causal models
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Causal Models Brady Neal causalcourse.com <latexit - PowerPoint PPT Presentation

Causal Models Brady Neal causalcourse.com <latexit


  1. <latexit sha1_base64="a6Z9Snb6PnEk9DLry8ApQe6sxH0=">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</latexit> <latexit sha1_base64="frZh0LXvPMBTwOYEetgnVkd7Mx4=">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</latexit> Conditioning vs. intervening Population Subpopulations T = 0 T = 1 / 38 Brady Neal The do -operator 5

  2. <latexit 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<latexit 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Conditioning vs. intervening Population Subpopulations Conditioning T = 0 T = 1 T = 1 T = 0 / 38 Brady Neal The do -operator 5

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<latexit 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Conditioning vs. intervening Population Subpopulations Conditioning Intervening do ( T = 1) T = 0 T = 1 T = 1 do ( T = 0) T = 0 / 38 Brady Neal The do -operator 5

  4. Some notation and terminology / 38 Brady Neal The do -operator 6

  5. <latexit sha1_base64="dwH/2hM6oy74wdvu4D+YuVEcJ6c=">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</latexit> Some notation and terminology Interventional distributions: P ( Y ( t ) = y ) / 38 Brady Neal The do -operator 6

  6. <latexit sha1_base64="dwH/2hM6oy74wdvu4D+YuVEcJ6c=">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</latexit> <latexit sha1_base64="dwH/2hM6oy74wdvu4D+YuVEcJ6c=">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</latexit> Some notation and terminology Interventional distributions: y ) , P ( Y = y | do ( T = t )) P ( Y ( t ) = y ) / 38 Brady Neal The do -operator 6

  7. <latexit sha1_base64="dwH/2hM6oy74wdvu4D+YuVEcJ6c=">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</latexit> <latexit sha1_base64="dwH/2hM6oy74wdvu4D+YuVEcJ6c=">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</latexit> <latexit sha1_base64="dwH/2hM6oy74wdvu4D+YuVEcJ6c=">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</latexit> Some notation and terminology Interventional distributions: y ) , P ( Y = y | do ( T = t )) )) , P ( y | do ( t )) P ( Y ( t ) = y ) / 38 Brady Neal The do -operator 6

  8. <latexit 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<latexit 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<latexit 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<latexit 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Some notation and terminology Interventional distributions: y ) , P ( Y = y | do ( T = t )) )) , P ( y | do ( t )) P ( Y ( t ) = y ) Average treatment effect (ATE): E [ Y | do ( T = 1)] − E [ Y | do ( T = 0)] / 38 Brady Neal The do -operator 6

  9. <latexit 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<latexit 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<latexit 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<latexit 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Some notation and terminology Interventional distributions: y ) , P ( Y = y | do ( T = t )) )) , P ( y | do ( t )) P ( Y ( t ) = y ) Average treatment effect (ATE): E [ Y | do ( T = 1)] − E [ Y | do ( T = 0)] Observational Interventional / 38 Brady Neal The do -operator 6

  10. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Some notation and terminology Interventional distributions: y ) , P ( Y = y | do ( T = t )) )) , P ( y | do ( t )) P ( Y ( t ) = y ) Average treatment effect (ATE): E [ Y | do ( T = 1)] − E [ Y | do ( T = 0)] Observational Interventional P ( Y, T, X ) / 38 Brady Neal The do -operator 6

  11. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Some notation and terminology Interventional distributions: y ) , P ( Y = y | do ( T = t )) )) , P ( y | do ( t )) P ( Y ( t ) = y ) Average treatment effect (ATE): E [ Y | do ( T = 1)] − E [ Y | do ( T = 0)] Observational Interventional P ( Y, T, X ) P ( Y | do ( T = t )) / 38 Brady Neal The do -operator 6

  12. <latexit 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<latexit 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Some notation and terminology Interventional distributions: y ) , P ( Y = y | do ( T = t )) )) , P ( y | do ( t )) P ( Y ( t ) = y ) Average treatment effect (ATE): E [ Y | do ( T = 1)] − E [ Y | do ( T = 0)] Observational Interventional P ( Y, T, X ) P ( Y | do ( T = t )) P ( Y | T = t ) / 38 Brady Neal The do -operator 6

  13. <latexit 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<latexit 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Some notation and terminology Interventional distributions: y ) , P ( Y = y | do ( T = t )) )) , P ( y | do ( t )) P ( Y ( t ) = y ) Average treatment effect (ATE): E [ Y | do ( T = 1)] − E [ Y | do ( T = 0)] Observational Interventional P ( Y, T, X ) P ( Y | do ( T = t )) P ( Y | T = t ) P ( Y | do ( T = t ) , X = x ) / 38 Brady Neal The do -operator 6

