Transfer Learning andTransportability
Brady Nealcausalcourse.com

Causal Insights for Transfer Learning

Transportability of Causal Effects Across Populations

Brady Neal

2 / 22

Causal Insights for Transfer Learning

Transportability of Causal Effects Across Populations

Brady Neal

Causal Insights for Transfer Learning

3 / 22

Transfer Learning
Task 1Model 1

Training Data 1

Transfer

Training Data 2

Model 2

Task 2
Brady Neal

Causal Insights for Transfer Learning

4 / 22

Domain Generalization
Task 1Model 1

Training Data 1

Transfer

Training Data 2

Model 2

Task 2
Brady Neal

Causal Insights for Transfer Learning

5 / 22

Domain Generalization
Task 1Model

Training Data 1

Transfer

Training Data 2

Task 2
Brady Neal

Causal Insights for Transfer Learning

5 / 22

Domain Generalization
Task 1Model

Training Data

Transfer

Test Data

Task 2
Brady Neal

Causal Insights for Transfer Learning

5 / 22

Domain Generalization
Task 1Model

Training Data<latexit sha1_base64="mfwn6V3fkGjP6QF8uYK4AHVh954=">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</latexit>

Ptrain (x, y)Transfer

Test Data

Task 2
Brady Neal

Causal Insights for Transfer Learning

5 / 22

Domain Generalization
Task 1Model

Training Data<latexit sha1_base64="mfwn6V3fkGjP6QF8uYK4AHVh954=">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</latexit>

Ptrain (x, y)Transfer<latexit sha1_base64="4LGBaeVNmnFYO6payq7p8AH+xDA=">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</latexit>

Ptest (x, y)

Test Data

Task 2
Brady Neal

Causal Insights for Transfer Learning

5 / 22

Domain Generalization
Task 1Model

Training Data<latexit sha1_base64="mfwn6V3fkGjP6QF8uYK4AHVh954=">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</latexit>

Ptrain (x, y)
6=
Ptest (x, y)Transfer<latexit sha1_base64="Sy1LQhoAZGc0wmbkzcPX4960OLU=">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</latexit><latexit sha1_base64="4LGBaeVNmnFYO6payq7p8AH+xDA=">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</latexit>

Test Data

Task 2
Brady Neal

Causal Insights for Transfer Learning

5 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)<latexit sha1_base64="v98VzaZHrSrL3ggKrtLeMjImVcg=">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</latexit>

Brady Neal

Causal Insights for Transfer Learning

6 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)
Goal: Model Etest (Y | x) only given access to Ptrain (x, y)<latexit 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Brady Neal

Causal Insights for Transfer Learning

6 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)
Goal: Model Etest (Y | x) only given access to Ptrain (x, y)
Covariate Shift Assumption: Ptrain (y | x) = Ptest (y | x)<latexit 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Brady Neal

