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Causal reasoning and inference with causal Bayes nets Alexander Gebharter Duesseldorf Center for Logic and Philosophy of Science Heinrich Heine University Duesseldorf 28.04.2016 Alexander Gebharter (DCLPS) Reasoning & inference with CBNs

  1. Causal Bayes nets Definition ( d -connection/ d -separation) X and Y are d -connected by Z ⊆ V \{ X , Y } if and only if X and Y are connected by a causal path π such that (i) no non-collider on π is in Z , and (ii) every collider on π is in Z or has an effect in Z . X and Y are d -separated by Z iff they are not d -connected by Z . Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 7 / 26

  2. Causal Bayes nets Definition ( d -connection/ d -separation) X and Y are d -connected by Z ⊆ V \{ X , Y } if and only if X and Y are connected by a causal path π such that (i) no non-collider on π is in Z , and (ii) every collider on π is in Z or has an effect in Z . X and Y are d -separated by Z iff they are not d -connected by Z . Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 7 / 26

  3. Causal Bayes nets Definition ( d -connection/ d -separation) X and Y are d -connected by Z ⊆ V \{ X , Y } if and only if X and Y are connected by a causal path π such that (i) no non-collider on π is in Z , and (ii) every collider on π is in Z or has an effect in Z . X and Y are d -separated by Z iff they are not d -connected by Z . Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 7 / 26

  4. Causal Bayes nets Definition ( d -connection/ d -separation) X and Y are d -connected by Z ⊆ V \{ X , Y } if and only if X and Y are connected by a causal path π such that (i) no non-collider on π is in Z , and (ii) every collider on π is in Z or has an effect in Z . X and Y are d -separated by Z iff they are not d -connected by Z . Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 7 / 26

  5. Causal Bayes nets Definition ( d -connection condition) A causal model satisfies the d -connection condition if and only if for all X , Y ∈ V and Z ⊆ V \{ X , Y } : If Dep ( X , Y | Z ) , then X and Y are d -connected by Z . Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 8 / 26

  6. Causal Bayes nets Definition (causal Markov condition) A causal model satisfies the causal Markov condition (CMC) if and only if every X is probabilistically independent of its non-effects conditional on its direct causes. (cf. Spirtes et al., 2000, p. 29) CMC determines the following Markov factorization: n � P ( x 1 , ..., x n ) = P ( x i | par ( X i )) (1) i = 1 Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 9 / 26

  7. Causal Bayes nets Definition (causal Markov condition) A causal model satisfies the causal Markov condition (CMC) if and only if every X is probabilistically independent of its non-effects conditional on its direct causes. (cf. Spirtes et al., 2000, p. 29) CMC determines the following Markov factorization: n � P ( x 1 , ..., x n ) = P ( x i | par ( X i )) (1) i = 1 Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 9 / 26

  8. Causal Bayes nets P ( a , b , c , d , e ) = P ( a ) · P ( b | a ) · P ( c | a ) · P ( d | b , c ) · P ( e | d ) Indep ( B , C | A ) Indep ( C , B | A ) Indep ( D , A |{ B , C } ) Indep ( E , { A , B , C }| D ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 10 / 26

  9. Causal Bayes nets P ( a , b , c , d , e ) = P ( a ) · P ( b | a ) · P ( c | a ) · P ( d | b , c ) · P ( e | d ) Indep ( B , C | A ) Indep ( C , B | A ) Indep ( D , A |{ B , C } ) Indep ( E , { A , B , C }| D ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 10 / 26

  10. Causal Bayes nets The causal Markov condition is assumed to be satisfied by causal models that satisfy the causal sufficiency condition. Definition (causal sufficiency condition) A causal model satisfies the causal sufficiency condition if and only if every common cause C of every pair X , Y ∈ V is in V or is fixed to a certain value c . Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 11 / 26

  11. Causal Bayes nets A causal model that satisfies CMC satisfies the causal faithfulness condition (CFC) if and only if the independencies implied by CMC are all the independencies in the model (cf. Spirtes et al., 2000, p. 31). Generalized: Definition (causal faithfulness condition) A causal model satisfies the causal faithfulness condition if and only if every d -connection implies a probabilistic dependence. (cf. Schurz & Gebharter, 2015, sec. 3.2) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 12 / 26

  12. Causal Bayes nets A causal model that satisfies CMC satisfies the causal faithfulness condition (CFC) if and only if the independencies implied by CMC are all the independencies in the model (cf. Spirtes et al., 2000, p. 31). Generalized: Definition (causal faithfulness condition) A causal model satisfies the causal faithfulness condition if and only if every d -connection implies a probabilistic dependence. (cf. Schurz & Gebharter, 2015, sec. 3.2) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 12 / 26