  14. <latexit sha1_base64="+DJ3EJYTMyzhKLbocg2dN5vtJsE=">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</latexit> Identifiability Causal Estimand Causal Model Identification Statistical Estimand / 38 Brady Neal The do -operator 7

  15. <latexit sha1_base64="EvBHguW/pkQPfKh/wiE+r9GjSkc=">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</latexit> <latexit sha1_base64="+DJ3EJYTMyzhKLbocg2dN5vtJsE=">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</latexit> Identifiability P ( y | do ( t )) Causal Estimand Causal Model Identification Statistical Estimand / 38 Brady Neal The do -operator 7

  16. <latexit 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<latexit 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<latexit 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Identifiability P ( y | do ( t )) Causal Estimand Causal Model Identification P ( y | t ) Statistical Estimand / 38 Brady Neal The do -operator 7

  17. <latexit 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<latexit 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<latexit 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Identifiability Accessible via experiment P ( y | do ( t )) Causal Estimand Causal Model Identification P ( y | t ) Statistical Estimand / 38 Brady Neal The do -operator 7

  18. <latexit 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Identifiability Accessible via experiment P ( y | do ( t )) Causal Estimand Causal Model Identification P ( y | t ) Statistical Estimand Accessible via observational data / 38 Brady Neal The do -operator 7

  19. <latexit 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<latexit 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Identifiability Accessible via experiment P ( y | do ( t )) Causal Estimand Causal Model Identification No confounding P ( y | t ) Statistical Estimand Accessible via observational data / 38 Brady Neal The do -operator 7

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Identifiability Accessible via experiment P ( y | do ( t )) Causal Estimand Causal Model Identification confounders X P ( y | t ) Statistical Estimand Accessible via observational data / 38 Brady Neal The do -operator 7

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Identifiability Accessible via experiment P ( y | do ( t )) Causal Estimand Causal Model Identification confounders X E X [ P ( y | t, X )] Statistical Estimand Accessible via observational data / 38 Brady Neal The do -operator 7

  22. The do -operator Main assumption: modularity Backdoor adjustment Structural causal models A complete example with estimation / 38 Brady Neal 8 Main assumption: modularity

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  25. Modularity assumption / 38 Brady Neal 10 Main assumption: modularity

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Modularity assumption If we intervene on a node X i , then only the mechanism P ( x i | pa i ) changes. All other mechanisms where remain unchanged. P ( x j | pa j ) i 6 = j / 38 Brady Neal 10 Main assumption: modularity

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<latexit 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Modularity assumption If we intervene on a node X i , then only the mechanism P ( x i | pa i ) changes. All other mechanisms where remain unchanged. P ( x j | pa j ) i 6 = j In other words, the causal mechanisms are modular . / 38 Brady Neal 10 Main assumption: modularity

