA i being the parents of X i in G 0 . The N i are required to be jointly independent. Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 13 / 30
The Alternative Route G ′′ independence G ′ X 1 ⊥ ⊥ X 2 Faithfuln. X 2 ⊥ ⊥ X 3 Markov tests X 1 ⊥ ⊥ X 4 | { X 3 } G X 1 X 1 ⊥ ⊥ X 2 | { X 3 } X 2 ⊥ ⊥ X 3 | { X 1 } X 2 X 3 P ( X 1 , ..., X 4 ) X 4 X 1 = f 1 ( N 1 ) X 2 = f 2 ( N 2 ) unique? a l i v i r t X 3 = f 3 ( X 1 , N 3 ) X 4 = f 4 ( X 2 , X 3 , N 4 ) N i jointly independent Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 14 / 30
The Alternative Route G ′′ independence G ′ X 1 ⊥ ⊥ X 2 Faithfuln. X 2 ⊥ ⊥ X 3 Markov tests X 1 ⊥ ⊥ X 4 | { X 3 } G X 1 X 1 ⊥ ⊥ X 2 | { X 3 } X 2 ⊥ ⊥ X 3 | { X 1 } X 2 X 3 P ( X 1 , ..., X 4 ) X 4 X 1 = f 1 ( N 1 ) X 2 = f 2 ( N 2 ) unique? a l i v i r t X 3 = f 3 ( X 1 , N 3 ) X 4 = f 4 ( X 2 , X 3 , N 4 ) N i jointly independent Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 14 / 30
Relating Causal Graph and Joint Distribution New assumptions: assumption Strong strong Faithfulness Restricted SEM Faithfulness assumption weak Causal SEM Markov Minimality Jonas Peters (ETH Z¨ urich) Identifiability of Restricted SEMs 3rd September 2012 15 / 30
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