identification
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Identification Brady Neal causalcourse.com The magic of - PowerPoint PPT Presentation

Identification Brady Neal causalcourse.com The magic of randomized experiments Frontdoor adjustment Pearls do -calculus Determining identifiability from the graph / 40 Brady Neal 2 The magic of randomized experiments Frontdoor


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  3. Few different perspectives on the magic Comparability and covariate balance Exchangeability No backdoor paths / 40 Brady Neal The magic of randomized experiments 6

  4. Comparability and covariate balance: intuition / 40 Brady Neal The magic of randomized experiments 7

  5. Comparability and covariate balance: intuition Treatment and control groups are the same in all aspects except treatment / 40 Brady Neal The magic of randomized experiments 7

  6. Comparability and covariate balance: intuition Treatment and control groups are the same in all aspects except treatment / 40 Brady Neal The magic of randomized experiments 7

  7. <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">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</latexit> Comparability and covariate balance: intuition Treatment and control groups are the same in all aspects except treatment ( T = 1) ( T = 0) drunk sober sober sober drunk sober drunk drunk sober sober sober drunk drunk drunk sober sober drunk sober sober sober drunk drunk sober drunk sober drunk sober sober / 40 Brady Neal The magic of randomized experiments 7

  8. <latexit sha1_base64="URwmej02i2AH9vdCN2XvjlTkL2s=">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</latexit> <latexit sha1_base64="Wb7BH7Q3ruoixRloLuaiKRormQ=">AE7nicbVRLaxRBEJ4kq8b1lejRS2sIRJisuyEQBRcCISLiIUJekA2hp6d2t9l+0V1jshnmR3jxoIhX/45H/401s5OYTdLDQHXV91V9XV0ziVMyYLv9d2Z2rnHn7r35+80HDx89frKw+HQ/2MwL2BNWX+Y8ABKGthDiQoOnQeuEwUHyWirjB98AR+kNbs4dnCs+cDIvhQcyXWwsu6rPqZGp3WpXi90OrWxFNVr52Rx7k8vtSLTYFAoHsJRp+3wOcepVBQNHtZAMfFiA/giEzDNYTjvNJbsGXypKxvPb0GWeW9ysi5DmGsE0JqjsNwPVY6b4sdZdh/c5xL4zIEIyaF+pliaFl5eJZKDwLVmAwuvCStTAy5wKpRc2pMokupvfWjpAnoWBsmb0n6UYKYORMK3vrE/5plGlp2w3lVhmu3J0zmrwNHPSnSkXEri4ZAVAlGYQmHUotTyvmyh4FrhiA8/dMLQI/DELZVfdeNXxgKRyKAOV9FjlLnMqmXjux3kYcgchTkFYX01EiLn39jTEgitR2y0NyO+xNjZIEsUiSCdZSJKlzcZrdVPHOEs5hna2Ni07HVAbuj03Q4TmnGTstKIA2hJgsQorngEQnjdCjhWBM1BKekCxCz1/DRmWhqpM81OZYpDGtV2a4NyFBOqs9LgJZXlQnqavQuqNAY8tcx1aZDXqXBfKjWRJmjqaHBDNy/qVImcjAakdb5uXp2nPlXKwxDSGgvqSs3/VdYcxowrOTBdBX2yg7KOSM3q/rboSqymylpTJ+jeDZzWm7y3XeS9cpiTJN8uiunYPvd1Ou83FyLS5OCI4QD71jvRfmwanMNZy6AxiKBboHT6Bz/ZO/aeyvtTrbef15c239W/g/noefQyWok60Ua0GX2IdqK9SESj6Gv0PfrRcI1vjZ+NXxPo7EzNeRZNrcbvf4d2oMo=</latexit> Comparability and covariate balance: intuition Treatment and control groups are the same in all aspects except treatment ( T = 1) ( T = 0) drunk drunk sober sober sober sober sober sober drunk drunk drunk drunk drunk drunk sober sober sober sober drunk drunk / 40 Brady Neal The magic of randomized experiments 7

  9. Covariate balance definition / 40 Brady Neal The magic of randomized experiments 8

  10. Covariate balance definition We have covariate balance if the distribution of covariates X is the same across treatment groups. More formally, / 40 Brady Neal The magic of randomized experiments 8

  11. <latexit sha1_base64="dj6y5XB1/IBZYdldoF1mYHNyeY=">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</latexit> Covariate balance definition We have covariate balance if the distribution of covariates X is the same across treatment groups. More formally, d P ( X | T = 1) = P ( X | T = 0) / 40 Brady Neal The magic of randomized experiments 8

  12. Randomization implies covariate balance / 40 Brady Neal The magic of randomized experiments 9

  13. <latexit sha1_base64="zvDRyd5i6QD3DtA8iP9JG1soWJM=">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</latexit> Randomization implies covariate balance Because T is not at all determined by X (solely by a coin flip), T ⊥ ⊥ X / 40 Brady Neal The magic of randomized experiments 9

  14. <latexit sha1_base64="zvDRyd5i6QD3DtA8iP9JG1soWJM=">ASmnicrVjZbtGFJ2km+tujvrYPrBRCvRBVmXgNMHA0HtBN0COIC8NHEQSBRlEeIWkrLqCPqLvrZ/0g/p3/TcM0OR1K4gFkSO7tx7p27zYzbkecmaPx35273/wYcfbX28/cmn3+xc69ynkSDmLbObNDL4wv263E8dzAOUvd1HMuo9hp+W3PuWj3j2X+4saJEzcMmult5Lz0W9eB23XtVgrS86Z15QYdJ7IuX+1UG/UG/6zZwZ4ZVJX5Ow3vbf2rlRHhcpWA+UrRwUqxdhTLZXg80LtqYaKQHupRqDFGLmcd9RYbUN2AC4HC1Q+3he49cLQw3wWzATStvQ4uEbQ9JS3xqeDsZdUvVb9FsF3kU6RsQWG2/xbhtMH9RU9UBdJZdxrisna0ph4UOuxYWdESmySru0oi7eHn6nsF+et+B0MOpAKsbIBs0DVNER4y39qusvEc/t8jnYCQ2LV5NG7YvjoTMh/j0gdXCOKGlYqulnhivB9Ts0Fbh8RixYh/YoXavmVY3ckaXEZVr0J4m6D01RuMysjLdBZzZzFXapDHc3QJf0qOANwJKCHz2oX/XHCUM9EGpuhoMZLXEvE/Kgb5F85n+VqhHju0p6EUbOYTS71RKZucrszOz2828SOIT/CXI96xBc12OMQO2ZuZl6skTvGryF/2bTRnqLXWXcSzxrWJNlVA26IGXeCpT2h/ZlZpK0bgWaZz676nfodxL1Gr0he1yAb0kKdxQl1BSb2R+ga4hcfT6GKZzNKjTp82tFjtvRBy/VpJNH3PVaRsH48gzrC26PXIqLUqF38M+TY5okogPqHmLcoR6RtDJ6urQ2DEuaRXfuOw9s1ot5p/Uru5701pFLmClWiYrj6ipoQ7MiUCYnfRa7bpdbrjJlzdeMqNu3Pu4bkaNk+kcrjU45Vh9nYo1QZV7w4f53z1rLPOqkxluL7a3AcUapr6AlrMzKatgv1d2yqJGQ8bL59kxO63kXbcGpmBLpvPOWBT9e57Dgy8xqYLebUFb1XnF+G+NjIZ72+jc+I1OVy56ysmyM0Wgys1ze5cihf/RIq7xIvo64lq+mXytwsy62O8GL5hBlIxNDdK6B3mxvyVN1UVvaCKrCr7UyItlJhnjwjBxPOB8AV3j61BKBIhY8m8+OVMTgBZWw81uVcVoe5DS47X4ecVybzNIeuqHSKQ6p6uVbJy3l5o+nLZTuTfC/LavpyWaHcMDd0w8SYl2ukEsZB5HwC1naXCEVck/QdG3zH2vYF3CvHZi8iI2FyskfYvarQVOXTlfal7IKx8YnIPH8LD96w/415AtrUj7lsupE3c7nbt/JpLj/c0LO5pL+hf3PJNyskHe4N7oQmMs9W1pRwna5Efj05l+gqvHZhndkL+6YjmOZfUSirXe9I9Z2zexh2S6WnT6TQn9zqT0/mY54ihiaTpidJ3apSa/tZ9TSU+zRMj5WP+B8sIfnk1LXBdVzukDU29F3H0gH6LfnbAvbo4bmdP0bunMXdbxGBqy79jcTnJuz5whFyFtonc+s71zsxZdlqTPq30wKVP8LcrEPqmY+3PL7zkfU9OuQJMj/Hvwvk4h1lPbw8ZxbfKFfdcxfviK8u3h67AiJ4T1h1nsDI46MzdEvfsez9x1JX/3URWSu6P7lSl9Zfzp29M1bz8DcOQ8+hScUlrTioiL7mERZVJz2kh4Z8lvZvXSTUvjTduWFPLKN3hHpls5pitv9qp7k3/b2Z2cL5f3zuo/jsoProfm/zZb6St1X38FPh+oRKvMUfrWB/pf6W/1T+bryU+WXym+a9e4dI/OlKv1Vmv8DgHmUpA=</latexit> <latexit sha1_base64="Oi6fX/XWZ5qD8XLb0XrehYgYtg=">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</latexit> Randomization implies covariate balance Because T is not at all determined by X (solely by a coin flip), T ⊥ ⊥ X d P ( X | T = 1) = P ( X ) / 40 Brady Neal The magic of randomized experiments 9

  15. <latexit 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<latexit 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Randomization implies covariate balance Because T is not at all determined by X (solely by a coin flip), T ⊥ ⊥ X d P ( X | T = 1) = P ( X ) d P ( X | T = 0) = P ( X ) / 40 Brady Neal The magic of randomized experiments 9

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<latexit sha1_base64="zck9KyrBRyVhrduVnarl8/M8+VQ=">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</latexit> Randomization implies covariate balance Because T is not at all determined by X (solely by a coin flip), T ⊥ ⊥ X d P ( X | T = 1) = P ( X ) d P ( X | T = 0) = P ( X ) d P ( X | T = 1) = P ( X | T = 0) / 40 Brady Neal The magic of randomized experiments 9