  28. <latexit 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<latexit 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<latexit sha1_base64="YM8Q81DrJEvndhZycMmhU0qfgs=">ASIHicrVhLb+NUFL4zvEp5NcOSjZkMEos0pKVSh0WlkdoZIcRIHakvmI5GieM0Jn5hOw0dKwv+AltY8WvYIZbwa/jOd69jO89mIFHs63P+c65+V704k8N0lbrb/v3H3jzbfefmfj3c3v/gw4+2avfOknAY286pHXphfNFpJ47nBs5p6qaecxHFTtveM5Z3Ao8+fXTpy4YXCS3kTOC79Fbg9126nIF241mXg/Gj98HKr3mq2+LFmBztmUFfmcxzWNn5Wl6qrQmWrofKVowKVYuyptkrwfa52VEtFoL1QGWgxRi7nHTVWm5AdgsBRxvUAa5XeHpuqAGeBTOhtA0tHn4xJC31meHpYtwjVd9Fv1XiXaQjI7bYeIN7x2D6oKaqD+oquZztnKyphQWPuRaXNgZkSKrtCsr6uHu4TmF/XK9AaeDURdSMUY2aB6omiI6Yty1X2Xlfq5T4HI7Fp8Wo6sH1xJGQ+xHcArDbGCS0VWy31xHg9oGaHtgqPx4gtRvwJK9T2LcPqTdbgMqp6FcJ7AspAvcKoirxMZzl3FnOlBnk8R5fwp+QIwJ2AEjKvXfjPBUc1E21gio42I3nFtUTMj6ZB/obzea5GiOc27UkYNYvZ5FJPZOqmsDu308O9Q+wY8hnm+tQjvmjAHofYMXMz92KD3DGeRnyaM9RW+y7iSeDaxJsqsB3BAz7gRLe0L7M7dIW5eBZpnvtvqW+h3EvUGvSF43IBvSQp3FCXUFJvYH6BriFx9XoYpnc0qDOnza0We2DEAr9Gk0fcFVpGwfjyDmuHu0WsRURrULv4ZcexzTRLRIXWPMO5Sj0hb6GRNtW/sGFe0im9c9p5ZrRbzT2pX971prSIXsFItk5UH1NRSe2bFEgGxu+w12/Q63XETrm48ZVWH9hdQ3K0ap9IFfGpxqrLbOxTqorXpy/znlr2WdNBhL8f0VOA4o1TP0hLUZGU2bpfo7NFUSMh427JCV3vom0NZOB/hgocs87cwfjNTxUrkz1kFVNsYom8wsl3c5kpVrjIjeiGD9pfp08rNKM8vxghlEyYHUIP0X9LKtjhkFrF6RjqlF503hiYgx1J3Qmcs5D/Nkyhe5D7vMgUydR8HdlTjYTknlBi7jEijB9MOB8AV3gH1BOAIpWcTebHK6N3BMrYxLHubzeChtc+qhLzkvakxgOXTnpFIdU73Kt4tV5Gafpy2W73EnNymr6clmhXDOrXVP3CbEuVsiljINI5Ds+kTpZIRWy92u6tvm7W9gX8J06NHkRGwvPV0j67CV6VaGp56cr7UvZ7WLjE5H5/jU8eM0aGXOns64fC9l0LW8Wcjev5dNCfrSmZwtJf03/FpKvVkg6fAe4E5rIPFtZU8J1bN4l+dsk3wUmpf7jEqPYIWZ8m49Mr8rf69uMldb9NXL9Kd6VMj5UX+I9vYPrk0pXuy2q7JeHph7KuLtA3kc/OmLfWh83Mrva7cret6rjMTkv7E5JRTcntnLUJaR+98ZJ2L3Zk95bQmvWvog0vpG9WIBbZPR9veXznI+vzbMidXLGf/j+Qy2eF2+EVObP4ZLfqvLn4FBTh3sPVY8UmhveIWexch1ak5q+u14OHPmlPzdRVI7mb3703pq+JPn2KuKMYgqPg0bvRlNKaVkZcdB6KJOa3UDCs0NxQmpWTjwab9q2pJRXvsE7MD1HdyXIvdyq70z/RzI7ONt7uw1v3q2V3/0Px/sqE+UfV5/DTvnqEyjyGX8XHv6hf1W+132t/1P6s/aVZ794xMh+ryqf2z7+98IFq</latexit> Modularity assumption If we intervene on a node X i , then only the mechanism P ( x i | pa i ) changes. All other mechanisms where remain unchanged. P ( x j | pa j ) i 6 = j In other words, the causal mechanisms are modular . Many names: independent mechanisms, autonomy, invariance, etc. / 38 Brady Neal 10 Main assumption: modularity

  29. Modularity assumption: more formal / 38 Brady Neal 11 Main assumption: modularity

  30. <latexit sha1_base64="tyC8bvHoHCEJCd+ptJHYsQJlIyg=">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</latexit> Modularity assumption: more formal If we intervene on a set of nodes , setting them to constants, then S ⊆ [ n ] for all i, we have the following: / 38 Brady Neal 11 Main assumption: modularity

  31. <latexit 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Modularity assumption: more formal If we intervene on a set of nodes , setting them to constants, then S ⊆ [ n ] for all i, we have the following: 1. If , then remains unchanged. P ( x i | pa i ) i 62 S / 38 Brady Neal 11 Main assumption: modularity

  32. <latexit 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Modularity assumption: more formal If we intervene on a set of nodes , setting them to constants, then S ⊆ [ n ] for all i, we have the following: 1. If , then remains unchanged. P ( x i | pa i ) i 62 S 2. If , then if is the value that was set to by the P ( x i | pa i ) = 1 i ∈ S X i x i intervention; otherwise, . P ( x i | pa i ) = 0 / 38 Brady Neal 11 Main assumption: modularity

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Modularity assumption: more formal If we intervene on a set of nodes , setting them to constants, then S ⊆ [ n ] for all i, we have the following: 1. If , then remains unchanged. P ( x i | pa i ) i 62 S 2. If , then if is the value that was set to by the P ( x i | pa i ) = 1 i ∈ S X i x i intervention; otherwise, . P ( x i | pa i ) = 0 consistent with the intervention / 38 Brady Neal 11 Main assumption: modularity

  34. <latexit sha1_base64="jvV01CkfNm/jfECptL01IDIO1M=">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</latexit> Manipulated graphs Observational data T 3 T 2 T Y / 38 Brady Neal 12 Main assumption: modularity

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  41. <latexit sha1_base64="jvV01CkfNm/jfECptL01IDIO1M=">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</latexit> What would it mean if modularity is violated? T 3 T 2 T Y / 38 Brady Neal 13 Main assumption: modularity

