Causal Semantics of Bayesian Networks Jirka Vomlel Institute of Information Theory and Automation Academy of Sciences of the Czech Republic http://www.utia.cz/vomlel Salzburg, 26 February 2010 J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 1 / 22
Preface (J. Pearl, Causality, 2000) Development of Western science is based on two great achievements: the invention of the formal logical system by the Greek philosophers, and the discovery of the possibility to find out causal relationship by systematic experiment during Renaissance. Albert Einstein (1953) J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 2 / 22
Outline Bayesian networks J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 3 / 22
Outline Bayesian networks Observations versus interventions J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 3 / 22
Outline Bayesian networks Observations versus interventions Causal semantics of Bayesian networks J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 3 / 22
Outline Bayesian networks Observations versus interventions Causal semantics of Bayesian networks Latent variables in causal models J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 3 / 22
Outline Bayesian networks Observations versus interventions Causal semantics of Bayesian networks Latent variables in causal models Tractable causal models with latent variables J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 3 / 22
Bayesian network (BN) See file fever1.net in Hugin. J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 4 / 22
Bayesian network (BN) See file fever1.net in Hugin. Acyclic directed graph G = ( V , E ) where V ⊂ { 1 , 2 , . . . , n } is a set of nodes and E is a set of directed edges - formally, a subset of V × V . J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 4 / 22
Bayesian network (BN) See file fever1.net in Hugin. Acyclic directed graph G = ( V , E ) where V ⊂ { 1 , 2 , . . . , n } is a set of nodes and E is a set of directed edges - formally, a subset of V × V . X i , i ∈ V are discrete random variables. J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 4 / 22
Bayesian network (BN) See file fever1.net in Hugin. Acyclic directed graph G = ( V , E ) where V ⊂ { 1 , 2 , . . . , n } is a set of nodes and E is a set of directed edges - formally, a subset of V × V . X i , i ∈ V are discrete random variables. Let pa ( i ) denote the set of nodes that are parents of i in G - formally, pa ( i ) = { j ∈ V : ( j → i ) ∈ E } . J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 4 / 22
Bayesian network (BN) See file fever1.net in Hugin. Acyclic directed graph G = ( V , E ) where V ⊂ { 1 , 2 , . . . , n } is a set of nodes and E is a set of directed edges - formally, a subset of V × V . X i , i ∈ V are discrete random variables. Let pa ( i ) denote the set of nodes that are parents of i in G - formally, pa ( i ) = { j ∈ V : ( j → i ) ∈ E } . Further, let for A ⊆ V symbol X A denotes a set of variables { X j } j ∈ A . J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 4 / 22
Bayesian network (BN) See file fever1.net in Hugin. Acyclic directed graph G = ( V , E ) where V ⊂ { 1 , 2 , . . . , n } is a set of nodes and E is a set of directed edges - formally, a subset of V × V . X i , i ∈ V are discrete random variables. Let pa ( i ) denote the set of nodes that are parents of i in G - formally, pa ( i ) = { j ∈ V : ( j → i ) ∈ E } . Further, let for A ⊆ V symbol X A denotes a set of variables { X j } j ∈ A . For each random variable X i , i ∈ V a conditional probability distribution P ( X i | X pa ( i ) ) is defined. J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 4 / 22
Bayesian network (BN) See file fever1.net in Hugin. Acyclic directed graph G = ( V , E ) where V ⊂ { 1 , 2 , . . . , n } is a set of nodes and E is a set of directed edges - formally, a subset of V × V . X i , i ∈ V are discrete random variables. Let pa ( i ) denote the set of nodes that are parents of i in G - formally, pa ( i ) = { j ∈ V : ( j → i ) ∈ E } . Further, let for A ⊆ V symbol X A denotes a set of variables { X j } j ∈ A . For each random variable X i , i ∈ V a conditional probability distribution P ( X i | X pa ( i ) ) is defined. Then the joint probability distribution defined by a Bayesian network is � P ( X V ) = P ( X i | X pa ( i ) ) . i ∈ V J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 4 / 22