  17. <latexit sha1_base64="Gc9HoWSZnFmkEcYODdN9QLZv2UE=">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</latexit> Covariate balance implies association is causation d P ( X | T = 1) = P ( X | T = 0) / 40 Brady Neal The magic of randomized experiments 10

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Covariate balance implies association is causation d P ( X | T = 1) = P ( X | T = 0) P ( y | do ( t )) X = P ( y | t ) / 40 Brady Neal The magic of randomized experiments 10

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<latexit sha1_base64="UrPWDCK97DrVqKyXpYcim4O9FI=">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</latexit> Covariate balance implies association is causation d Let X be a sufficient adjustment set P ( X | T = 1) = P ( X | T = 0) P ( y | do ( t )) X = P ( y | t ) / 40 Brady Neal The magic of randomized experiments 10

  20. <latexit 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<latexit sha1_base64="UrPWDCK97DrVqKyXpYcim4O9FI=">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</latexit> <latexit 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Covariate balance implies association is causation d Let X be a sufficient adjustment set P ( X | T = 1) = P ( X | T = 0) X P ( y | do ( t )) )) = P ( y | t, x ) P ( x ) x X / 40 Brady Neal The magic of randomized experiments 10

  21. <latexit 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<latexit 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<latexit 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<latexit 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Covariate balance implies association is causation d Let X be a sufficient adjustment set P ( X | T = 1) = P ( X | T = 0) X X P ( y | do ( t )) )) = P ( y | t, x ) P ( x ) x P ( y | t, x ) P ( t | x ) P ( x ) X X = P ( t | x ) x X / 40 Brady Neal The magic of randomized experiments 10

  22. <latexit 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Covariate balance implies association is causation d Let X be a sufficient adjustment set P ( X | T = 1) = P ( X | T = 0) X X P ( y | do ( t )) )) = P ( y | t, x ) P ( x ) x P ( y | t, x ) P ( t | x ) P ( x ) X X X = P ( t | x ) | x P ( y, t, x ) X X = P ( t | x ) x X / 40 Brady Neal The magic of randomized experiments 10

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Covariate balance implies association is causation d Let X be a sufficient adjustment set P ( X | T = 1) = P ( X | T = 0) 0) = ⇒ T ⊥ ⊥ X X X P ( y | do ( t )) )) = P ( y | t, x ) P ( x ) x P ( y | t, x ) P ( t | x ) P ( x ) X X X = P ( t | x ) | x P ( y, t, x ) X X = P ( t | x ) x X / 40 Brady Neal The magic of randomized experiments 10

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Covariate balance implies association is causation d Let X be a sufficient adjustment set P ( X | T = 1) = P ( X | T = 0) 0) = ⇒ T ⊥ ⊥ X X X P ( y | do ( t )) )) = P ( y | t, x ) P ( x ) x P ( y | t, x ) P ( t | x ) P ( x ) X X X = P ( t | x ) | x P ( y, t, x ) X X X = P ( t | x ) | x P ( y, t, x ) X X = P ( t ) x X / 40 Brady Neal The magic of randomized experiments 10

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Covariate balance implies association is causation d Let X be a sufficient adjustment set P ( X | T = 1) = P ( X | T = 0) 0) = ⇒ T ⊥ ⊥ X X X P ( y | do ( t )) )) = P ( y | t, x ) P ( x ) x P ( y | t, x ) P ( t | x ) P ( x ) X X X = P ( t | x ) | x P ( y, t, x ) X X X = P ( t | x ) | x P ( y, t, x ) X X X = P ( t ) x X X = P ( y, x | t ) x / 40 Brady Neal The magic of randomized experiments 10

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Covariate balance implies association is causation d Let X be a sufficient adjustment set P ( X | T = 1) = P ( X | T = 0) 0) = ⇒ T ⊥ ⊥ X X X P ( y | do ( t )) )) = P ( y | t, x ) P ( x ) x P ( y | t, x ) P ( t | x ) P ( x ) X X X = P ( t | x ) | x P ( y, t, x ) X X X = P ( t | x ) | x P ( y, t, x ) X X X = P ( t ) x X X X = P ( y, x | t ) x = P ( y | t ) / 40 Brady Neal The magic of randomized experiments 10

  27. <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> <latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">AFh3icfVTbjhNHEB0uDmCSsBCJl7x0MCvlYTD2asVFiUitFEU5YFIuwtox1r19NTYLfdN3TUspmfyWvyQ/mbVI8Hdr1EacujupyqOlVd6tIpGXAy+efK1WvXB1/duHlrePvrb769s3P3nGwjRdwJKy/k3JAyhp4AglKnjPHBdKnhdrl4m/+t34IO05hDXDuaL4yspeBIptOd+8XByVtWaFmxQzZjkzl91qeT053RZDzpDvtSmPbCKOvPq9O7198WlRWNBoNC8RBOphOH8g9SqGgHRZNAMfFi/ghETDNYR57Bpo2S5ZKlZbT3+DrLNejIhch7DWJSE1x2W47EvG/KdNFg/m0dpXINgxKZQ3SiGlqVpsEp6EKjWJHDhJXFlYsk9F0gzG26VKXW7rVu7Ql6GlrFd9gtRN1IAI4uCbX7va8q3jUqWNH8qscsO5eoD68HbkZvpbJmQwO3nqACI0iwCsw6lh/6IQreBK7YwnO3DGMC/9aENFW3fuR4QGK5lIFKeuxyp5xKlp7dQxL7iDkFQjruxUJOfenoVcCV6eawBeV5LzJ0NMqGIBPFMiShdHDI6j37nCO9z3qDNja3SrANyQ93Ppkxoxk3FkpAH0JIiVXexREI4fE4FoRNIJS0gXIWeX5Wc60NFI3mp3JCpdpYcdPKUe7CXVWGvwcyqKQnbvU6g0BjyNzM1okfepcC2V2lATtHW0uGEW2z5VKTerAVWfbxa7fvquKh6WUPVYUBdqnlfZc5gzruTCzBTUJAdlHQUNu/t7SVdiNVXWmiZB927grFdicdDGIi1zWcaDt32HXPfe72OSbnkl6YCRwgH3rHih/RjnXIJZz4BjUC/T98kxXoY5BSe4s0m46E4qWEM6N58jDnkViWEF90TOajvb6JkDFwivu4sNkfNgWegXe7OkmJr2lYHqMpefni+F473xdH/8/I/90Yuf+mfpZvZ9iD7MZtmT7MX2a/Zq+woE9nH7M/sr+zvwa3B48GTwbMN9OqVPua7bOsMfv4XHAzX0g=</latexit> Exchangeability s t d a a i l e s h T = 0 T = 1 E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 / 40 Brady Neal The magic of randomized experiments 11

  28. <latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">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</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Exchangeability heads tails T = 0 T = 1 E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 / 40 Brady Neal The magic of randomized experiments 11

  29. <latexit sha1_base64="hsz8RjcljbDBaGAYNVc09xwW9nw=">AFdHicfVRLjxtFEJ6ENQTzSuAIhwZnJQ4Tx16tFJCwFCkKQohDkHY3QetVNTY7fcL3XsHFa8xu4wk/j3CmejzJrjeItjyqx1dVX1WXuvJaRZrN/r51+72D0fsf3Plw/NHn3z62d17n59F1waJp9JpF15UEFEri6ekSOMLHxBMpfF5tXmS/c9/xCVsye09XhYGVoyQm05PxELMX96dzKaz/oh3hfkgTIrhPHt57+C3Ze1ka9CS1BDj+Xzm6SJBICU1duNlG9GD3MAKz1m0YDBepJ5tJw7ZUovGBf5bEr31ekQCE+PWVIw0QOt405eN/+U7b6n57iIp61tCK3eFmlYLciK3LmoVUJLesgAyKOYq5BoCSOIBjfKVKb153bEFSxE+JQ/MjUrZIo2KJxn9+rhvPto7IlD5tLHIoTtXktBvB+5G46eyZicPc2KiKRsqsonCdl1OthiBLaCFqsAvh1nDL45zbmqfrtAw+RmOVaRS4ZqM+dc2pVBQjbFNfgMZY1Shf6fYglhOAuYylBy0GeGiQoG0Wld1FlFJNgnjkRp0tjwefBL0D4qoSWXGldnWcdCSx3v5gLaQTYWmShjGgUE5Kbso9jEOHDaStZmhCrZWPWIo6wGUpjLKtEZcqprWvKmz6SPO0e1CvVOW3oaKJFXg3XsTqzFwCPzC17kYy7cK131CRvHS9uXKRuSFWp3WpgPeRbpL6foasa4hrAYv6Ws2rKkeSgFarexCY8Ny1M5z0Li/vyd8Jc5wZWN4EnzvFi8HJS2fdmZl7mq0tOu2/edQRi8waSs3PArW6NnhMfgxfLr/BO9cgNn3wCtIwb9P3yXFfljiVMHRzybnoQHzUuIV8Yr5MnAIjOsbnumcwnR0MTqNMyaPDpfjbe75Zmg8EemTZlveNgfozmN5+ed4Wzo+n8ePr9r8eTxz8Mz9Kd4svim+LbYl48Kh4XPxXPitNCFqr4o/iz+Ovgn9FXo8nocAe9fWuI+aLYO6Ppv/DG0uo=</latexit> <latexit sha1_base64="+csBynRxFr6koeweOUGm1ZEASRU=">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</latexit> <latexit sha1_base64="u9JVBphx7MmuBVj2PHmsIsdN7Y=">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</latexit> <latexit sha1_base64="B6SDsM+n6ge+raWORAaVSuAj6XE=">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</latexit> Exchangeability heads tails T = 0 T = 1 E [ Y | T = 1] = y 1 E [ Y | T = 0] = y 0 / 40 Brady Neal The magic of randomized experiments 11

  30. Question: Write down the formal definition of (mean) exchangeability. Then, prove that this yields “association is causation.”