  42. <latexit 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<latexit 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What would it mean if modularity is violated? T 3 Intervention on not T T 2 only changes P ( T | pa( T )) T Y / 38 Brady Neal 13 Main assumption: modularity

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What would it mean if modularity is violated? T 3 Intervention on not T T 2 only changes P ( T | pa( T )) T but also changes other Y mechanisms such as P ( T 2 | pa( T 2 )) / 38 Brady Neal 13 Main assumption: modularity

  44. <latexit 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<latexit 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<latexit 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Truncated factorization Recall the Bayesian network factorization: Y Y P ( x 1 , . . . , x n ) = ) = P ( x i | pa i ) i / 38 Brady Neal 14 Main assumption: modularity

  45. <latexit 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<latexit 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<latexit 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Truncated factorization Truncated factorization: Y Y P ( x 1 , . . . , x n | do ( S = s )) ) = P ( x i | pa i ) i / 38 Brady Neal 14 Main assumption: modularity

  46. <latexit 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<latexit 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<latexit 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Truncated factorization Truncated factorization: Y Y P ( x 1 , . . . , x n | do ( S = s )) ) = P ( x i | pa i ) i 62 S / 38 Brady Neal 14 Main assumption: modularity

  47. <latexit 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Truncated factorization Truncated factorization: Y Y P ( x 1 , . . . , x n | do ( S = s )) ) = P ( x i | pa i ) i 62 S if is consistent with the intervention. x / 38 Brady Neal 14 Main assumption: modularity

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Truncated factorization Truncated factorization: Y Y P ( x 1 , . . . , x n | do ( S = s )) ) = P ( x i | pa i ) i 62 S if is consistent with the intervention. x Otherwise, P ( x 1 , . . . , x n | do ( S = s )) = 0 / 38 Brady Neal 14 Main assumption: modularity

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  50. <latexit 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<latexit 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<latexit 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Simple identification via truncated factorization X Goal: identify P ( y | do ( t )) T Y Bayesian network factorization: P ( y, t, x ) = P ( x ) P ( t | x ) P ( y | t, x ) / 38 Brady Neal 15 Main assumption: modularity

  51. <latexit 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Simple identification via truncated factorization X Goal: identify P ( y | do ( t )) T Y Bayesian network factorization: P ( y, t, x ) = P ( x ) P ( t | x ) P ( y | t, x ) Truncated factorization: P ( y, x | do ( t )) = P ( x ) P ( y | t, x ) / 38 Brady Neal 15 Main assumption: modularity

  52. <latexit 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Simple identification via truncated factorization X Goal: identify P ( y | do ( t )) T Y Bayesian network factorization: P ( y, t, x ) = P ( x ) P ( t | x ) P ( y | t, x ) Truncated factorization: P ( y, x | do ( t )) = P ( x ) P ( y | t, x ) Marginalize: X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x / 38 Brady Neal 15 Main assumption: modularity

  53. <latexit sha1_base64="YXOaEGfuIJpeheTujZW3/w/ZRoU=">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</latexit> <latexit sha1_base64="yWiF2RQ7hTVPr7cZisCl4fO65vs=">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</latexit> Association vs. causation revisited X X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x T Y / 38 Brady Neal 16 Main assumption: modularity

  54. <latexit 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<latexit 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<latexit 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Association vs. causation revisited X X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x T Y P ( y | do ( t )) 6 = P ( y | t ) / 38 Brady Neal 16 Main assumption: modularity

  55. <latexit 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<latexit 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<latexit 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Association vs. causation revisited X X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x T Y P ( y | do ( t )) 6 = P ( y | t ) X P ( y | t, x ) P ( x ) x / 38 Brady Neal 16 Main assumption: modularity

  56. <latexit 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<latexit 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Association vs. causation revisited X X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x T Y P ( y | do ( t )) 6 = P ( y | t ) X P ( y | t, x ) P ( x | t ) = x / 38 Brady Neal 16 Main assumption: modularity

  57. <latexit 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Association vs. causation revisited X X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x T Y P ( y | do ( t )) 6 = P ( y | t ) X X t ) = P ( y, x | t ) P ( y | t, x ) P ( x | t ) = x x / 38 Brady Neal 16 Main assumption: modularity

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<latexit 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<latexit 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<latexit 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Association vs. causation revisited X X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x T Y P ( y | do ( t )) 6 = P ( y | t ) X X X t ) = P ( y, x | t ) P ( y | t, x ) P ( x | t ) = x x = P ( y | t ) / 38 Brady Neal 16 Main assumption: modularity