Conditional independence Let C ⊂ V denote the set with indexes corresponding to variables whose state is known. J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 5 / 22
Conditional independence Let C ⊂ V denote the set with indexes corresponding to variables whose state is known. Definition (Path blocked by evidence) A path in a acyclic directed graph is blocked by a set C if there is a node n ∈ V in the path such that J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 5 / 22
Conditional independence Let C ⊂ V denote the set with indexes corresponding to variables whose state is known. Definition (Path blocked by evidence) A path in a acyclic directed graph is blocked by a set C if there is a node n ∈ V in the path such that the arrows do not meet head-to-head in n and n ∈ C or J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 5 / 22
Conditional independence Let C ⊂ V denote the set with indexes corresponding to variables whose state is known. Definition (Path blocked by evidence) A path in a acyclic directed graph is blocked by a set C if there is a node n ∈ V in the path such that the arrows do not meet head-to-head in n and n ∈ C or the arrows meet head-to-head in n and neither n nor any of its descendants belong to C . J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 5 / 22
Conditional independence Let C ⊂ V denote the set with indexes corresponding to variables whose state is known. Definition (Path blocked by evidence) A path in a acyclic directed graph is blocked by a set C if there is a node n ∈ V in the path such that the arrows do not meet head-to-head in n and n ∈ C or the arrows meet head-to-head in n and neither n nor any of its descendants belong to C . Definition (Conditional independence) Let A , B , C be pairwise disjoint subsets of V . X A is independent of X B given X C ( X A ⊥ ⊥ X B | X C ) iff all paths between A and B are blocked by set C . J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 5 / 22
Conditional independence Let C ⊂ V denote the set with indexes corresponding to variables whose state is known. Definition (Path blocked by evidence) A path in a acyclic directed graph is blocked by a set C if there is a node n ∈ V in the path such that the arrows do not meet head-to-head in n and n ∈ C or the arrows meet head-to-head in n and neither n nor any of its descendants belong to C . Definition (Conditional independence) Let A , B , C be pairwise disjoint subsets of V . X A is independent of X B given X C ( X A ⊥ ⊥ X B | X C ) iff all paths between A and B are blocked by set C . See file fever1.net in Hugin again. J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 5 / 22
BN example: Removing Kidney stones (Charig et al., 1986) Let V = { X 1 , X 2 , X 3 } where X 1 is Stones’ size taking values s (small) and l (large), J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 6 / 22
BN example: Removing Kidney stones (Charig et al., 1986) Let V = { X 1 , X 2 , X 3 } where X 1 is Stones’ size taking values s (small) and l (large), X 2 is Treatment taking values A (all open procedures) and B (percutaneous nephrolithotomy), J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 6 / 22
BN example: Removing Kidney stones (Charig et al., 1986) Let V = { X 1 , X 2 , X 3 } where X 1 is Stones’ size taking values s (small) and l (large), X 2 is Treatment taking values A (all open procedures) and B (percutaneous nephrolithotomy), X 3 is Success taking values 0 (False) and 1 (True). J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 6 / 22
BN example: Removing Kidney stones (Charig et al., 1986) Let V = { X 1 , X 2 , X 3 } where X 1 is Stones’ size taking values s (small) and l (large), X 2 is Treatment taking values A (all open procedures) and B (percutaneous nephrolithotomy), X 3 is Success taking values 0 (False) and 1 (True). J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 6 / 22
BN example: Removing Kidney stones (Charig et al., 1986) Let V = { X 1 , X 2 , X 3 } where X 1 is Stones’ size taking values s (small) and l (large), X 2 is Treatment taking values A (all open procedures) and B (percutaneous nephrolithotomy), X 3 is Success taking values 0 (False) and 1 (True). P ( X 1 , X 2 , X 3 ) = P ( X 3 | X 1 , X 2 ) · P ( X 2 | X 1 ) · P ( X 1 ) J. Vomlel (´ UTIA AVˇ CR) Causal Semantics of Bayesian Networks 26/Feb/2010 6 / 22
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