  31. <latexit sha1_base64="mCBGRKLIqysEtUgJ84kBMYJ/1I=">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</latexit> No backdoor paths X T Y / 40 Brady Neal The magic of randomized experiments 13

  32. <latexit sha1_base64="mu6oZyxoxSR78a5QdGAnFoA9Y=">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</latexit> No backdoor paths Confounding association X T Y / 40 Brady Neal The magic of randomized experiments 13

  33. <latexit sha1_base64="mu6oZyxoxSR78a5QdGAnFoA9Y=">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</latexit> No backdoor paths Confounding association X T Y / 40 Brady Neal The magic of randomized experiments 13

  34. <latexit sha1_base64="+xY8cbVzkZB4g8VIEArhV5uRrc=">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</latexit> No backdoor paths X T Y / 40 Brady Neal The magic of randomized experiments 13

  35. <latexit sha1_base64="+xY8cbVzkZB4g8VIEArhV5uRrc=">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</latexit> Question: What previous result tells us that association is causation in this graph? X T Y

  36. The magic of randomized experiments Frontdoor adjustment Pearl’s do -calculus Determining identifiability from the graph / 40 Brady Neal Frontdoor adjustment 15

  37. <latexit sha1_base64="e1lOcgeC/uNMrGHnQ+sWTUehG4=">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</latexit> <latexit sha1_base64="ox3/eARyg5DmkB3v3xif3G1icUY=">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</latexit> Recall the backdoor adjustment P ( Y | t ) W 2 W 1 W 3 C T M Y Causal association / 40 Brady Neal Frontdoor adjustment 16

  38. <latexit sha1_base64="0uFmwvyNqMaIY0MPd3xbNmsdXkI=">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</latexit> <latexit sha1_base64="Z7rlzBpTi+dPmukn+wKc7khHqvg=">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</latexit> Recall the backdoor adjustment P ( Y | t, c, w 2 ) W 2 W 1 W 3 C T M Y Causal association / 40 Brady Neal Frontdoor adjustment 16

  39. <latexit sha1_base64="LQXcL0mQbxU7nAhuYqjJ+F5HZQ=">AWI3icrVjdjtEFJ7SBcry1xbtFVyY7i5qJW9IQtFWqiJV2rZCiKJW2nYLm1Xl2JPEiv9qTxq2UZ6Bl+A1eAEuETdc8Bpc84349jO/5Ymij0+c853fuac45l0ksDPVL3+96V3Lm+9+97Vz7Y/vCjz/59Oq168+yeJi68qkbB3H6vONkMvAj+VT5KpDPk1Q6YSeQJ53BEc+fvJp5sfRsTpP5Fno9CK/67uOItKLa5d/bXdkz4/Gyh+8TnxXDVM52bMZ9/6MfZkNn1uR/R4milHSdtzsr70zqybTeOuvEw8kjNLctR1s26Xb9ljfKE3uTu4tQWDyUnu+oOL1lnXZkEI+sVty1yrJnjJVzrQBS5LgKZaQIKZBdpYFyQaBMWVbAxEPlxqEkNTv9RehGI4coxSuB16vHK7EUf3ZAEliqRh7dyP+3Kp57pLbhnca0XnmIthLcfeXIdPyRJ6FoLSs2/WzhdIlmCPjh/1LCd1i5l+DqlbEv1fXdgW8julugHIBiW7CDV7h1WLetI4zGcksazVrzfVGOcPMCf6PQComKJD0vz2zEImIYspUVyjKsti10e9Teasa5N8pfZeXN2t1+r4WPODhnsCvN5HF+78rtoC0/EwhVDEQopIqFoHAhHZPQ9FQ1RFwnRzsSYaCmNfMxLMRHbJDskLkcDlEHdO3R06mhRvTMmBmkXdIS0C8lSUvsGx6Pxl1Q9Z31WyXeZTrGwGYbz+neMZghUZXoE3WdXM65qRz7pMjCO/DFJzsTUNhLt+JRl+4BPSuyn6/nxClp5JFUSiOXaAFRNYV1pHTXcWXP+4izAz5JI7ZpuTcdsn35SvB8TN8BYTk0zmAp2qJhybqETRL2Mo8AVZsOeIv5KG2bxVWd+qDj1XVXjDvMVEG4jWNqsirdJZzZzmXMsiTBbqYX4EjIu6MKDHy2qf4+cRzUSXMFmHg5XswZcE+VEzyN9jPs/VhNbzAPZkWDUL2eRDT2LqprA7tzOgewfYKcmPa4PRwLm+yRwE6Rm3kUbXCn9DTCkwsb3Rl6DXH62mT5xdNuHGNONPsXQkdDxzi7R1Y6JZ5nsgfoB+SetuIyqc1zbJxrBQZ3EGXZFZ+xZ1DY5LSFemcmRzig0dIezoI1sGRCv0aSTW9zV5kaF+AoM6pnuAqCVAsaGd4zPCOIRPvKJD6B7R2IMelraok9XEobFjUtHKsfHRe+a1Wsg/rl3d92a1slyESrVMVragqS5uG495BdjuctRc0+t0x83g3WTGqg7sL7oG52jVPpYq1qe6Vh6ysQ+pKi5HcbGfi3xpok5srCXHvkcLUh1DT1DbSZG03ap/o5MlcRYDxf30OSErnfWNpqZGRM9NJEKiE/XOb9xeOYlYTrIqTaiV5fhfjAyOe9vkPfMair5Z6hsqyKY3G05nV8j5GEvHRI15xjZcg1gl8+XL6s0ozm2K/HbxoDpEzVhmkTdE95MZiz4/FLvWCXcqajx5pZmSYu+R0HhvyrlHuMw7gJaIKFzh4+n8ZO0a3CfKxESsi7m8DgsbfHQ+D5xtk3maQ1eUmuHgql6tlfNyUd5o+mpZb5rvVlNXy3LlFfITd/0gwxYz9fIKawDS4SlLD1eIxXjnaDp2uafNrAvwrt2aPIiNRaerJEM0WO0V7Gpykdr7VPogqmJCcv8/AYRfIX+N8EO6KJxLGTVhaJZyJ2/UwL+dEFI1tIheMbyH5eo2kxLvBn9JY5snamKux2uRX073JbqKbFw7FB1+F3um41jmPcKrd96LdS2bd5h+Vs31mpf7mQ3uxMx1jFzEynTDfTxAk/btO6qlR/SO5vGR+Ib2Bw26Pqx0zU1ReZ8+NPVWxm0S8iH1u/voixfHTcxu+qCy567qeEAa8t/EnE4K7sDsIZchXUTvYmSd697cXnZWk96t9IlL7+DP1yAW1bMYb/X6LkbW5+gYO8hiH/82kMtnlM3wipxZfqJcd85dfvpK6N6la4COkBne+8i6AJ1BiqfmhKjfvkdzZ13O3yZVBefu+Mb1GX1V/NnTUw+nyFxFDx6F6wgrWlxGXnsAQyuw2MpxZipNZrXLS0niztmWlvAoNXst0K2m60vaLq7uN2f9m5gcnzVrjdq3ReNLcvXfH/HFzRXwuboibFKhDcY9K8zEF1r3879YXW/tbX+38tvPHzp87f2nWdy4Zmc9E5bPz3/7cLPe</latexit> <latexit sha1_base64="Z7rlzBpTi+dPmukn+wKc7khHqvg=">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</latexit> <latexit sha1_base64="0uFmwvyNqMaIY0MPd3xbNmsdXkI=">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</latexit> Recall the backdoor adjustment P ( Y | t, c, w 2 ) W 2 W W 1 W 3 C T M Y T M Y Causal association / 40 Brady Neal Frontdoor adjustment 16

  40. <latexit sha1_base64="rmzf2dnvrtQDlEmd/bgMDday4FU=">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</latexit> Frontdoor adjustment: big picture confounding association W T M Y causal association / 40 Brady Neal Frontdoor adjustment 17

  41. <latexit sha1_base64="+9t/9tyzRe4QnrnlIJ4LymjoViE=">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</latexit> Frontdoor adjustment: big picture W focus T M Y only causal association / 40 Brady Neal Frontdoor adjustment 17

  42. <latexit sha1_base64="97OV9zRW5X1PA5ILGqON3gwUec=">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</latexit> Frontdoor adjustment: big picture W T M Y / 40 Brady Neal Frontdoor adjustment 17

  43. <latexit sha1_base64="jaG7k2a3PUqW/u74qXTDqaNxUVs=">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</latexit> Frontdoor adjustment: big picture 1. Identify the causal effect of T on M W T M Y Step 1 / 40 Brady Neal Frontdoor adjustment 17

  44. <latexit sha1_base64="s/lzlHOyX5PtbL+vjpSExh86iHY=">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</latexit> Frontdoor adjustment: big picture 1. Identify the causal effect of T on M 2. Identify the causal effect of M on Y W T M Y Step 1 Step 2 / 40 Brady Neal Frontdoor adjustment 17

  45. <latexit sha1_base64="W2GtS6RWfVw5rxfaIh3GklPLAi0=">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</latexit> Frontdoor adjustment: big picture 1. Identify the causal effect of T on M 2. Identify the causal effect of M on Y 3. Combine the above steps to identify W the causal effect of T on Y T M Y Step 1 Step 2 Step 3 / 40 Brady Neal Frontdoor adjustment 17