  59. <latexit 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Association vs. causation revisited X X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x T Y P ( y | do ( t )) 6 = P ( y | t ) X X X t ) = P ( y, x | t ) P ( y | t, x ) P ( x | t ) = x x = P ( y | t ) / 38 Brady Neal 16 Main assumption: modularity

  60. <latexit 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Association vs. causation revisited X X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x T Y P ( y | do ( t )) 6 = P ( y | t ) X X X t ) = P ( y, x | t ) P ( y | t, x ) P ( x | t ) = x x = P ( y | t ) / 38 Brady Neal 16 Main assumption: modularity

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Association vs. causation revisited X X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x T Y P ( y | do ( t )) 6 = P ( y | t ) X X X t ) = P ( y, x | t ) P ( y | t, x ) P ( x | t ) = x x = P ( y | t ) / 38 Brady Neal 16 Main assumption: modularity

  62. <latexit 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Association vs. causation revisited X X P ( y | do ( t )) = P ( y | t, x ) P ( x ) x T Y P ( y | do ( t )) 6 = P ( y | t ) X X X t ) = P ( y, x | t ) P ( y | t, x ) P ( x | t ) = x x = P ( y | t ) / 38 Brady Neal 16 Main assumption: modularity

  63. The do -operator Main assumption: modularity Backdoor adjustment Structural causal models A complete example with estimation / 38 Brady Neal 17 Backdoor adjustment

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  68. <latexit 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Blocking backdoor paths P ( Y | t ) P ( Y | do ( t )) W 2 W 2 W 1 W 3 W 1 W 3 C C T M Y T M Y Causal association Causal association / 38 Brady Neal 18 Backdoor adjustment

  69. <latexit 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<latexit 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<latexit 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Blocking backdoor paths P ( Y | t ) P ( Y | do ( t )) W 2 W 2 W 1 W 3 W 1 W 3 C C T M Y T M Y Causal association Causal association / 38 Brady Neal 18 Backdoor adjustment

  70. <latexit 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<latexit 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<latexit 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Blocking backdoor paths P ( Y | t, c, w 2 ) P ( Y | do ( t )) W 2 W 2 W 1 W 3 W 1 W 3 C C T M Y T M Y Causal association Causal association / 38 Brady Neal 18 Backdoor adjustment

  71. <latexit 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<latexit 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<latexit 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<latexit 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Blocking backdoor paths P ( Y | do ( t )) P ( Y | t, c, w 1 ) W 2 W 2 W 1 W 3 W 1 W 3 C C T M Y T M Y Causal association Causal association / 38 Brady Neal 18 Backdoor adjustment

  72. <latexit 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<latexit 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<latexit 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<latexit 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Blocking backdoor paths P ( Y | t, c, w 1 , w 3 ) P ( Y | do ( t )) W 2 W 2 W 1 W 3 W 1 W 3 C C T M Y T M Y Causal association Causal association / 38 Brady Neal 18 Backdoor adjustment

  73. Backdoor criterion and backdoor adjustment / 38 Brady Neal 19 Backdoor adjustment

  74. <latexit 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<latexit 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Backdoor criterion and backdoor adjustment A set of variables satisfies the backdoor criterion relative to and W T Y if the following are true: / 38 Brady Neal 19 Backdoor adjustment

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Backdoor criterion and backdoor adjustment A set of variables satisfies the backdoor criterion relative to and W T Y if the following are true: 1. blocks all backdoor paths from to W T Y 2. / 38 Brady Neal 19 Backdoor adjustment

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Backdoor criterion and backdoor adjustment A set of variables satisfies the backdoor criterion relative to and W T Y if the following are true: 1. blocks all backdoor paths from to W T Y 2. does not contain any descendants of W T / 38 Brady Neal 19 Backdoor adjustment

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Backdoor criterion and backdoor adjustment A set of variables satisfies the backdoor criterion relative to and W T Y if the following are true: 1. blocks all backdoor paths from to W T Y 2. does not contain any descendants of W T Given the modularity assumption and that satisfies the backdoor W criterion, we can identify the causal effect of on : Y T X P ( y | do ( t )) = P ( y | t, w ) P ( w ) w / 38 Brady Neal 19 Backdoor adjustment