Causal Insights for Transfer Learning

6 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)
Goal: Model Etest (Y | x) only given access to Ptrain (x, y)
Covariate Shift Assumption: Ptrain (y | x) = Ptest (y | x)<latexit 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Ptrain (x) 6= Ptest (x)<latexit sha1_base64="78fnabqyPHMIObd2bl6JG18IMzs=">AAAVlHicrVjZbttGFJ2kW+puTgr3oX1h4xRIAVmV3BTOQw0EsB0URVM4gLc2DgyJomzC3EJSVhxBQL+mr+3v9G967pmhSGqjZFiCyNGdu81dZ6YdeW6SNhr/3bn73vsffPjRvY9XPvn0s8+/WL3/4CgJe7HtHNqhF8Yn7VbieG7gHKZu6jknUey0/LbnHLcvd2T++MqJEzcMDtLryHntt84Dt+varRSgs9WvT6M0brnB47ffW9Zp4LyxAHCSFP/PVtcb9QY/1uSgaQbrynz2w/srTXWqOipUtuopXzkqUCnGnmqpBN9XqqkaKgLstRoAFmPkct5RQ7UC2h6wHGC0AL3E8xz/XhlogP/CMyG1DSkefjEoLfWdwelg3CVUv0W+VcCdJWNA3qLjNd5tw9MHNFUXgFbRZZiL07Xx9StWnWINT7laFyuJCBE72KU1d/H28D/FCuV5DUwHow6oYoxswDxANURkxHhry4ttLuiJFvEcjETreXrP01rmQ3wvwauFcUJNRVdLPTd+CSjZoa6C49Gnszm+xQq1fvN4dUdrcOl3vQrBPQDkUr3DqMx5nsxidM3GSg3n4RRZgp8SIwB2AkjIyHdhPxcY5Vi1wVNktOjJc64lYgTVDedfOZ9FcwR/blCfhF6zGG8u5UQms3K9Mz09vNvkHYN+gLkLyhFb1KCPQ94xozezYo3YMf71+c+mjvYYvM7MFH/WsCaJrhr4hphxR7y0JbQ9M420dgPALPPdUL9RvgO/12gViesaaENqqKM4oazA+H4bdUXs4uMpULFsBqlRhk89Lhgtl4Dl8jQnkfcDVpEwfzzDdYC3R6tF5FKjdLFPn2OfaxKP9ii7j3GHcoTaQq2rqy2jx7AkVWzjsjpNSrUYf5K7ujKOSxW6gJlqmajcpqSGemJWLB4QvYtWs0011DU54eqGY1q1qX9eNSRGy/oJVe6fsq86jMYLUpX5ihWnr3PaWjaZJzX6Umx/DoxtUnUNPGFuRkbSSiH/dkyWhPSHzbdvYkLnu0jrj80MAPeNpTzg6TyXniQzb8CzxZg6pfWK8/M47hn6rBtIxR8QOp/uiJlVpo0xGoxm5tO7HDm0jx6JxzW/iLaOuJZvRz+rMLMo79vhF0xwlIhNDadFuXcYG9NXfqDWUQvWEVVle4qnBRJzdxJh/GiE+Qh8BfeSUgJAJMMHo/lhpQ92ARkai3U5l+VhroPLytch5qmJPI2hMyodw5Csni9V4nJa3Gj4fNrOKN7LtBo+n1YgV4xN19SDhLxOKuhS+kEo/EKUHlRQhewJGq51/mMB/QL22p6Ji9hoeFxB6bPG6FWFJitfVOqXsgrGxiZC8+cNLHjF+jfkDmhZO+a06VLWzOmub2TTnL6/pGVzSn9J++aU7yooHfYGdwQTmpeVOSVY+5VYOp6qeoKDbpLtX3S21fhsw4rSszumMlmm30hU6O64zRpQW6g+T/YdyeRmZR73WH3brEMxJXfm5M7hrXHLPdhbYH0J4z0/Xy6SX9PoFo+bHfphWnXMZqq6XGhOArP6ssawJvrbTftqxLW2KDPr1GeFzjIozVfzcgoRO8kpm9X7sHwvlp2hkkKXdiknP18NuBfum1Vlu+IN5oGO5F/QEV5gpzmkvX/ELreJ5/NS71+Uq5w2e6ZrFPlugvMWuvYuu/vyfCNzJtwonRzLMvYgIfsNzRk7x5b9bWp69TROy8idzlnnXmfiRDYuSe+5L4Clz6HXFRzzbJ7Ob75/p3PW90Uhz0H5afQ2OBdP2ovxy2Nm9r1I1W3N7DuECO8unh4rQGJwdxl1Hvubg0qr7zn0HnJn4sZG4ncTWSGxO3j4YExemf/4HcA5z/A9YOQ4+iyXklrDihxn3SZEpEnNnjnhyTu/X6iX7gs0v3HdkkJc+YbftqlLWY1ZOVtdb47fQU4OjjbrzZ/qjZdP1p89NfeT99Q36qF6DDttqWfIzH3Y1VZ/qb/VP+rfta/Wfl7bWdvTqHfvGJovVemz9vv/Z1gwbw==</latexit>

Brady Neal

Causal Insights for Transfer Learning

6 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)
Goal: Model Etest (Y | x) only given access to Ptrain (x, y)
Covariate Shift Assumption: Ptrain (y | x) = Ptest (y | x)<latexit 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Ptrain (x) 6= Ptest (x)<latexit sha1_base64="78fnabqyPHMIObd2bl6JG18IMzs=">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</latexit>x
Brady Neal

Causal Insights for Transfer Learning

6 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)
Goal: Model Etest (Y | x) only given access to Ptrain (x, y)
Covariate Shift Assumption: Ptrain (y | x) = Ptest (y | x)<latexit 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Ptrain (x) 6= Ptest (x)<latexit sha1_base64="78fnabqyPHMIObd2bl6JG18IMzs=">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</latexit>

Etrain (Y | x)<latexit sha1_base64="sCgSnqrFVjfsK2VBUnK+yYCDTro=">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</latexit>x
Brady Neal