  46. <latexit sha1_base64="jaG7k2a3PUqW/u74qXTDqaNxUVs=">AVznicrVhLb9tGEN7EfaTuK07hUy9sbAMJQKuSHSBAgEBnARF0bQJ6sRJLSOgyJVEiK+QqyiOIPTav9Ef0nOv7bH/pjPfLkVSbweRIHI5O/PNY2eGu2ongZ+pev2/S5c3Pvr4k0+vfLb5+RdfvX1a1rz7N4kLrymRsHcfqi7WQy8CP5TPkqkC+SVDphO5An7f4Rz5+8kWnmx9GxOk/kWeh0I7/ju4i0qutjbutuz60Uj5/XeJ76pBKseblvnsWT/Hnswmz62IHk8z5Shpe07Wk96ZdaPlxlEnHkQeqblpOcq6UbfrN63Rbnlid3xvHgqLh9LzHRWnN63TtgziodWMO1Z9oyxcq4lQIocV6GMFCEFsqM0UC4IlAnLEph4oNw4lAS+t3ePBTDkWOUwvXQ65bDlTiqNx0gSwVY+txZ9bNctdchu8AaRJGN/VTKxGuNymAuEvRyiWIFCAfjuSbsLbJhroF2uYBl0MaBH6SyZJOmNfxVbOstwRbANqW1NK25aXO0LbeZq4TyGa9dse2zvW4UWvYVuCQu024fBcOH1ImZDQ4pEQ2y9ySkVcpk1dXd+q1Oj7W7KBhBjvCfJ7EW1f+Ei3hiVi4YiBCIUkFI0D4YiMvqeiIeoiIdqZGBEtpZGPeSnGYpNkB8QlicMhap+uXo6NdSInhkzg7RLWgL6pSRpiT3D49G4A6q+s36rxLtIxwjYbOM53dsGMySqEj2irpLOdeVY58UWXgHvhkZwIKe+lWPOrQPaBnRfbz9Zw4JY08kp5BItIKqmsI6U7jqu7HkPcXbAJ2nENi32pk2L14Jno/p2ycsh8YZLGVbLfHIRD2CZglbmSfAi1GfEseavuWYXUmPvhYVe0F8x4TpS/e0aiKvExnOXcWcymDPJ6ji/kVOCLizogSI69ip9PHNVMdAmTdThYyS58SZAfNYP8I+bzXE1oPfdhT4ZVs5BNPvQkpm4Ku3M7A7q3gZ2S/IjmetDsbDJHgnsFLmZR9EGd0pPQzy5sNGdotdQd7yeNvnE2WUTbkwz/gRLR0LHM7dIWzcimW+In6Je07jaiwnltk2wMC3UWZ9AVmbVvUtfguIR0ZSpHNqfY0BHCjh6ypU+0Qp9GYn3fkxcZ6icwqCO6B4haAhQb2jk+Q4xD+MQrOoDuIY096GFpizpZTdw2dowrWjk2PnrPrFYL+ce1q/vetFaWi1CplsnKJjTVxS3jMa8A212Omt6ne64GbwbT1nVhv1F1+AcrdrHUsX6VNfKQzb2IFXF5SjO93OeLweoExtrybHvEkcTUh1Dz1CbidG0Waq/I1MlMdbDxT0OaHrnbUNp2ZGRA9NpALi03XObxyeU2YDnKqheiV5chPjTyea9v03cE6nK56isqmxKo9FkZrm8j5FEfPSIV1zjJYh1Al+m/ys0sy62B8GL5pB5IxVBmldA+5Md/zY7FDvWCHsqoaT15pqTYeyQ03p1w7hIu8/ahJSIKV/hoMj9euQYPiDI2EetgLq/DwgYfnc8DZ8tknubQFaWmOLiql2vlvJyXN5q+XNab5HtVtOXyzLlDXLTN/0gA9aLFXIK68ASYSlLj1dIxXgnaLq2+eUa9kV41w5MXqTGwpMVkiF6jPYqNlX5eKV9Cl0wNTFhmd/eI4Jv0P/G2AFdNI6FrLpQNAu58/eKaSE/vGBkC8nwgvEtJN+tkJR4N/gTGs8XVlTzPVkJfLryb5EV5GNa5uiw+9iz3Qcy7xHeLX1W6+J2rbNOyx/i+W7z6zU3xoL3amI+wihqYT5vuJfWjSv1AtfSY3tE8PhKHtD9o0PVRpWui8r79IGptzLuASHfpn73AH3x4riJ2U3vV/bcVR0PSUP+G5vTScEdmD3kIqSL6J2PrHPdm9nLTmvSu5Uecekd/PkKxKJ65uMtX9/5yPocHWMHWezjPwRy+YyHl6RM4tPlKvOuYtPXwndO3QN0BEyw/sAWRegM0jxzJwQ9dv3aOasy/l7QFXBuTu6fm1KXxV/+vTUxelnQBwFj94FK0hrWhlx0TksgYwyu40MZ5biZFarnLQ03rRtWSmvQoPXN1Kmq60+erqTmP6v5nZwclBrXGr1mg8Pdi5f8f8cXNFfCuixsUqNviPpXmEwqsu/Hnxt8b/2z8u/3L9mB7vP27Zr18ych8Iyqf7T/+B3xymB8=</latexit> Frontdoor adjustment: step 1 Identify the causal effect of T on M W T M Y Step 1 / 40 Brady Neal Frontdoor adjustment 18

  47. <latexit sha1_base64="jaG7k2a3PUqW/u74qXTDqaNxUVs=">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</latexit> Frontdoor adjustment: step 1 Identify the causal effect of T on M <latexit sha1_base64="G8X2gIMu+wYdwsRzOv8TGJFb5Q=">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</latexit> P ( m | do ( t )) W T M Y Step 1 / 40 Brady Neal Frontdoor adjustment 18

  48. <latexit sha1_base64="jaG7k2a3PUqW/u74qXTDqaNxUVs=">AVznicrVhLb9tGEN7EfaTuK07hUy9sbAMJQKuSHSBAgEBnARF0bQJ6sRJLSOgyJVEiK+QqyiOIPTav9Ef0nOv7bH/pjPfLkVSbweRIHI5O/PNY2eGu2ongZ+pev2/S5c3Pvr4k0+vfLb5+RdfvX1a1rz7N4kLrymRsHcfqi7WQy8CP5TPkqkC+SVDphO5An7f4Rz5+8kWnmx9GxOk/kWeh0I7/ju4i0qutjbutuz60Uj5/XeJ76pBKseblvnsWT/Hnswmz62IHk8z5Shpe07Wk96ZdaPlxlEnHkQeqblpOcq6UbfrN63Rbnlid3xvHgqLh9LzHRWnN63TtgziodWMO1Z9oyxcq4lQIocV6GMFCEFsqM0UC4IlAnLEph4oNw4lAS+t3ePBTDkWOUwvXQ65bDlTiqNx0gSwVY+txZ9bNctdchu8AaRJGN/VTKxGuNymAuEvRyiWIFCAfjuSbsLbJhroF2uYBl0MaBH6SyZJOmNfxVbOstwRbANqW1NK25aXO0LbeZq4TyGa9dse2zvW4UWvYVuCQu024fBcOH1ImZDQ4pEQ2y9ySkVcpk1dXd+q1Oj7W7KBhBjvCfJ7EW1f+Ei3hiVi4YiBCIUkFI0D4YiMvqeiIeoiIdqZGBEtpZGPeSnGYpNkB8QlicMhap+uXo6NdSInhkzg7RLWgL6pSRpiT3D49G4A6q+s36rxLtIxwjYbOM53dsGMySqEj2irpLOdeVY58UWXgHvhkZwIKe+lWPOrQPaBnRfbz9Zw4JY08kp5BItIKqmsI6U7jqu7HkPcXbAJ2nENi32pk2L14Jno/p2ycsh8YZLGVbLfHIRD2CZglbmSfAi1GfEseavuWYXUmPvhYVe0F8x4TpS/e0aiKvExnOXcWcymDPJ6ji/kVOCLizogSI69ip9PHNVMdAmTdThYyS58SZAfNYP8I+bzXE1oPfdhT4ZVs5BNPvQkpm4Ku3M7A7q3gZ2S/IjmetDsbDJHgnsFLmZR9EGd0pPQzy5sNGdotdQd7yeNvnE2WUTbkwz/gRLR0LHM7dIWzcimW+In6Je07jaiwnltk2wMC3UWZ9AVmbVvUtfguIR0ZSpHNqfY0BHCjh6ypU+0Qp9GYn3fkxcZ6icwqCO6B4haAhQb2jk+Q4xD+MQrOoDuIY096GFpizpZTdw2dowrWjk2PnrPrFYL+ce1q/vetFaWi1CplsnKJjTVxS3jMa8A212Omt6ne64GbwbT1nVhv1F1+AcrdrHUsX6VNfKQzb2IFXF5SjO93OeLweoExtrybHvEkcTUh1Dz1CbidG0Waq/I1MlMdbDxT0OaHrnbUNp2ZGRA9NpALi03XObxyeU2YDnKqheiV5chPjTyea9v03cE6nK56isqmxKo9FkZrm8j5FEfPSIV1zjJYh1Al+m/ys0sy62B8GL5pB5IxVBmldA+5Md/zY7FDvWCHsqoaT15pqTYeyQ03p1w7hIu8/ahJSIKV/hoMj9euQYPiDI2EetgLq/DwgYfnc8DZ8tknubQFaWmOLiql2vlvJyXN5q+XNab5HtVtOXyzLlDXLTN/0gA9aLFXIK68ASYSlLj1dIxXgnaLq2+eUa9kV41w5MXqTGwpMVkiF6jPYqNlX5eKV9Cl0wNTFhmd/eI4Jv0P/G2AFdNI6FrLpQNAu58/eKaSE/vGBkC8nwgvEtJN+tkJR4N/gTGs8XVlTzPVkJfLryb5EV5GNa5uiw+9iz3Qcy7xHeLX1W6+J2rbNOyx/i+W7z6zU3xoL3amI+wihqYT5vuJfWjSv1AtfSY3tE8PhKHtD9o0PVRpWui8r79IGptzLuASHfpn73AH3x4riJ2U3vV/bcVR0PSUP+G5vTScEdmD3kIqSL6J2PrHPdm9nLTmvSu5Uecekd/PkKxKJ65uMtX9/5yPocHWMHWezjPwRy+YyHl6RM4tPlKvOuYtPXwndO3QN0BEyw/sAWRegM0jxzJwQ9dv3aOasy/l7QFXBuTu6fm1KXxV/+vTUxelnQBwFj94FK0hrWhlx0TksgYwyu40MZ5biZFarnLQ03rRtWSmvQoPXN1Kmq60+erqTmP6v5nZwclBrXGr1mg8Pdi5f8f8cXNFfCuixsUqNviPpXmEwqsu/Hnxt8b/2z8u/3L9mB7vP27Zr18ych8Iyqf7T/+B3xymB8=</latexit> <latexit sha1_base64="G8X2gIMu+wYdwsRzOv8TGJFb5Q=">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</latexit> Frontdoor adjustment: step 1 Identify the causal effect of T on M <latexit sha1_base64="G8X2gIMu+wYdwsRzOv8TGJFb5Q=">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</latexit> P ( m | do ( t )) )) = P ( m | t ) W T M Y Step 1 / 40 Brady Neal Frontdoor adjustment 18

  49. <latexit sha1_base64="4qCWONnkrp2T9BdIkwGTnidGsek=">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</latexit> Frontdoor adjustment: step 2 Identify the causal effect of M on Y W T M Y Step 2 / 40 Brady Neal Frontdoor adjustment 19