  78. <latexit sha1_base64="rqyCTkDHF7VoOlhnfJT/NWUg5Eg=">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</latexit> <latexit sha1_base64="rqyCTkDHF7VoOlhnfJT/NWUg5Eg=">ATvnic1VhbT+NGFB7ojdIbLI9cZdtBVJIA0VitxISEuyqroSK3Fr1yuUOA6x8E2A8tGehv6a/qv+l3vhnHsXMD2pcmSjw5Pt93zpzLeCat2PfSrNH4e2Hxo48/+fSzpc+Xv/jyq6+/WVl9cpZGvcRxT53Ij5KLVjN1fS90TzMv892LOHGbQct3z1vXh3L/MZNUi8KT7K72H0XNK9Cr+M5zQyiy9WFv+x4w456mRMF7k3Tt+zAa1v2UbRhZ+DJAjfMIN7cXLbM64d9y057waXtRGEn6oVtsBN4L56aVcFtWnaN2CrdFD8s236cK6M0/y8vNi9X1hv1Bl/W+GDbDNaVeR1Hq0t/Klu1VaQc1VOBclWoMox91VQp3m/VtmqoGLJ3qg9ZgpH+64aqGVge9ByodGE9BrfV/j1khD/BbOlGgHVnx8EiAt9b3RaWPcoVRfxb41ojvNRp/c4uMdri3DGUCaqS6k83C5n1xMqcMHj7nXDz4GVMis3RKM+rg6uN3Bv/l+w6aLkZtoBKMHMh8SLVEbCS46rjKzLuMc5N6Lkbi0/TZtOD79EzI/Qjva3A1MU7pqfhqVcm6iEtu/RVdHxmbDrje8xQ+zeLqzOcg8es6lmI7gk1+oDRmXmWTZHa2e6VmaYBxNsiX5GjRDaKSQR69pD/DxolCvRAafYaDKTV5xLzPqoG+ZfeT+v1Rj53KI/KbNmsZo82olN3xR+5376uLbInQDfx70u7UgsavDHJXfC2syjWKN2gl+3/OXQR6cir7PvJ81zEmqwbeCHe8IZeOhI5n7pH2rg+Zd5b6jfad5H3GqMidV0DNqKHuopT2gpN7vexakhcAnyLVCKbS2q0EdCPLqvlGrLCnmYSez9iFin7xzesfVx9Ri0mS43WJT63HAeck2S0R9u3GLdpR9AWVrK62jN+DEpWJTYe15xqxbrT3pXr3tVq4IL2amWqcp9WmqoXTNjyYD4PRo1x6x1esVNObtBxasW/S9WDanRsn+CKvJTzlWb1dglqswrUZw8z0lz2WGf1JhLif0VNPaJ6h5yt6MjaXlkf47NF0SMR8Or4GpCd3vYu2cqcP+UuwyDVfmVt49ykdzMSdsQ/K2ASj/vDObLzHkcxc8SMRgzvbfXd8GON3JnNF4xSg1khunfsI/6pRyO4VdEIrum6KSMTMoV4J3YmakzhPKrHIY9hmDUzDrKPn1E95UxI7Yk4R4jxvjZUPMZeEX3mnZCSKST+8P7g7nZO4JkYPLY4b283wofPMaoTU2b/qRGQ3dOVtGQ7p1tVaI6qeK0fDa2zZ3UOFbLZ2NFcsOq9kzfp+S6mIPLmAdB5Ds+QZ3MQUVc+7Vc+/z7PfwL+UztmbpIjIfnc5AB1xI9q8j08+u5/mVc7RITE8H8YgI3rBHBtzpPDSOBTZ7UDQL3N2jYlrgbx8Y2QIZPDC+BfLDHKTLZ4A3lAnmzdyeEq1j8yzJnyb5LjAdWX8chQ7xD6f5rdmrcqf61vMlb9C2r9NZ6VMj5UP+E5vY3vV6V7b6sl/umX4Y5d0B8x7WoyOuWw/njc2udqu09y3beAkL+WdgTgmFtm/2ctOYHmJ3MrOuxfbYnrJqSe8autDSO+m7OYxFdU/m53fycz6PBtxJ1fsp/8L5tGzwv34ipqZfrKbd96cfgqKce3g2fHpkb3iFXns3NdWpOavrpeDh25pT63UFXSO32nz6p2CvzV08xV9xR9KBR6OjdaEa0lo0yTjsPxcRkZjeQ8uxQnJDqpROP5qv6lo7UVWD49s2ao1cl4C5X1rer/5GMD8526tu79RdvdtcPnpv/T5bUt+qp2kCc9tQBOvMYcXUWlxcbiy8Wf147WOusBWuRVl1cMJg1VXqtvf8Ht+IQHw=</latexit> Proof of backdoor adjustment P ( y | do ( t )) X w X = P ( y | t, w ) P ( w ) w / 38 Brady Neal 20 Backdoor adjustment