  13. Causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 13 / 26

  14. Causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 13 / 26

  15. Causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 13 / 26

  16. Causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 13 / 26

  17. Causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 13 / 26

  18. Causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 13 / 26

  19. Causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 13 / 26

  20. Causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 13 / 26

  21. Causal Bayes nets Introduction Causal Bayes nets Intervention and observation Causal reasoning with causal Bayes nets Causal discovery with causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 14 / 26

  22. Causal Bayes nets Introduction Causal Bayes nets Intervention and observation Causal reasoning with causal Bayes nets Causal discovery with causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 14 / 26

  23. Intervention and observation CBNs allow for distinguishing intervention from observation (cf. Pearl, 2009, sec. 1.3.1; Spirtes et al., 2000, sec. 3.7.2). Sprinkler Season Wet Slippery Rain Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 15 / 26

  24. Intervention and observation CBNs allow for distinguishing intervention from observation (cf. Pearl, 2009, sec. 1.3.1; Spirtes et al., 2000, sec. 3.7.2). Sprinkler = on Season Wet Slippery Rain Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 15 / 26

  25. Intervention and observation CBNs allow for distinguishing intervention from observation (cf. Pearl, 2009, sec. 1.3.1; Spirtes et al., 2000, sec. 3.7.2). Sprinkler = on Season Wet Slippery Rain Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 15 / 26

  26. Intervention and observation Introduction Causal Bayes nets Intervention and observation Causal reasoning with causal Bayes nets Causal discovery with causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 16 / 26

  27. Intervention and observation Introduction Causal Bayes nets Intervention and observation Causal reasoning with causal Bayes nets Causal discovery with causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 16 / 26

  28. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation: P ( sl 1 | sp on ) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 17 / 26

  29. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation: P ( sl 1 | sp on ) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 17 / 26

  30. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation: P ( sl 1 | sp on ) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 17 / 26

  31. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation: P ( sl 1 | sp on ) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 17 / 26

  32. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation: P ( sl 1 | sp on ) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 17 / 26

  33. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation: P ( sl 1 | sp on ) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 17 / 26

  34. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation: P ( sl 1 | sp on ) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 17 / 26

  35. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation: P ( sl 1 | sp on ) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 17 / 26

  36. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation: P ( sl 1 | sp on ) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( sp on | se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 17 / 26

  37. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation generalized: � u P ( y , x , u ) P ( y | x ) = w P ( x , w ) , where U = V \{ X , Y } and W = V \{ X } � Note: X and Y can also be sets of variables! Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 18 / 26

  38. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation generalized: � u P ( y , x , u ) P ( y | x ) = w P ( x , w ) , where U = V \{ X , Y } and W = V \{ X } � Note: X and Y can also be sets of variables! Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 18 / 26

  39. Causal reasoning with causal Bayes nets Sprinkler Season Wet Slippery Rain Observation generalized: � u P ( y , x , u ) P ( y | x ) = w P ( x , w ) , where U = V \{ X , Y } and W = V \{ X } � Note: X and Y can also be sets of variables! Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 18 / 26

  40. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention: P ( sl 1 | do ( sp on )) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 19 / 26

  41. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention: P ( sl 1 | do ( sp on )) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 19 / 26

  42. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention: P ( sl 1 | do ( sp on )) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 19 / 26

  43. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention: P ( sl 1 | do ( sp on )) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 19 / 26

  44. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention: P ( sl 1 | do ( sp on )) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 19 / 26

  45. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention: P ( sl 1 | do ( sp on )) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 19 / 26

  46. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention: P ( sl 1 | do ( sp on )) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 19 / 26

  47. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention: P ( sl 1 | do ( sp on )) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 19 / 26

  48. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention: P ( sl 1 | do ( sp on )) = P ( sl 1 , sp on ) P ( sp on ) � P ( sl , sp on ) = P ( sl 1 , sp on , u ) , where U = V \{ Sl , Sp } u � � P ( sl 1 , sp on , u ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we � P ( sp on , w ) , where W = V \{ Sp } P ( sp on ) = w � � P ( sp on , w ) = P ( se ) · P ( ra | se ) · P ( we | sp on , ra ) · P ( sl 1 | we ) u se , ra , we , sl Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 19 / 26

  49. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention generalized: � u P ( y , x , u ) P ( y | do ( x )) = w P ( x , w ) , where U = V \{ X , Y } and W = V \{ Y } � Note: X and Y can also be sets of variables! Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 20 / 26

  50. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention generalized: � u P ( y , x , u ) P ( y | do ( x )) = w P ( x , w ) , where U = V \{ X , Y } and W = V \{ Y } � Note: X and Y can also be sets of variables! Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 20 / 26

  51. Causal reasoning with causal Bayes nets Sprinkler = on Season Wet Slippery Rain Intervention generalized: � u P ( y , x , u ) P ( y | do ( x )) = w P ( x , w ) , where U = V \{ X , Y } and W = V \{ Y } � Note: X and Y can also be sets of variables! Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 20 / 26