  50. <latexit sha1_base64="QG57jZbv7M0gSwvR6kQ0evEFBxI=">ATFHicrVhLb9tGEF6nL9dtWkcF2kMvbJQCDkCrkmvAuQgIYCcoigZwADt2GwUGRa4sQnyFpKw6hA79Ef01RU9Fr73F/RvdObpUjqrSAWRK5mZ76Zndfuht5bpI2m/9u3Xnv/Q8+/Gj7451Pr372e792ovknAY2/LcDr0wvuxaifTcQJ6nburJyiWlt/15EV3cMzFzcyTtwOEtvI/nKt64Dt+faVkqkq93fOtFeJxymdujLG8szOr7rGJ2TcK/jS8e10jAm6sOHRtvoJEP/qpMSeOrLIAXzPNmSnGlU+B8aHRMyFeLVbr3ZaOLPmB209KAu9N9peG/7T9ERjgiFLYbCF1IEIqWxJyR0OelaImiIj2SmREi2nkYl6Ksdgh2SFxSeKwiDqg5zX9eqmpAf1mzATSNmnx6BuTpCG+1TwOjXugqjfrN0q8i3RkwGYb+nd1Zg+UVPRJ+oquZxzXTleU0oWPsJaXLIzAoVXaVdW1KO3R79Tsp+ft8QpaeSQVEwjm2geURWFdcT0Vn7lfhZwt8kZs0+LVdMn2xZHg+ZA+A8KyaJzAUrbVE+1wNolrCVeTxEbDHir7RCZd8yrN5kDS6iqlbBvGdEGYg3NKoiL9NZzp3FXKlGHs/RxfwpOALiTogSIq9d8p9LHNVMtAmTdViI5DXWEiE/Ghr5R8znuRpRPdhT4KoGcgmF3oiXTeF3bmdHr27wI5JPqO5PvSwL0yRwI7Rm7mXjTBHdOvEX7ZsNGeojdQdxPk9bE2WUSbkgz7gRLeUL5M7dIWZcRzdCfET9EuKuwmvcF6bJBvCQpXFCXQFOvZt6hrsF5+eTGXP5hQTOnzY0Ue2DIhW6FNIrO87WkWC+vE0akZvD16LgGJCO/tnhLGPNXFEh9A9orEDPSxtUCdriCNtx7ilX3jovfMajWQf1y7qu9Na2W5AJVq6KxsQ1NTHOoVcwTY7rLXbN3rVMdNsLrxlFVd2F90Dc7Rqn0sVcSnGisH2diHVBWXvTh/nfPWcoA6MRFL9v01cbQh1dP0BLUZaU07pfo71lUSIh423r7OCVXvrG0NZMR3de8ohP1TnvODzmjAt5FQH3ivPL0N8ouXzXt+lTwbqcrkXqKyqbEyjbDKzXN7FSMI/asQRV3gRfB1hLd9MvkZpZl3sd4MXzCByxqYaV10B7kxf+Vnok69oE5ZVfUnR5opMc4eEY0fTDgfEC7zDqAlIApXeDaZH6+MwQlRxtpjPczldVjY4KLzOeDs6MxTHKqi0ikOrurlWjkv5+WNoi+XdSb5XpV9OWyTLlBbrq6HyTAulwhlyIOLOGXsvRshVSIPUHRlc0/r2FfgL12qPMi1hZerJD0WPUqkJdlc9W2peiC8baJyzy1t48Ab9b4wT0KZ+LGTjbxZyN2+lU8L+dGni0k/Q39W0i+WSEpsTe4ExrLPF9ZU8x1uhL59eRcoqrIxLNL3uG92NEdx9D7CEdb7Xpt1Lap97B8F8tPn0mpv7nQXpxM5wiRroT5ueJfWhSa/uBaukZ7dE8Phbf0/mgRc+nla65Liqf04e63sq4B4R8RP3uBH1xc9xIn6b3K2fuqo4npCH/jvXtpOD29BlyEdImeucjq1x3Zs6y05rUaVPXOoEf7sCsaie+XjL4zsfWd2jQ5wgi3P8u0Au31HWwytyZvGNctU9d/HtK6J3j54eOkKieU+QdR46gxTn+oaodt/jmbsu5+8BVQXnbna/NqWvij9e7rG7WdIHAWPOgWnkFa0MuKie1gEmVSfNhLcWYqbWaNy01J407YlpbzyNV5bdyupu9LO1W69Nf2/mdnBxUGjdhotZ4f1h8/0v+42RZfi/tijx1JB5TaZ6SY23x39bdrS+3vqr9Xvuj9lftb8V6Z0vLfCEqf7V/gf50sLu</latexit> <latexit sha1_base64="4qCWONnkrp2T9BdIkwGTnidGsek=">AVznicrVhLb9tGEN7EfaTuK07hUy9sbAMJQKuSHSBAgEBnARF0bQJ6sRJLSOgyJVEiK+QqyiOIPTav9Ef0nOv7bH/pjPfLkVSbweRIHI5O/PNY2eGu2ongZ+pev2/S5c3Pvr4k0+vfLb5+RdfvX1a1rz7N4kLrymRsHcfqi7WQy8CP5TPkqkC+SVDphO5An7f4Rz5+8kWnmx9GxOk/kWeh0I7/ju4i0qutjbutuz60Uj5/XeJ76pBKseblvnsWT/Hnswmz62IHk8z5Shpe07Wk96ZdaPlxlEnHkQeqblpOcq6UbfrN63Rbnlid3xvHgqLh9LzHRWnN63TtgziodWMO1Z9oyxcq4lQIocV6GMFCEFsqM0UC4IlAnLEph4oNw4lAS+t3ePBTDkWOUwvXQ65bDlTiqNx0gSwVY+txZ9bVXDv5ewlv8EMdxBKsvZXJROrMS7HeVZhsQLA7Gc01YZMFcZXsLtM0DLoc0CPwkyUYmNfxVbOsuARbANqW1NK25aXO0LbeZq4TyGa9dse2zvW4UWvYVuCQu024fBcOH1ImZDQ4pEQ2y9ySkVcpk1dXd+q1Oj7W7KBhBjvCfJ7EW1f+Ei3hiVi4YiBCIUkFI0D4YiMvqeiIeoiIdqZGBEtpZGPeSnGYpNkB8QlicMhap+uXo6NdSInhkzg7RLWgL6pSRpiT3D49G4A6q+s36rxLtIxwjYbOM53dsGMySqEj2irpLOdeVY58UWXgHvhkZwIKe+lWPOrQPaBnRfbz9Zw4JY08kp5BItIKqmsI6U7jqu7HkPcXbAJ2nENi32pk2L14Jno/p2ycsh8YZLGVbLfHIRD2CZglbmSfAi1GfEseavuWYXUmPvhYVe0F8x4TpS/e0aiKvExnOXcWcymDPJ6ji/kVOCLizogSI69ip9PHNVMdAmTdThYyS58SZAfNYP8I+bzXE1oPfdhT4ZVs5BNPvQkpm4Ku3M7A7q3gZ2S/IjmetDsbDJHgnsFLmZR9EGd0pPQzy5sNGdotdQd7yeNvnE2WUTbkwz/gRLR0LHM7dIWzcimW+In6Je07jaiwnltk2wMC3UWZ9AVmbVvUtfguIR0ZSpHNqfY0BHCjh6ypU+0Qp9GYn3fkxcZ6icwqCO6B4haAhQb2jk+Q4xD+MQrOoDuIY096GFpizpZTdw2dowrWjk2PnrPrFYL+ce1q/vetFaWi1CplsnKJjTVxS3jMa8A212Omt6ne64GbwbT1nVhv1F1+AcrdrHUsX6VNfKQzb2IFXF5SjO93OeLweoExtrybHvEkcTUh1Dz1CbidG0Waq/I1MlMdbDxT0OaHrnbUNp2ZGRA9NpALi03XObxyeU2YDnKqheiV5chPjTyea9v03cE6nK56isqmxKo9FkZrm8j5FEfPSIV1zjJYh1Al+m/ys0sy62B8GL5pB5IxVBmldA+5Md/zY7FDvWCHsqoaT15pqTYeyQ03p1w7hIu8/ahJSIKV/hoMj9euQYPiDI2EetgLq/DwgYfnc8DZ8tknubQFaWmOLiql2vlvJyXN5q+XNab5HtVtOXyzLlDXLTN/0gA9aLFXIK68ASYSlLj1dIxXgnaLq2+eUa9kV41w5MXqTGwpMVkiF6jPYqNlX5eKV9Cl0wNTFhmd/eI4Jv0P/G2AFdNI6FrLpQNAu58/eKaSE/vGBkC8nwgvEtJN+tkJR4N/gTGs8XVlTzPVkJfLryb5EV5GNa5uiw+9iz3Qcy7xHeLX1W6+J2rbNOyx/i+W7z6zU3xoL3amI+wihqYT5vuJfWjSv1AtfSY3tE8PhKHtD9o0PVRpWui8r79IGptzLuASHfpn73AH3x4riJ2U3vV/bcVR0PSUP+G5vTScEdmD3kIqSL6J2PrHPdm9nLTmvSu5Uecekd/PkKxKJ65uMtX9/5yPocHWMHWezjPwRy+YyHl6RM4tPlKvOuYtPXwndO3QN0BEyw/sAWRegM0jxzJwQ9dv3aOasy/l7QFXBuTu6fm1KXxV/+vTUxelnQBwFj94FK0hrWhlx0TksgYwyu40MZ5biZFarnLQ03rRtWSmvQoPXN1Kmq60+erqTmP6v5nZwclBrXGr1mg8Pdi5f8f8cXNFfCuixsUqNviPpXmEwqsu/Hnxt8b/2z8u/3L9mB7vP27Zr18ych8Iyqf7T/+B3fmB8=</latexit> Frontdoor adjustment: step 2 Identify the causal effect of M on Y P ( y | do ( m )) W T M Y Step 2 / 40 Brady Neal Frontdoor adjustment 19