  79. <latexit sha1_base64="rqyCTkDHF7VoOlhnfJT/NWUg5Eg=">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</latexit> <latexit sha1_base64="NxG/T+YTJKi09RnX7PnkG6v8+qo=">ATz3icrVhbT+NGFB7ojdIbLI/74i6sBFJIA12JrVZIWwGratutQOLWEoQSe5JY8U32hBSsVO1j/0/TP9Nz/lmHNu5wraJYo/PnPOd+3gmzchzE1Wr/bOw+MGH38ydKny59/sWX62sPjlPwl5syzM79ML4stlIpOcG8ky5ypOXUSwbftOTF83uAc9f3Mo4cPgVN1F8tpvtAO35doNRaSb1YW/603ZdoNUud37yLVL5aDZct8nls/h45Mhs/1gB6vEtVQ8trarNth0Ap7gUP4W1ZDWZu1Sm3LSjeKExuDV9PEFRmqfBmoLeuqKb2wb3mypaz9sGUVEa4Zcg7Ay/sKTv05RAtdtudKXCGNQMrOHzktIsORw3VGfVUEkvJ/FcP4s/MG+cuBGIib10GTilBNyvrtWoNH2t8sGMG68J8jsPVpT9FXTgiFLboCV9IEQhFY080RELfK7EjaiIi2rVIiRbTyMW8FAOxTLI94pLE0SBql65teroy1ICeGTOBtE1aPrFJGmJ54bHoXELVH1n/VaBd5qOFNhs4x3dmwbTJ6oSHaLOk8s4HyrHPimy8CV8cnOCBT20i51K7R8+K7OfrHXFKGjkFdPIJpHVE1hHTHdVzZ8w7i3ACfpBHbN2bJtk+PRM8H9K3S1gNGiewlG21xBsT9QCaJWxlHg8Zm474G3mo7ZuF1Rr64CKr2gvmPSVKV9zTqIw8S2exdqZzKYM8mKCL+RU4AuJOiBKirl2Kn0sc5Uq0CZN1NJDJNnyJUB9Vg/wW81mtRpTPbdiTIGsWqsmFnsj0TW53ZqdH9yawY5JPa4DPRyLCtkjgR2jNrMoVsAd01MfTzZstEfoVfQd57NCPnF1VQg3pBl3iKUjoeOZWaStS4lme+2+An6JeW9gqhwXVdINoSFuoT6ApM7vdp1eC4+HRlKkc2o1Sgw4cdHVRLl2i5Po3E+r4hLxL0j2dQU7p7iFoElAq0c3z6GPvwiTPag+4+jR3oYWmLVrKq2DN2DEpaOTYu1p5xrRbqj3tXr3ujWlkuQKdapir3oakmXhiPOQNsdzFqtlnr9IqbwLvBiFVN2J+vGlyjZftYKs9POVcOqrEDqTIuR3Gyn5N82UWfVJBLjn2bOPYh1TL0BL0ZGU3Lhf47MF0SIh827r6pCd3vrK0/MpMS/YhQ+J6tzE36pqAOZsqdow/KsjGN0uHMbHkXI/ZcY0SIRkTW18Xw59VmJmNF4whcg0og/Rf0Iu2SjMK0L0sHUOLrps8EhFyqFdCOZFzEubpSCyGDqogWky69Tz61Q95Uxw7TElxh4jovHGkHODcJm3Cz0BUbiT0+H8YG72DokyMHlsYS7rt9wGFzFywFmHPYnh0J2jRji4e2dr5ahOqjhNny3rYCc1Lqvps2WZcouqdk3fJ8C6nCOnkAeWyHZ8LHU6RyrE2q/p2uZfHmBfgHdqz9RFbCy8mCPpYy3RXoWmn9/NtU9htYtNTFjm1/eI4C16ZICdzmPjmMuqR0Uzl7t7r5jm8v1HRjaX9B8Z31zyfo6kxDvAHdJY5mRuTzHXsXmXZG+TbBeYFNYfFxj5DjHF27xv1qrsvb6NXGndP1Ctv6N3JY8PxLf0nt6h65vSqvZQVN4v90w/FHF3CXmP1qNDrFuPx43Mrna7tPct6zgiDdlvYE4JObdn9nLTkB6jdzKyrkVnbE85qknvGjrEpXfSd3MQ8+qejDc7v5OR9Xk2xE4u30/H8jFs8LD8PKamX6ym3fenH4KiujeoquHjk0M7yGqzkPnSnFmTmr67Xgwdubk+t2lruDaTZ89GdFXxh89xbSxo+gR86jd6MK0pWRJx2Hogo8xuIMHZIT8hVUsnHo03altSqCvf4O2bNUevSiR3s7K+M/ofyfjgfLe686L63cnu+uX5v+TJfFUPBObFKc98Zo685jiai8+Xfx+8e3ij2sna/2139f+0KyLC0ZmTZQ+a3/9C/xrB+w=</latexit> Proof of backdoor adjustment P ( y | do ( t )) Example graph: W T Y / 38 Brady Neal 20 Backdoor adjustment