Causal Insights for Transfer Learning

6 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)
Goal: Model Etest (Y | x) only given access to Ptrain (x, y)
Covariate Shift Assumption: Ptrain (y | x) = Ptest (y | x)<latexit 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Ptrain (x) 6= Ptest (x)<latexit sha1_base64="78fnabqyPHMIObd2bl6JG18IMzs=">AAAVlHicrVjZbttGFJ2kW+puTgr3oX1h4xRIAVmV3BTOQw0EsB0URVM4gLc2DgyJomzC3EJSVhxBQL+mr+3v9G967pmhSGqjZFiCyNGdu81dZ6YdeW6SNhr/3bn73vsffPjRvY9XPvn0s8+/WL3/4CgJe7HtHNqhF8Yn7VbieG7gHKZu6jknUey0/LbnHLcvd2T++MqJEzcMDtLryHntt84Dt+varRSgs9WvT6M0brnB47ffW9Zp4LyxAHCSFP/PVtcb9QY/1uSgaQbrynz2w/srTXWqOipUtuopXzkqUCnGnmqpBN9XqqkaKgLstRoAFmPkct5RQ7UC2h6wHGC0AL3E8xz/XhlogP/CMyG1DSkefjEoLfWdwelg3CVUv0W+VcCdJWNA3qLjNd5tw9MHNFUXgFbRZZiL07Xx9StWnWINT7laFyuJCBE72KU1d/H28D/FCuV5DUwHow6oYoxswDxANURkxHhry4ttLuiJFvEcjETreXrP01rmQ3wvwauFcUJNRVdLPTd+CSjZoa6C49Gnszm+xQq1fvN4dUdrcOl3vQrBPQDkUr3DqMx5nsxidM3GSg3n4RRZgp8SIwB2AkjIyHdhPxcY5Vi1wVNktOjJc64lYgTVDedfOZ9FcwR/blCfhF6zGG8u5UQms3K9Mz09vNvkHYN+gLkLyhFb1KCPQ94xozezYo3YMf71+c+mjvYYvM7MFH/WsCaJrhr4hphxR7y0JbQ9M420dgPALPPdUL9RvgO/12gViesaaENqqKM4oazA+H4bdUXs4uMpULFsBqlRhk89Lhgtl4Dl8jQnkfcDVpEwfzzDdYC3R6tF5FKjdLFPn2OfaxKP9ii7j3GHcoTaQq2rqy2jx7AkVWzjsjpNSrUYf5K7ujKOSxW6gJlqmajcpqSGemJWLB4QvYtWs0011DU54eqGY1q1qX9eNSRGy/oJVe6fsq86jMYLUpX5ihWnr3PaWjaZJzX6Umx/DoxtUnUNPGFuRkbSSiH/dkyWhPSHzbdvYkLnu0jrj80MAPeNpTzg6TyXniQzb8CzxZg6pfWK8/M47hn6rBtIxR8QOp/uiJlVpo0xGoxm5tO7HDm0jx6JxzW/iLaOuJZvRz+rMLMo79vhF0xwlIhNDadFuXcYG9NXfqDWUQvWEVVle4qnBRJzdxJh/GiE+Qh8BfeSUgJAJMMHo/lhpQ92ARkai3U5l+VhroPLytch5qmJPI2hMyodw5Csni9V4nJa3Gj4fNrOKN7LtBo+n1YgV4xN19SDhLxOKuhS+kEo/EKUHlRQhewJGq51/mMB/QL22p6Ji9hoeFxB6bPG6FWFJitfVOqXsgrGxiZC8+cNLHjF+jfkDmhZO+a06VLWzOmub2TTnL6/pGVzSn9J++aU7yooHfYGdwQTmpeVOSVY+5VYOp6qeoKDbpLtX3S21fhsw4rSszumMlmm30hU6O64zRpQW6g+T/YdyeRmZR73WH3brEMxJXfm5M7hrXHLPdhbYH0J4z0/Xy6SX9PoFo+bHfphWnXMZqq6XGhOArP6ssawJvrbTftqxLW2KDPr1GeFzjIozVfzcgoRO8kpm9X7sHwvlp2hkkKXdiknP18NuBfum1Vlu+IN5oGO5F/QEV5gpzmkvX/ELreJ5/NS71+Uq5w2e6ZrFPlugvMWuvYuu/vyfCNzJtwonRzLMvYgIfsNzRk7x5b9bWp69TROy8idzlnnXmfiRDYuSe+5L4Clz6HXFRzzbJ7Ob75/p3PW90Uhz0H5afQ2OBdP2ovxy2Nm9r1I1W3N7DuECO8unh4rQGJwdxl1Hvubg0qr7zn0HnJn4sZG4ncTWSGxO3j4YExemf/4HcA5z/A9YOQ4+iyXklrDihxn3SZEpEnNnjnhyTu/X6iX7gs0v3HdkkJc+YbftqlLWY1ZOVtdb47fQU4OjjbrzZ/qjZdP1p89NfeT99Q36qF6DDttqWfIzH3Y1VZ/qb/VP+rfta/Wfl7bWdvTqHfvGJovVemz9vv/Z1gwbw==</latexit>