  51. <latexit sha1_base64="QG57jZbv7M0gSwvR6kQ0evEFBxI=">ATFHicrVhLb9tGEF6nL9dtWkcF2kMvbJQCDkCrkmvAuQgIYCcoigZwADt2GwUGRa4sQnyFpKw6hA79Ef01RU9Fr73F/RvdObpUjqrSAWRK5mZ76Zndfuht5bpI2m/9u3Xnv/Q8+/Gj7451Pr372e792ovknAY2/LcDr0wvuxaifTcQJ6nburJyiWlt/15EV3cMzFzcyTtwOEtvI/nKt64Dt+faVkqkq93fOtFeJxymdujLG8szOr7rGJ2TcK/jS8e10jAm6sOHRtvoJEP/qpMSeOrLIAXzPNmSnGlU+B8aHRMyFeLVbr3ZaOLPmB209KAu9N9peG/7T9ERjgiFLYbCF1IEIqWxJyR0OelaImiIj2SmREi2nkYl6Ksdgh2SFxSeKwiDqg5zX9eqmpAf1mzATSNmnx6BuTpCG+1TwOjXugqjfrN0q8i3RkwGYb+nd1Zg+UVPRJ+oquZxzXTleU0oWPsJaXLIzAoVXaVdW1KO3R79Tsp+ft8QpaeSQVEwjm2geURWFdcT0Vn7lfhZwt8kZs0+LVdMn2xZHg+ZA+A8KyaJzAUrbVE+1wNolrCVeTxEbDHir7RCZd8yrN5kDS6iqlbBvGdEGYg3NKoiL9NZzp3FXKlGHs/RxfwpOALiTogSIq9d8p9LHNVMtAmTdViI5DXWEiE/Ghr5R8znuRpRPdhT4KoGcgmF3oiXTeF3bmdHr27wI5JPqO5PvSwL0yRwI7Rm7mXjTBHdOvEX7ZsNGeojdQdxPk9bE2WUSbkgz7gRLeUL5M7dIWZcRzdCfET9EuKuwmvcF6bJBvCQpXFCXQFOvZt6hrsF5+eTGXP5hQTOnzY0Ue2DIhW6FNIrO87WkWC+vE0akZvD16LgGJCO/tnhLGPNXFEh9A9orEDPSxtUCdriCNtx7ilX3jovfMajWQf1y7qu9Na2W5AJVq6KxsQ1NTHOoVcwTY7rLXbN3rVMdNsLrxlFVd2F90Dc7Rqn0sVcSnGisH2diHVBWXvTh/nfPWcoA6MRFL9v01cbQh1dP0BLUZaU07pfo71lUSIh423r7OCVXvrG0NZMR3de8ohP1TnvODzmjAt5FQH3ivPL0N8ouXzXt+lTwbqcrkXqKyqbEyjbDKzXN7FSMI/asQRV3gRfB1hLd9MvkZpZl3sd4MXzCByxqYaV10B7kxf+Vnok69oE5ZVfUnR5opMc4eEY0fTDgfEC7zDqAlIApXeDaZH6+MwQlRxtpjPczldVjY4KLzOeDs6MxTHKqi0ikOrurlWjkv5+WNoi+XdSb5XpV9OWyTLlBbrq6HyTAulwhlyIOLOGXsvRshVSIPUHRlc0/r2FfgL12qPMi1hZerJD0WPUqkJdlc9W2peiC8baJyzy1t48Ab9b4wT0KZ+LGTjbxZyN2+lU8L+dGni0k/Q39W0i+WSEpsTe4ExrLPF9ZU8x1uhL59eRcoqrIxLNL3uG92NEdx9D7CEdb7Xpt1Lap97B8F8tPn0mpv7nQXpxM5wiRroT5ueJfWhSa/uBaukZ7dE8Phbf0/mgRc+nla65Liqf04e63sq4B4R8RP3uBH1xc9xIn6b3K2fuqo4npCH/jvXtpOD29BlyEdImeucjq1x3Zs6y05rUaVPXOoEf7sCsaie+XjL4zsfWd2jQ5wgi3P8u0Au31HWwytyZvGNctU9d/HtK6J3j54eOkKieU+QdR46gxTn+oaodt/jmbsu5+8BVQXnbna/NqWvij9e7rG7WdIHAWPOgWnkFa0MuKie1gEmVSfNhLcWYqbWaNy01J407YlpbzyNV5bdyupu9LO1W69Nf2/mdnBxUGjdhotZ4f1h8/0v+42RZfi/tijx1JB5TaZ6SY23x39bdrS+3vqr9Xvuj9lftb8V6Z0vLfCEqf7V/gf50sLu</latexit> <latexit sha1_base64="etULFHFN9sWdeuOCtSayGN9DhY=">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</latexit> Frontdoor adjustment: step 2 Identify the causal effect of M on Y P ( y | do ( m )) W T M Y Step 2 / 40 Brady Neal Frontdoor adjustment 19

  52. <latexit sha1_base64="QG57jZbv7M0gSwvR6kQ0evEFBxI=">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</latexit> <latexit sha1_base64="etULFHFN9sWdeuOCtSayGN9DhY=">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</latexit> <latexit sha1_base64="QG57jZbv7M0gSwvR6kQ0evEFBxI=">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</latexit> Frontdoor adjustment: step 2 Identify the causal effect of M on Y X P ( y | do ( m )) )) = P ( y | m, t ) P ( t ) t W T M Y Step 2 / 40 Brady Neal Frontdoor adjustment 19

  53. <latexit sha1_base64="2uDt9fKu4fa/U8cv7ZLSawAUzd4=">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</latexit> Frontdoor adjustment: step 3 Combine steps 1 and 2 to identify the causal effect of T on Y W T M Y Step 3 / 40 Brady Neal Frontdoor adjustment 20

  54. <latexit sha1_base64="TimGMbSX9Ug74mo/kChlzWTeYlU=">ATIXicrVjbtGEF2nN9e9Oepb+8JGLhADsiq5BpwXAQHsBEXRA5gx26jwJAoyiLEW0jKriPoF/oR/Y+97HoW9Hf6Af0zNmVSMq6BpEgcjk7c2Z2btxVO/LcJK3V/tm497H3z40ebHW598+tnX2zfL71IwkFsO2d26IXxRbuVOJ4bOGepm3rORQ7Lb/tOeft/pHMn187ceKGwWl6Gzmv/NZV4HZdu5WCdLn9WzN62AwHqR36znXLs5q+27Gax+HDZgqY1HeCFOTdXathNZOBf9n0nY7bSsOYzJAtPM8TblasuXpyALu7l9vlWrXGj3V3UDeDsjKfk/D+5h+qToqVLYaKF85KlApxp5qQTfl6quaioC7ZUaghZj5HLeUSO1BdkBuBxwtEDt43qFp5eGuBZMBNK29Di4RdD0lLfGp4Oxl1S9V30WzneTqGxBYb3FvG0wf1FT1QF0mN+ZcVU7WlMLCR1yLCzsjUmSVdmFXdw9PKewX634HQw6kAqxsgGzQNVU0RHjLv2q6y8Rz+3yOdgJDbNX0bts+PhMyH+PaB1cI4oaViq6WeGq8H1OzQVuHxGLH5iL9ihdq+RVjdyRpcRlWvQnhPQemrNxgVkRfpzOfOfK7UI9m6BL+lBwBuBNQua1C/+54Chmog1M0dFiJK+4loj5UTXIP3J+nKsR4rlHexJGzWI2udQTmbrJ7B7b6eHeJnYM+SHmetQjvqjAHofYMXNz7MUKuWM83fDJpo32FL3KupN4VrAmya4KcEPMuBMs7Qntz7F2rohaJb57qmfqN9B3Cv0iuR1BbIhLdRZnFBXYGLfQNcQv/i4ClU8O6ZUqMOnHT1mSx+0TJ9GEn3fYRUJ68czqEPcPXotIkqF2sU/Nxz7XJNEdEDdNxh3qEekLXSyqjo0dowKWsU3LnvPXa0W809qV/e9a0iF7BSLZOVDWqQOzYomA2J3m16ne64CVc3mrKqTfuzriE5WrRPpL4FGPVYTb2KFXEFS/OXuesteyzTiqMpfj+ChwNSnUNPWFtRkbTVq7+jkyVhIyHzbtvckLXu2i7mZoZgu4bT3ng03UubxyZeQ3MFnOqSe/l5xchPjHy417fxndI6mK5F6ysomyM0XAys1je5cihf/RIq7xIvo64lq+mfys3Myq2O8GL7iDKBmbGqRV0TvMjdkrP1Vl9IysqroT4m0UGLuPSKMdyacO8AV3j61BKBIhQ8n86OlMTgGZWQ81uXcuA4zG1x2vg45mybzNIeuqHSKQ6p6sVbJy1l5o+mLZTuTfC/KavpiWaFcMzd0w8SYl0skUsZB5Hwc1l6ukQq5DtB07XNP69gX8B37cDkRWwsPF8i6bPH6FWFpiqfLbUvZReMjU9E5pe38OA1+9+IO6B1/ZjJpmt5M5O7fSufZvI3a3o2k/TX9G8m+WaJpMN3gzuhiczpTUlXCdLkV9P9iW6iq8tuEdeRd3TMexzHtEoq3feg3WdsW8w8ZvsfHuM8n1N5fas53pkLuIG9MJx/uJPWrSa/sBtfQM72gZH6nvsT+o4/q0DVXRZV9+sDUWx53H8iH6HfH7Ivr40ZmN71X2HMXdTyBhvFvZE4nGbdn9pDzkNbROxtZ53rnzl52WpPerfTApXfwt0sQs+qZjbc4vrOR9Tk65A4y28e/C+T8GWU1vCxn5p8ol51z5+Ity7uHrsCInhPWbWewMjozJ0T9j26c9aV/N1HVUjuDh+UpvQV8adPT1c8/QzAkfHoXBKaU3LI847h0WUSc1uI+GZJTuZVQsnLY03bVuSyvf4DVMt3JMV9q63C7Xp/+buTs436/WD6r1+vOD8uNH5o+bTfW1eqAewlGH6jFK8wSOtdV/G19tlDd2Sr+X/iz9Vfpbs97bMDJfqsKn9O/HlTHOw=</latexit> <latexit sha1_base64="2uDt9fKu4fa/U8cv7ZLSawAUzd4=">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</latexit> Frontdoor adjustment: step 3 Combine steps 1 and 2 to identify the causal effect of T on Y Goal P ( y | do ( t )) W T M Y Step 3 / 40 Brady Neal Frontdoor adjustment 20

  55. <latexit 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<latexit 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<latexit 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Frontdoor adjustment: step 3 Combine steps 1 and 2 to identify the causal effect of T on Y Goal P ( y | do ( t )) P ( m | do ( t )) W T M Y Step 1 Step 3 / 40 Brady Neal Frontdoor adjustment 20