  80. <latexit sha1_base64="NxG/T+YTJKi09RnX7PnkG6v8+qo=">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</latexit> <latexit sha1_base64="rqyCTkDHF7VoOlhnfJT/NWUg5Eg=">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</latexit> <latexit sha1_base64="rqyCTkDHF7VoOlhnfJT/NWUg5Eg=">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</latexit> Proof of backdoor adjustment X P ( y | do ( t )) )) = P ( y | do ( t ) , w ) P ( w | do ( t )) w X Example graph: W T Y / 38 Brady Neal 20 Backdoor adjustment

  81. <latexit 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<latexit sha1_base64="rqyCTkDHF7VoOlhnfJT/NWUg5Eg=">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</latexit> <latexit sha1_base64="rqyCTkDHF7VoOlhnfJT/NWUg5Eg=">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</latexit> <latexit sha1_base64="NxG/T+YTJKi09RnX7PnkG6v8+qo=">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</latexit> Proof of backdoor adjustment X X P ( y | do ( t )) )) = P ( y | do ( t ) , w ) P ( w | do ( t )) w X X = P ( y | t, w ) P ( w | do ( t )) w X Example graph: W T Y / 38 Brady Neal 20 Backdoor adjustment

  82. <latexit 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<latexit 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<latexit 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<latexit 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<latexit 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Proof of backdoor adjustment X X P ( y | do ( t )) )) = P ( y | do ( t ) , w ) P ( w | do ( t )) w X X X = P ( y | t, w ) P ( w | do ( t )) w w X X = P ( y | t, w ) P ( w ) w Example graph: W T Y / 38 Brady Neal 20 Backdoor adjustment

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<latexit 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Backdoor criterion as d-separation non-causal association G W 2 1. blocks all backdoor paths from to W T Y 2. W 1 W 3 causal association M 1 M 2 T Y X 1 X 3 X 2 non-causal association / 38 Brady Neal 21 Backdoor adjustment

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Backdoor criterion as d-separation G W 2 1. blocks all backdoor paths from to W T Y 2. W 1 W 3 causal association M 1 M 2 T Y X 1 X 3 X 2 non-causal association / 38 Brady Neal 21 Backdoor adjustment

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Backdoor criterion as d-separation G W 2 1. blocks all backdoor paths from to W T Y 2. does not contain any descendants of W T W 1 W 3 causal association M 1 M 2 T Y X 1 X 3 X 2 non-causal association / 38 Brady Neal 21 Backdoor adjustment

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Backdoor criterion as d-separation G W 2 1. blocks all backdoor paths from to W T Y 2. does not contain any descendants of W T W 1 W 3 causal association M 1 M 2 T Y X 1 X 3 X 2 / 38 Brady Neal 21 Backdoor adjustment

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<latexit 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<latexit 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<latexit 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Backdoor criterion as d-separation G T W 2 1. blocks all backdoor paths from to W T Y 2. does not contain any descendants of W T W 1 W 3 M 1 M 2 T Y X 1 X 3 X 2 / 38 Brady Neal 21 Backdoor adjustment

  89. <latexit 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<latexit 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Backdoor criterion as d-separation G T W 2 1. blocks all backdoor paths from to W T Y 2. does not contain any descendants of W T W 1 W 3 M 1 M 2 T Y ⊥ G T T | W Y ⊥ X 1 X 3 X 2 / 38 Brady Neal 21 Backdoor adjustment

  90. <latexit sha1_base64="ptO8jOfXsQWAoZipYRhDJHMjuUg=">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</latexit> <latexit sha1_base64="0PvH4ibBO1Zc0tsFiWy23pC2fmM=">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</latexit> Question: How does this backdoor adjustment relate to the adjustment formula we saw in the potential outcomes lecture? Backdoor adjustment: X P ( y | do ( t )) = P ( y | t, w ) P ( w ) w Adjustment formula from before: E [ Y (1) − Y (0)] = E W [ E [ Y | T = 1 , W ] − E [ Y | T = 0 , W ]]

  91. <latexit sha1_base64="ptO8jOfXsQWAoZipYRhDJHMjuUg=">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</latexit> <latexit sha1_base64="0PvH4ibBO1Zc0tsFiWy23pC2fmM=">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</latexit> Question: How does this backdoor adjustment relate to the adjustment formula we saw in the potential outcomes lecture? Section 4.4.1 of the ICI book Backdoor adjustment: X P ( y | do ( t )) = P ( y | t, w ) P ( w ) w Adjustment formula from before: E [ Y (1) − Y (0)] = E W [ E [ Y | T = 1 , W ] − E [ Y | T = 0 , W ]]

  92. The do -operator Main assumption: modularity Backdoor adjustment Structural causal models A complete example with estimation / 38 Brady Neal 23 Structural causal models

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