Etrain (Y | x)

Etest (Y | x)xx<latexit sha1_base64="sCgSnqrFVjfsK2VBUnK+yYCDTro=">AAAVk3icrVhZb9tGEN6kV+peTgo9FH1hoxRIAVqV3BQOUBgIajsoihpwAF9JFBgUSVmEeJWkLDuE0F/T1/b39N905tulSOogJcMSRK5m59o5d7cXuk6ctNv/3bv/wYcfffzJg083Pvv8iy+/2nz46DQORpFpn5iBG0TnPSO2Xce3TxInce3zMLINr+faZ73hHs+fXdlR7AT+cXIT2u8849J3+o5pJAS62Pyme3DRTezrJE0iw/EnT19rXc+xtOsfLjab7VYbH21+0FGDplCfo+DhRkd0hSUCYYqR8IQtfJHQ2BWGiOn7VnREW4QEeydSgkU0cjBvi4nYINoRYdmEYRB0SM9L+vdWQX36zzxjUJskxaVfRJSa+F7hWDTuAyrfLF8r4C6TkYI363hD757i6RE0EQOC1tFlmKvT9ejr1aw6oTU8x2odWkkICNvBLK25T2+X/ie0Qn7eEKZNI4uoIhqZBHMJKiEsI6K3tDzbZgBPGMCzacRaV+ldpTXPB/QdEi+DxjE0ZV018VL5xYdkG7oyjgufLud4TSuU+lXx6k/X4MDvchWMe0yQoXhPozLnKpnF6FqOlSjOkwWyGD8Bhk/YMUECRL5D9nMIoxyrJvFkGQY8eYm1hIigluL8O+azaA7Jn1vQJ4bXNMSbAzmhyqxc70xPl9498I6IPqW5AeSwLXTSxwbvCNGbWVEHdkT/xvhnQkdzBt5CZrI/dVoTR5dOfAOacaa8pCWkPTONpHYpwTT13RJ/QL5NftdhFY5rnWgDaCijOIYsX/l+l+oK28WjJ0PZshlEhwwPegwQLUOC5fIkJ5b3I60iRv64imtKbxdWC8FFh3S2zxhjD2tij44ge0xjC3KYWqNa1xI7So9JSSrbxkF1mpeqIf44d2VlnJXKdD4yVVNRuQtJbfFMrZg9wHoXrWaqaihrcozVTWa06kH/vGpwjJb1Y6rcP2VfWYjGAajKfNmKi9e5aC3byBMdvmTbXxLGLqj6Ch4jN0MlaaOQf3sqSwL4w8TbUzEh852ljWdmUoJ7ylIu4ck8557EM38STwMx1YX1ivNVHA8UfdYNuOKngFbTnSKzyrQRjdLpTDW9g5EN+8gRe1zyC2HrEGv5bvrTCjOr8r4bfv4cR47YRHFalbuF2Fi88mPRpFrQpKgq25M9zZAIu5OQxk+mmE+IL+MOIcUnCGd4Op2f1PpgnyATZbE+5rI8zHVwUPksYHZV5EkMmVHJDAZndbVUjstFcSPh1bTWNN7LtBJeTcuQK8Smo+pBDF7nNXQJ/MAUXiFKj2uoAvQECZc6v15BPx+9dqTiIlIantVQeqgxclWBysrDWv0SVMFI2YRp3tzCgleofxPsgNa1Y06brGXNnO7mVjbN6cdrWjan9Na0b075vobSRm9wpjCmeVWbU4x1VIsl46muJ9jUTbL9i8w2Hc8eWZF7tqUqk6b6DUeF7I67qAH6SvV5vu9wJndq83iE6ttDHYog2arInZM745Z7cLTC+mLEe36+XCW/FtGtHjd78MOi6pjN1HW5QJ0ElvVliaHN9bfb9tUQazUgM+vUF4XOkpbm63nZhYid55TNyn1YvhfLzlBxoUs7kJOfr1LshcdqVdmueAt5ICP5N+oIh7TTnMDeP9Eut0PPl6XevypXPm2OVNco8t0mzjvUtffR3dfnG6oz4Vbp5FiWcUASst9EnbFzbN7fJqpXL+K0jtzFnGXuWXMnsllJcs89ICx5Dr2p4Zhn82J+1f5dzFneFwU4B+Wn0bvgXDxpr8Yvj5nl9yJ1tzXL7xBCevfp6aICxAp3H1Hnor/ZVGnlPYfcQ+7N3dhw/G5TVnDspo8fzcgr85+9A7jEGX5EGDmOPMsloJawIsdltwkhaBK1Z45x8s7vF1ql+wLJb1a3uBBXnuK3q+pSVmM2Ljabndk7yPnB6Xar83Or/epZ88VzdT/5QHwrHounZKcd8YIy84jsaoq/xN/iH/Fvo9H4pfFrY1+i3r+naL4WpU/j8H9l+TCM</latexit>