  56. <latexit 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<latexit 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<latexit 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Frontdoor adjustment: step 3 Combine steps 1 and 2 to identify the causal effect of T on Y Goal P ( y | do ( t )) P ( m | do ( t )) )) P ( y | do ( m )) W T M Y Step 1 Step 2 Step 3 / 40 Brady Neal Frontdoor adjustment 20

  57. <latexit 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<latexit 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<latexit 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<latexit 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Frontdoor adjustment: step 3 Combine steps 1 and 2 to identify the causal effect of T on Y Goal X P ( y | do ( t )) )) = P ( m | do ( t )) )) P ( y | do ( m )) m W T M Y Step 1 Step 2 Step 3 / 40 Brady Neal Frontdoor adjustment 20

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<latexit sha1_base64="TimGMbSX9Ug74mo/kChlzWTeYlU=">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</latexit> Frontdoor adjustment: step 3 Combine steps 1 and 2 to identify the causal effect of T on Y Goal X P ( y | do ( t )) )) = P ( m | do ( t )) )) P ( y | do ( m )) m X X P ( y | m, t 0 ) P ( t 0 ) = P ( m | t ) W m t 0 T M Y Step 1 Step 2 Step 3 / 40 Brady Neal Frontdoor adjustment 20

  59. <latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit> The frontdoor adjustment and criterion X X P ( y | m, t 0 ) P ( t 0 ) P ( y | do ( t )) = P ( m | t ) m t 0 / 40 Brady Neal Frontdoor adjustment 21

  60. <latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit> The frontdoor adjustment and criterion If (T, M, Y) satisfy the frontdoor criterion, and we have positivity, then X X P ( y | m, t 0 ) P ( t 0 ) P ( y | do ( t )) = P ( m | t ) m t 0 / 40 Brady Neal Frontdoor adjustment 21

  61. <latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit> The frontdoor adjustment and criterion If (T, M, Y) satisfy the frontdoor criterion, and we have positivity, then X X P ( y | m, t 0 ) P ( t 0 ) P ( y | do ( t )) = P ( m | t ) m t 0 A set of variables M satisfies the frontdoor criterion relative to T and Y if the following are true: / 40 Brady Neal Frontdoor adjustment 21

  62. <latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit> The frontdoor adjustment and criterion If (T, M, Y) satisfy the frontdoor criterion, and we have positivity, then X X P ( y | m, t 0 ) P ( t 0 ) P ( y | do ( t )) = P ( m | t ) m t 0 A set of variables M satisfies the frontdoor criterion relative to T and Y if the following are true: 1. M completely mediates the effect of T on Y (i.e. all causal paths from T to Y go through M). / 40 Brady Neal Frontdoor adjustment 21

  63. <latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit> The frontdoor adjustment and criterion If (T, M, Y) satisfy the frontdoor criterion, and we have positivity, then X X P ( y | m, t 0 ) P ( t 0 ) P ( y | do ( t )) = P ( m | t ) m t 0 A set of variables M satisfies the frontdoor criterion relative to T and Y if the following are true: 1. M completely mediates the effect of T on Y (i.e. all causal paths from T to Y go through M). 2. There is no unblocked backdoor path from T to M. / 40 Brady Neal Frontdoor adjustment 21

  64. <latexit sha1_base64="2tkLk9Yiao3kHPCdzl9N1f7aGpo=">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</latexit> The frontdoor adjustment and criterion If (T, M, Y) satisfy the frontdoor criterion, and we have positivity, then X X P ( y | m, t 0 ) P ( t 0 ) P ( y | do ( t )) = P ( m | t ) m t 0 A set of variables M satisfies the frontdoor criterion relative to T and Y if the following are true: 1. M completely mediates the effect of T on Y (i.e. all causal paths from T to Y go through M). 2. There is no unblocked backdoor path from T to M. 3. All backdoor paths from M to Y are blocked by T. / 40 Brady Neal Frontdoor adjustment 21

  65. See proof of frontdoor adjustment using the truncated factorization in Section 6.1 of the course book

  66. Question: What is the intuition for why the frontdoor criterion gives us identifiability?

  67. The magic of randomized experiments Frontdoor adjustment Pearl’s do -calculus Determining identifiability from the graph / 40 Brady Neal Pearl’s do -calculus 24

  68. Can we identify the causal effect if neither the backdoor criterion nor the frontdoor criterion is satisfied?

  69. Yes, and do -calculus tells us how.

  70. Pearl’s do -calculus Will allow us to identify any causal quantity that is identifiable / 40 Brady Neal Pearl’s do -calculus 27

  71. Pearl’s do -calculus Will allow us to identify any causal quantity that is identifiable <latexit sha1_base64="PbGifmL7z2FVrihKI14Mtwy4tG4=">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</latexit> P ( Y | do ( T = t ) , X = x ) / 40 Brady Neal Pearl’s do -calculus 27

  72. Pearl’s do -calculus Will allow us to identify any causal quantity that is identifiable <latexit sha1_base64="PbGifmL7z2FVrihKI14Mtwy4tG4=">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</latexit> P ( Y | do ( T = t ) , X = x ) where Y, T, and X are arbitrary sets / 40 Brady Neal Pearl’s do -calculus 27

  73. Pearl’s do -calculus Will allow us to identify any causal quantity that is identifiable <latexit sha1_base64="PbGifmL7z2FVrihKI14Mtwy4tG4=">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</latexit> P ( Y | do ( T = t ) , X = x ) where Y, T, and X are arbitrary sets Multiple treatments and/or multiple outcomes / 40 Brady Neal Pearl’s do -calculus 27

  74. <latexit sha1_base64="0qan9QIDmpdGkebkgeb20VOC5sc=">AU/HicrVhLb9tGEN6oL9d9xE5964WNHMAGZFWSDbhAYCA7aAomsIB7Fi1ZRh8rCRCfIVcWbUJ/Zoei15762/ouT+k17Yzs8uXHpQURILI5cx8M7OzM8NdGYFjR6LR+PtR5YMP/r4k7VP1z/7/IsvH29sPnkT+cPQ5Bem7/h29Aj7tgevxC2cHg7CLnuGg6/NAbHyL+842Fk+965uA/4jav3PLtrm7oA0u1mpd8xeM/2YmEPHgLbFMOQj9c19Xm/eRbPEqfOx48XkdCF/xG2nvarQdhq1xq4Wb7e3x8/nCF42d7Vr3fDvuObwrtCO/K7WvgHM5W2zBNVKUaHd6xdgrfmwK0AZ3PFHmfxViZH9VDx17Ura2C9BHaSozDUFO0hguSieWr18FANd9GVUODAwkM+nmK0SZjvhXZXwYGZTzKvlmAdl/szk7k9xO9yzCkl1u1Ft1Bv0aYHTWoMvU58zfX/mQdZjGfmWzIXMaZxwSMHazCL7XrMkaLADaDYuBFsLIJj5nY7YO2CFIcZDQgTqAaw+erhXVg2fUGRHaBCsO/EJAauyZkrFg3CWqvKN9LSc7z0ZMutHe7gbSqcLVMH6QF2ESySXxeGcBHj4Hc3FBj8DouAszcKMunB34FmA/3i9B0kOIwtQIYxMoDlAlRS0EcJdxhVn3qc46yTHYQ+zZ+NAb7PXwnk+/AdgC4dxhF5ir5q7KWKukeWOfmKMg6t2HyNv8AMpX9lurpHGxaVTkLlD0HyoA9wKioucxmPnfmSwmleTzDFsoLkvBAOgKT3ltQ/xskChmogk60YZOK9mjuQSUH3Wl+QfiJ7kawHrukT8RrZpG2WSTnUDVTeZ34qcDd4N0h4CPgdcnOxiLGvjDSXdIuZlEsUbSITyN6MkH80Jep3qDtezBnPC7KqBXh84dqpLRkLGM/FIehcDTVPfPfYj2ew7jWKCuZ1DbA+eSizOCJbnlr7I+gaGBcXrkjFyCaUGtlwyY8+ZcsAaJk9qQntfQuziKh+HKU1hrtDUQtIS42sY3xGNHZpTriQ7I9grFdhCtQSers0Plx7hgFWNjU+ZtqpR/mHtyr43aRVxHlWqprLyiCw12IGaMa4A+p2Pmql6ney4Ec1uPOGVQf5nXQNztOgforL1Ka6VRdnYJ1RL0Zx9jxnzaVFdVKjtcTY90DiFBdRY+oNgNlaT1Xf8eqSnxaD5PursoJWe9obTBiYHuqkg5ICfrHN84yHkLOnXKqQ5FL8v03iq8EmvN+AbE7Uc94Yq4gNYRSnHK8TSNO8ZEjXHGpL6BYBzSXb9KfluMsq/v96POmNGLGCqVpWe0W5cbsmZ+zKvSCKmRVMZ640kgJae8RwHg7ldwGvSg7ICseULDC45Q/XrgGJ0AZq4h1iZfUYeaDTZ3PIsmOyjwpIStKTEhgVZdbxbyclTeSXo610nwvYiW9HIuUO8pNW/WDiHS1F+AErQMi3FyWni9A+fROkHTp89L+OfRu3ao8iJUHl4uQLrUY+SsfFWVrxb6J6gLhiomiLl6hwjeUf8b0w5o1ThmWLFSNDPc/TvFNMOPVoxshnRXjG+GfFiA5PRusFMaYl4vrCmUOluo+W26L5FVKOrAdHBd7GlOo6m3iO42vKtd0S1XVPvsOQtluw+o1x/s8l6tjONaRcxUp0w2U/skSU5t+hl7BOxrHx2wf9gdNuL4sdM1lteI+fajqLa+3BZoPod+dUF9cXW+gdtN7hT130cYpWEh+Y3U6yaQdtYecp2kVu7M1y1y3pvayk5bkbqUPUnIHf79AY1Y9s/WVr+9szfIc7dMOMtvHvw/N+TPKcvqynJl/olx0zp1/+grg3oWrQx0hUrInlHUOdQbOLtQJUb59j6fOupi/LagKzN346ZMJe0X9k6enHp1+hiCRychdsC0pOU1zjuHBYQRarcR0ZklO5nVCyctqW/StyiXV67Sd6S6FVdaf12o9qc/G9menDZqjcP6s3m61b1xYH642aNfc2esh0I1CF7AaV5BoE1K39V/qn8W/lva7z169ZvW79L0cojhfmKFT5bf/wP7X86qA=</latexit> Notation for Pearl’s do -calculus <latexit sha1_base64="olJGHQ+TbdRQC38+mVX+dyzlS+s=">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</latexit> G W 1 W 2 X Z W 3 W 4 / 40 Brady Neal Pearl’s do -calculus 28