  52. Intervention and observation Introduction Causal Bayes nets Intervention and observation Causal reasoning with causal Bayes nets Causal discovery with causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 21 / 26

  53. Intervention and observation Introduction Causal Bayes nets Intervention and observation Causal reasoning with causal Bayes nets Causal discovery with causal Bayes nets Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 21 / 26

  54. Causal discovery There is a multitude of search algorithms for all kinds of causal scenarios available in the literature (e.g., Spirtes et al., 2000). I will present one of these algorithms: the SGS algorithm. SGS presupposes acyclicity as well as the causal Markov condition and the faithfulness condition to hold. Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 22 / 26

  55. Causal discovery There is a multitude of search algorithms for all kinds of causal scenarios available in the literature (e.g., Spirtes et al., 2000). I will present one of these algorithms: the SGS algorithm. SGS presupposes acyclicity as well as the causal Markov condition and the faithfulness condition to hold. Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 22 / 26

  56. Causal discovery There is a multitude of search algorithms for all kinds of causal scenarios available in the literature (e.g., Spirtes et al., 2000). I will present one of these algorithms: the SGS algorithm. SGS presupposes acyclicity as well as the causal Markov condition and the faithfulness condition to hold. Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 22 / 26

  57. Causal discovery SGS algorithm (cf. Spirtes et al., 2000, p. 82) S1: Form the complete undirected graph over vertex set V . S2: Check for every X — Y for which there is a Z ⊆ V \{ X , Y } such that Indep ( X , Y | Z ) , remove the edge between X and Y . S3: For all X — Z — Y (or X − → Z — Y ) without an edge between X and Y : Orient the edges as X − → Z ← − Y iff Dep ( X , Y | M ) holds for all M ⊆ V \{ X , Y } with Z ∈ M . S4: (a) For all X − → Z — Y without an edge between X and Y : Orient Z — Y as Z − → Y . (b) If X − → ... − → Y and X — Y , then orient X — Y as X − → Y . Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 23 / 26

  58. Causal discovery SGS algorithm (cf. Spirtes et al., 2000, p. 82) S1: Form the complete undirected graph over vertex set V . S2: Check for every X — Y for which there is a Z ⊆ V \{ X , Y } such that Indep ( X , Y | Z ) , remove the edge between X and Y . S3: For all X — Z — Y (or X − → Z — Y ) without an edge between X and Y : Orient the edges as X − → Z ← − Y iff Dep ( X , Y | M ) holds for all M ⊆ V \{ X , Y } with Z ∈ M . S4: (a) For all X − → Z — Y without an edge between X and Y : Orient Z — Y as Z − → Y . (b) If X − → ... − → Y and X — Y , then orient X — Y as X − → Y . Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 23 / 26

  59. Causal discovery SGS algorithm (cf. Spirtes et al., 2000, p. 82) S1: Form the complete undirected graph over vertex set V . S2: Check for every X — Y for which there is a Z ⊆ V \{ X , Y } such that Indep ( X , Y | Z ) , remove the edge between X and Y . S3: For all X — Z — Y (or X − → Z — Y ) without an edge between X and Y : Orient the edges as X − → Z ← − Y iff Dep ( X , Y | M ) holds for all M ⊆ V \{ X , Y } with Z ∈ M . S4: (a) For all X − → Z — Y without an edge between X and Y : Orient Z — Y as Z − → Y . (b) If X − → ... − → Y and X — Y , then orient X — Y as X − → Y . Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 23 / 26

  60. Causal discovery SGS algorithm (cf. Spirtes et al., 2000, p. 82) S1: Form the complete undirected graph over vertex set V . S2: Check for every X — Y for which there is a Z ⊆ V \{ X , Y } such that Indep ( X , Y | Z ) , remove the edge between X and Y . S3: For all X — Z — Y (or X − → Z — Y ) without an edge between X and Y : Orient the edges as X − → Z ← − Y iff Dep ( X , Y | M ) holds for all M ⊆ V \{ X , Y } with Z ∈ M . S4: (a) For all X − → Z — Y without an edge between X and Y : Orient Z — Y as Z − → Y . (b) If X − → ... − → Y and X — Y , then orient X — Y as X − → Y . Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 23 / 26

  61. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  62. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  63. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  64. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  65. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  66. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  67. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  68. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  69. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  70. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  71. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  72. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  73. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  74. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

  75. Causal discovery Step 1 Step 2 Step 3 Step 4 A & B : Indep ( A , B ) A & D : Indep ( A , D | C ) Indep ( A , D |{ B , C } ) B & D : Indep ( B , D | C ) Indep ( B , D |{ A , C } ) Alexander Gebharter (DCLPS) Reasoning & inference with CBNs 28.04.2016 24 / 26

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