Brady Neal<latexit sha1_base64="LAl2+/mPU0f4bLGXzMo62R1l0hA=">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</latexit>

Causal Insights for Transfer Learning

6 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)
Goal: Model Etest (Y | x) only given access to Ptrain (x, y)
Covariate Shift Assumption: Ptrain (y | x) = Ptest (y | x)<latexit 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Ptrain (x) 6= Ptest (x)<latexit sha1_base64="78fnabqyPHMIObd2bl6JG18IMzs=">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</latexit>

Etrain (Y | x)

Etest (Y | x)xx<latexit sha1_base64="sCgSnqrFVjfsK2VBUnK+yYCDTro=">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</latexit>

Brady Neal<latexit sha1_base64="LAl2+/mPU0f4bLGXzMo62R1l0hA=">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</latexit>

Causal Insights for Transfer Learning

6 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)
Goal: Model Etest (Y | x) only given access to Ptrain (x, y)
Covariate Shift Assumption: Ptrain (y | x) = Ptest (y | x)<latexit 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Ptrain (x) 6= Ptest (x)supptrain (x) = supptest (x)<latexit sha1_base64="78fnabqyPHMIObd2bl6JG18IMzs=">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</latexit><latexit sha1_base64="XPFc2cceYUdIGL8nCFTwSTIM+N0=">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</latexit>

Etrain (Y | x)

Etest (Y | x)xx<latexit sha1_base64="sCgSnqrFVjfsK2VBUnK+yYCDTro=">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</latexit>

Brady Neal<latexit sha1_base64="LAl2+/mPU0f4bLGXzMo62R1l0hA=">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</latexit>

Causal Insights for Transfer Learning

6 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)
Goal: Model Etest (Y | x) only given access to Ptrain (x, y)
Covariate Shift Assumption: Ptrain (y | x) = Ptest (y | x)<latexit 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Ptrain (x) 6= Ptest (x)supptrain (x) = supptest (x)<latexit sha1_base64="78fnabqyPHMIObd2bl6JG18IMzs=">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</latexit><latexit sha1_base64="XPFc2cceYUdIGL8nCFTwSTIM+N0=">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</latexit>

Etrain (Y | x)

Etest (Y | x)xx<latexit sha1_base64="sCgSnqrFVjfsK2VBUnK+yYCDTro=">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</latexit>

Brady Neal<latexit sha1_base64="LAl2+/mPU0f4bLGXzMo62R1l0hA=">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</latexit>

Causal Insights for Transfer Learning

6 / 22

Covariate Shift
Setting: Ptrain (x, y) 6= Ptest (x, y)
Goal: Model Etest (Y | x) only given access to Ptrain (x, y)
Covariate Shift Assumption: Ptrain (y | x) = Ptest (y | x)<latexit 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Ptrain (x) 6= Ptest (x)supptrain (x) = supptest (x)<latexit sha1_base64="78fnabqyPHMIObd2bl6JG18IMzs=">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</latexit><latexit sha1_base64="XPFc2cceYUdIGL8nCFTwSTIM+N0=">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</latexit>

Etrain (Y | x)

Etest (Y | x)xx<latexit sha1_base64="sCgSnqrFVjfsK2VBUnK+yYCDTro=">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</latexit>

Brady Neal<latexit sha1_base64="LAl2+/mPU0f4bLGXzMo62R1l0hA=">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</latexit>

Causal Insights for Transfer Learning

7 / 22Predicting Y from an Unstructured Vector

Etrain (Y | x)<latexit sha1_base64="sCgSnqrFVjfsK2VBUnK+yYCDTro=">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</latexit>x

Brady Neal

Causal Insights for Transfer Learning

8 / 22Predicting Y from an Unstructured Vector
X<latexit sha1_base64="0UKvpaWcUqfatZqCK8h0yaFNTis=">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</latexit>

Etrain (Y | x)<latexit sha1_base64="sCgSnqrFVjfsK2VBUnK+yYCDTro=">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</latexit>x

Brady Neal

Causal Insights for Transfer Learning

8 / 22Predicting Y from an Unstructured Vector
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Etrain (Y | x)<latexit sha1_base64="sCgSnqrFVjfsK2VBUnK+yYCDTro=">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</latexit>

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Brady Neal

Causal Insights for Transfer Learning

8 / 22Use the Causal Structure<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

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Brady Neal

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Causal Insights for Transfer Learning

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Causal Insights for Transfer Learning