  75. <latexit sha1_base64="0qan9QIDmpdGkebkgeb20VOC5sc=">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</latexit> Notation for Pearl’s do -calculus <latexit sha1_base64="MJkfQFqvZYC2fGlQPObRv2lIEAg=">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</latexit> G X W 1 W 2 X Z W 3 W 4 / 40 Brady Neal Pearl’s do -calculus 28

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  78. <latexit sha1_base64="0qan9QIDmpdGkebkgeb20VOC5sc=">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</latexit> Notation for Pearl’s do -calculus <latexit sha1_base64="AEKi/9Pzy+V3NF4QaJa4raLK94=">ASo3icrVjbtGEN2kN9e9OepjX9TIBfogK5JrwHkxEMBOWgQN4BR27DYODImiLEK8haSsOoT+pK/tX/RD+jc9c3YpkroriAWRq9mZM7Nz213QteJk2bzv3v3P/r4k08/2/p8+4sv/r6m50HlVdxMIws+9wK3C67LRj23V8+zxEte+DCO7XVc+6IzOJb5i1s7ip3AP0vuQvuN17xnZ5jtROQrnd2fr5OrwJwCEB6OR5f79SajSb/qrODlhnUlPk7DR5s/auVFcFylJD5Slb+SrB2FVtFePzWrVU4WgvVEpaBFGDudtNVbkB2CywZHG9QBnjf49dpQfwWzJjSFrS4+EaQrKofDE8X4x6p+i36qwXeRTpSYouNd3h3DKYHaqL6oK6SyzjXlZM1JbDwMdfiwM6QFmlVpRD28XvxPYL87cNoYdSEVYWSB5oKqKaIjwlv7Vbep5/b5LMxEpsWr6YD2xdHQuYDfAbAamMc01KxtaqeGa/71GzTVuFxGbHFiH9ihdq+ZVi9yRocRlWvQnjPQBmodxiVkZfpLObOYq7EI/n6BL+hBw+uGNQAua1A/854ChnogVM0dFmJG+4lpD50TDIzmf5WqIeO7RnphRqzKbHOoJTd3kdmd2unh3iB1BPsVcn3rEF3XYxM7Ym5mXqyTO8KvEX9ZtNGaojdYdxLPOtYk2VUHboAZ4KlPaH9mVmkrUtBq5rPnvqV+m3EvU6vSF7XIRvQp3FMX5JvZH6BriFw9PoYpnM0qdOjza0We2DEDL9Wk0fcIq4hZP65BTfF26bWQKHVqF/+MOPa4JonokLpHGHepR6Sr6GQNdWjsGJe0im8c9p5ZrVXmn9Su7nvTWkXOZ6VWTVYeUVNTHZgVSwTE7qLXLNPrdMeNubrxlFUd2p93DcnRsn0ilcenHKsus7FPqTKueH+OuetZ91Umcsxfc34DiVM/QY9ZmaDRtF+rv2FRJwHhYfHsmJ3S9i7bR1EwKumc85YJP17nsODLzFpht5tQVvVecX4b41Mhnvb6DT0rqcrlXrKybIROplZLu9wZNM/eiQR13ghfR1yLd9PvtXCzLrYHwbPn0GUjE0M0roXebG/JWfqRp6Q1ZVfanRFoEc8eIca7E85d4ArvgFp8UKTC08n8eGUMTkAZG4/1OJfVYW6Dw87XJeVyTzNoSsqmeKQql6uVfJyXt5o+nLZ7iTfy7KavlxWKLfMTcf0g5hYlyvkEsZBJLxClp6tkAq4J2i6tvn3NezudcOTV5ExsKLFZIe4xeVWCq8sVK+xJ2wcj4RGT+eA8P3rL/jXkC2tSPuWykTdzubv38mkuP9rQs7mkt6F/c8l3KyRt7g3OhCYyL1fWlHCdrkR+OzmX6Cq89mBd2Qv7pqOUzX7iERb73pHrO262cOyXSw7fcaF/uZQe34yTXmKGJlOmJ0n9qhJr+0X1NIL7NEyPlY/4XzQwvNZqWuiyrn9KGptyLuPpAP0e9O2Bc3xw3NaXqvdOYu63gKDdl3bG4nObdrzpCLkDbROx9Z53p35iw7rUmfVvrg0if4uxWIefXMx1se3/nI+h4d8ASZn+M/BHLxjrIeXp4zi2+Uq+65i29fId49PF12hNjwnjDrXHYGW52bG6LefY9n7rqSv/uoCsnd9GFlSl8Zf/r2dMPbzxAcOY8+BSeU1rQi4qJ7WEiZxJw2Yt5Z8ptZo3T0njTtsWFvPIM3pHpVrbpStvXO7XW9P9mZgcX+43WQaPVenlQe/LY/ONmS32nHqof4ahD9QSleQrHWmhf6m/1T+V3crzym+VM816/56R+VaV/ipv/gekP5iU</latexit> G X W 1 W 2 X Z W 3 W 4 / 40 Brady Neal Pearl’s do -calculus 28

  79. <latexit sha1_base64="v7YqGaHluauYp9hle+K9DFDOdk=">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</latexit> Notation for Pearl’s do -calculus <latexit sha1_base64="AEKi/9Pzy+V3NF4QaJa4raLK94=">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</latexit> G X W 1 W 2 X Z W 3 W 4 / 40 Brady Neal Pearl’s do -calculus 28

  80. <latexit sha1_base64="v7YqGaHluauYp9hle+K9DFDOdk=">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</latexit> Notation for Pearl’s do -calculus <latexit sha1_base64="8DuAaGmIN4aWOK/zbYOjY2iYg6I=">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</latexit> G XZ W 1 W 2 X Z W 3 W 4 / 40 Brady Neal Pearl’s do -calculus 28

  81. <latexit sha1_base64="YXc9aV4OG6vYNGgAH83uVy9eJ0c=">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</latexit> Notation for Pearl’s do -calculus <latexit sha1_base64="8DuAaGmIN4aWOK/zbYOjY2iYg6I=">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</latexit> G XZ W 1 W 2 X Z W 3 W 4 / 40 Brady Neal Pearl’s do -calculus 28

  82. Rule 1 of do -calculus / 40 Brady Neal Pearl’s do -calculus 29

  83. Rule 1 of do -calculus <latexit sha1_base64="5aSrsPTDmBuevDY57v75xBQ81M=">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</latexit> P ( y | do ( t ) , z, w ) = P ( y | do ( t ) , w ) if Y ⊥ ⊥ G T Z | T, W / 40 Brady Neal Pearl’s do -calculus 29

  84. Rule 1 of do -calculus <latexit sha1_base64="5aSrsPTDmBuevDY57v75xBQ81M=">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</latexit> P ( y | do ( t ) , z, w ) = P ( y | do ( t ) , w ) if Y ⊥ ⊥ G T Z | T, W Question: What concept does this remind you of? / 40 Brady Neal Pearl’s do -calculus 29

  85. Rule 1 of do -calculus <latexit sha1_base64="5aSrsPTDmBuevDY57v75xBQ81M=">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</latexit> P ( y | do ( t ) , z, w ) = P ( y | do ( t ) , w ) if Y ⊥ ⊥ G T Z | T, W Question: What concept does this remind you of? / 40 Brady Neal Pearl’s do -calculus 29

  86. Rule 1 of do -calculus <latexit sha1_base64="5aSrsPTDmBuevDY57v75xBQ81M=">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</latexit> P ( y | do ( t ) , z, w ) = P ( y | do ( t ) , w ) if Y ⊥ ⊥ G T Z | T, W Question: What concept does this remind you of? Rule 1 with do(t) removed: <latexit sha1_base64="djynxIpYlUCnIslXk0sHxpNh1Q4=">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</latexit> P ( y | z, w ) = P ( y | w ) if Y ⊥ ⊥ G Z | W / 40 Brady Neal Pearl’s do -calculus 29

  87. Rule 1 of do -calculus <latexit sha1_base64="5aSrsPTDmBuevDY57v75xBQ81M=">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</latexit> P ( y | do ( t ) , z, w ) = P ( y | do ( t ) , w ) if Y ⊥ ⊥ G T Z | T, W Question: What concept does this remind you of? Rule 1 with do(t) removed: <latexit sha1_base64="djynxIpYlUCnIslXk0sHxpNh1Q4=">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</latexit> P ( y | z, w ) = P ( y | w ) if Y ⊥ ⊥ G Z | W Generalization of d-separation to interventional distributions / 40 Brady Neal Pearl’s do -calculus 29

  88. Rule 2 of do -calculus / 40 Brady Neal Pearl’s do -calculus 30

  89. Rule 2 of do -calculus <latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">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</latexit> P ( y | do ( t ) , do ( z ) , w ) = P ( y | do ( t ) , z, w ) if Y ⊥ ⊥ G T ,Z Z | T, W / 40 Brady Neal Pearl’s do -calculus 30

  90. Rule 2 of do -calculus <latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">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</latexit> P ( y | do ( t ) , do ( z ) , w ) = P ( y | do ( t ) , z, w ) if Y ⊥ ⊥ G T ,Z Z | T, W Question: What concept does this remind you of? / 40 Brady Neal Pearl’s do -calculus 30

  91. Rule 2 of do -calculus <latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">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</latexit> P ( y | do ( t ) , do ( z ) , w ) = P ( y | do ( t ) , z, w ) if Y ⊥ ⊥ G T ,Z Z | T, W Question: What concept does this remind you of? / 40 Brady Neal Pearl’s do -calculus 30

  92. Rule 2 of do -calculus <latexit sha1_base64="HcYOjFqs6PGL84DcuD/U7PeV+Bc=">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</latexit> P ( y | do ( t ) , do ( z ) , w ) = P ( y | do ( t ) , z, w ) if Y ⊥ ⊥ G T ,Z Z | T, W Question: What concept does this remind you of? Rule 2 with do(t) removed: <latexit sha1_base64="2jPfRjazUWY+VS9eOWylbCzqCUM=">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</latexit> P ( y | do ( z ) , w ) = P ( y | z, w ) if Y ⊥ ⊥ G Z Z | W / 40 Brady Neal Pearl’s do -calculus 30

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