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Causal Insights for Transfer Learning

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Training data from Ptrain (x, y)<latexit sha1_base64="J9pFcTESUofN5l/RrRE0h5uO7Zk=">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</latexit>

X3
X7

Goal: in-distribution
prediction of Y fromout-of-sample data for X

Brady Neal

X8

X9

X11

X6

X10

Y

X12

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X15

X16

Causal Insights for Transfer Learning

X14

10 / 22

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X3
X7

Goal: in-distribution
prediction of Y fromout-of-sample data for X

Brady Neal

X8

X9

X11

X6

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

X14

10 / 22

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X4

X5

Training data from Ptrain (x, y)<latexit sha1_base64="J9pFcTESUofN5l/RrRE0h5uO7Zk=">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</latexit>

X3
X7

Goal: in-distribution
prediction of Y fromout-of-sample data for X

Question: What is the minimum
set of variables that will give us
optimal prediction?
Brady Neal

X8

X9

X11

X6

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

X14

10 / 22

In-distribution Prediction of Y – Markov Blanket<latexit sha1_base64="JJjKGFrdH9bZYpu0YNqkLJJKPxs=">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</latexit>

X1

X2

X4

X5

Training data from Ptrain (x, y)<latexit sha1_base64="J9pFcTESUofN5l/RrRE0h5uO7Zk=">AAAWcXicrVhrb9tUGD7buIxw64aohPhimiEN4WZJGZQvlSa1mxBiopN6g7WqHMdprPiG7TTLrPwOvsInfhO/gz/A+z7nOLYTx06qJop98p73fjuXbuDYUdxu/3vn7r133n3v/fsfND786ONPPt148PAk8kehaR2bvuOHZ10jshzbs45jO3assyC0DLfrWKfd4T7Pn15bYWT73lE8CawL17jy7L5tGjGBLs6DODRs7/EbXZt8c7nRbLfa+GiLg44aNIX6HPoPGv+Ic9ETvjDFSLjCEp6IaewIQ0T0fS06oi0Cgl2IhGAhjWzMW2IqGkQ7IiyLMAyCDul5Rf9eK6hH/5lnBGqTpDj0C4lSE18rnB6N+4DKN8vXcrjLZCTgzTpO6N1VPF2CxmJA0Dq6FHN1ui593RqrY7LhR1hrkyUBIOwHs2Bzn94O/Y/JQn5OCNOiUY+oQhqZBHMIKiEsI6S39Dz7ZoBIGMCzaMRaV+ldpTXP+/QdEi+DxhE0ZV018ULFxYNkC7oyjoOYLuf4hiyU+lXx6s9ssBF3aQXjHhFkKN7SqMi5SmY+u5ZjxYrztEQW48fA8Ag7IoiPzLfJfzZhFHPVJJ4sw0Akr2BLgAxqKc4/Yz7N5oDiuQ19IkRNQ77ZkBOoysr0TvV06N0F75DoE5obQA77Qid9LPAOkb2pF3Vgh/RvjH8mdDTn4C1UJsdTJ5s4u3Ti69OMPeMlPSH9mWoktUsIpqnvtvgF8i2Kuw6vcF7rROtDQ5nFEWR5KvZ71FfYLy49GcqeTSE6ZLjQY4BsGRIskyc5sbwnZEWE+nEU14TeDrwWgIsO6eyfMcYubOKIjiB7TOMe5DC1Rr2uJXaVHtOCVPaNje60KFVD/nHtys44L5XpPFSqprJyD5La4qmymCPAeue9ZqpuKHtyBOumc1p1oX/WNThHi/oxVRafYqx6yMYBqIp82YvldpbZsoM60RFL9v0VYeyBqq/gEWozUJIaufrbV1XiIx4m3q7KCVnvLG08N5MQ3FWecghP1jmvSTzzB/E0kFPn8F5+vorjc0Wfrgbc8RNAq+lOUFlF2pBGyWymmt7GyIJ/5IgjLvkF8HUAW76a/bTczKq8b4eft8CRMzZWnFbl3kNulFt+JJrUC5qUVUV/cqQZEmJ3EtD40QzzEfFl3CGkeAThCk9m89PaGBwQZKo81sdcWoeZDjY6Xw+Y5yrzJIasqHgOg6u6WirnZVneSHg1bW+W70VaCa+mZcg1ctNW/SACr7MauhhxYAo3l6VHNVQ+1gQJlzr/toJ+HtbakcqLUGl4WkPposdIq3xVlS9r9YvRBUPlE6b5/QYevEb/m2IHtK4fM9p4LW9mdJMb+TSjH6/p2YzSXdO/GeXbGkoLa4M9gzHNq9qaYqzDWiyZT3VrgkWrSbp/kdWm49klL/Ka3VOdSVPrDWeFXB330AP0lfrz4rrDldypreMRum8XfSiE5F5F7RzfGrcsgqMV7IuQ79n5cpX6KqNbPW/2EYey7pjO1K1yvjoJLFuXJYa2sL7ddF0NYKsBmelKfZlbWZLCfD0vK5exi5zSWcnnALtHB7VpUbeU/vpVaZ7W+znRjAgW4BwiR9V6pFjzlqXwm9qX8c1buYxr3tZGbt+Znhej3I7EhszsLJlg3z9WEUxPANvwk6zan2j1e0m76ily6zva0Xfo+aKwz1mVK5+sR2qFzPPdIc67FKUD7GTW5xuo8+924ZRclPGcJKS/qbpPyLB5Lx+rfUkZp3XklnOWfaa3cPqclyTPFwPCkmfuSQ3HrHOV86uObzlneTfm48yXnbxvg3P+VmE1flnOLL8DqruZWn5fEtC7T08H3S4q7RfHQt7pyP3y/sLtFOfvDlUF526y9XBOXpH//H3HFe4rZF2nOPLcGoNawvIcl92cBKCJ1fkgwi1DdpfSKtyNSH7zukW5vHIVvz3Vo9Ie07jcaHbm71sXByc7rc73rfarp81nP6i72PviS7ElHpOfdsUzqsxD8qtJe48/xV/i78//2/xiU9vckqh37yiaz0Ths/nt/2YnXUo=</latexit>

X3
X7

Goal: in-distribution
prediction of Y fromout-of-sample data for X

Question: What is the minimum
set of variables that will give us
optimal prediction?
Brady Neal

X8

X9

X11

X6

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

X14

10 / 22

In-distribution Prediction of Y – Markov Blanket<latexit sha1_base64="Gv2HFe6nNM7CHnn5TfmT/ayGn3M=">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</latexit>

X1

X2

X4

X5

Training data from Ptrain (x, y)<latexit sha1_base64="J9pFcTESUofN5l/RrRE0h5uO7Zk=">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</latexit>

X3
X7

Goal: in-distribution
prediction of Y fromout-of-sample data for X

Question: What is the minimum
set of variables that will give us
optimal prediction?
Brady Neal

X8

X9

X11

X6

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

X14

10 / 22

In-distribution Prediction of Y – Markov Blanket<latexit sha1_base64="v4n8HZziDTFX3SCn8ycSdfYOEKA=">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</latexit>

X1

X2

X4

X5

Training data from Ptrain (x, y)<latexit sha1_base64="J9pFcTESUofN5l/RrRE0h5uO7Zk=">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</latexit>

X3
X7

Goal: in-distribution
prediction of Y fromout-of-sample data for X

Question: What is the minimum
set of variables that will give us
optimal prediction?
Brady Neal

X8

X9

X11

X6

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

X14

10 / 22Other Tasks Generated via Interventions<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

X1

X2

X4

X5

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X3
X7

Goal: in-distribution
prediction of Y fromout-of-sample data for X

Brady Neal

X8

X9

X11

X6

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

X14

11 / 22Other Tasks Generated via Interventions<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

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X2

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X5

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X3
X7

Goal: prediction of Y
from X sampled from
Ptest (x, y)
X11Brady Neal

X8

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X6

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

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11 / 22Other Tasks Generated via Interventions<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

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Training data from Ptrain (x, y)<latexit sha1_base64="J9pFcTESUofN5l/RrRE0h5uO7Zk=">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</latexit>

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Goal: prediction of Y
from X sampled from
Ptest (x, y)
X11Consider that are all test
distributions are generated byinterventions on this graph
Brady Neal

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Causal Insights for Transfer Learning

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Recall Modularity<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

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Brady Neal

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Causal Insights for Transfer Learning

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Recall Modularity<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

Intervening on avariable only changes
the causal mechanism(structural equation) forthat variable. All othercausal mechanisms
remain unchanged.

X3

X2

X4

X5

X7

X6
X8

X9

X11

Brady Neal

X1

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

X14

12 / 22

Recall Modularity<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

Intervening on avariable only changes
the causal mechanism(structural equation) forthat variable. All othercausal mechanisms
remain unchanged.
Mechanism for Y:
P (y | x4 , x5 )<latexit sha1_base64="dC/8wg8jD5F6qfJ3T3WscN3315k=">AAAWeXicrVhJb9tGFJ6kW6puToL60gsTpYBb0KrkOHUvBgLYCYqiQR3AS9IoMCSSsghxK0lZcQj9lh7b/qT+ll763jdDkZQoUjIsQeTozdu3WfqBY0dxu/3vrdsffPjRx5/c+bTx2edffPnVxt17p5E/Dg3rxPAdP3zV70WWY3vWSWzHjvUqCK2e23ess/7ogOfPLq0wsn3vOL4KrLdu78KzB7bRiwl0vnG/G2xdaV3XNrV357s6PZ58d77RbLfa+GiLg44aNIX6HPl3G3+JrjCFLwwxFq6whCdiGjuiJyL6vhEd0RYBwd6KhGAhjWzMW2IqGkQ7JiyLMHoEHdHzgv69UVCP/jPPCNQGSXHoFxKlJr5VOCaNB4DKN8vXcrjLZCTgzTpe0buveLoEjcWQoHV0KebqdH36ujVWx2TDT7DWJksCQNgPRsHmAb0d+h+Thfy8IkyLRiZRhTQyCOYQVEJYRkhv6Xn2zRCR6AHPohFrXaV3ldY879N3RLx6NI6gKeuqiecqLh4kW9CVcRzEdDnHd2Sh1K+K12Bmg424SysY95ggI/GeRkXOVTLz2bUcK1acpyWyGD8GhkfYEUF8ZL5N/rMJo5irBvFkGT1E8gK2BMigluL8C+bTbA4ontvQJ0LUNOSbDTmBqqxM71RPh9598A6JPqG5IeSwL3TSxwLvENmbelEHdkj/JvhnQEdjDt5CZXI8dbKJs0snvj7N2DNe0hPSn6lGUruEYJr6botfId+iuOvwCue1TrQ+NJRZHEGWp2K/T32F/eLSk6Hs2RSiQ4YLPYbIlhHBMnmSE8v7gayIUD+O4prQ24HXAnDRIZ39M8HYhU0c0TFkT2hsQg5Ta9TrWmJP6TEtSGXf2OhOi1I15B/XruyM81KZzkOlaior9yGpLXaVxRwB1jvvNUN1Q9mTI1g3ndOqD/2zrsE5WtSPqbL4FGNlIhuHoCryZS+W21lmyw7qREcs2fcXhLEPqoGCR6jNQElq5OrvQFWJj3gYeLsqJ2S9s7TJ3ExCcFd5yiE8Wee8JvHMH8Szh5zqwnv5+SqOzxR9uhpwx08AraY7RWUVaUMaJbOZanobIwv+kSOOuOQXwNcBbHkw+2m5mVV53ww/b4EjZ2ysOK3K3URulFt+LJrUC5qUVUV/cqQZEmJ3EtD40QzzEfFl3BGkeAThCk9m89PaGBwSZKo8NsBcWoeZDjY6nwnMrso8iSErKp7D4Kqulsp5WZY3El5Na87yvUgr4dW0DLlEbtqqH0Tg9aqGLkYcmMLNZelxDZWPNUHCpc6vV9DPw1o7VnkRKg3Paihd9Bhpla+q8kWtfjG6YKh8wjS/X8ODl+h/U+yA1vVjRhuv5c2M7upaPs3oJ2t6NqN01/RvRvm+htLC2mDPYEzzsramGOuoFkvmU92aYNFqku5fZLXpePbJi7xmm6ozaWq94ayQq+M+eoC+Un9eXHe4kju1dTxG9+2jD4WQbFbUzsmNccsiOF7Bvgj5np0vV6mvMrrV8+YAcSjrjulM3Srnq5PAsnVZYmgL69t119UAtvYgM12pz3MrS1KYr+dl5TJ2kVM6K/kcYvfooDYt6pbSX78pzdN67xLNmGABziFyVK1HijVvWQq/rn0Z37yVy7jmbW3k9p3peTHK7UhsyMzOkgn2/RMVwfQEsA0/yar9mVa/F7SrniK3HtOOvkPP54V9zqpc+WQ9Vitknu8Ocd6jKB1iJ7M+30Cdf7cLp+SijGckIf1N1X1Chs17+VjtS8o4rSO3nLPsM+bC6XNekjxfDAlLnrmvajhmnaucX3V8yznLuzEfZ77s5H0TnPO3Cqvxy3Jm+R1Q3c3U8vuSgN4DejrodlFpvzgR8k5H7pcPFm6nOH93qCo4d5OH9+bkFfnP33dc4L5C1nWKI8+tMaglLM9x2c1JAJpYnQ8i3DJkdymtwt2I5DevW5TLK1fx21c9Ku0xjfONZmf+vnVxcLrT6jxptV/uNp/+qO5i74hvxEOxRX7aE0+pMo/IrwZp/qf4W/zz9X+bDza3Nr+XqLdvKZr7ovDZfPw/SZRfDg==</latexit>

Brady Neal

X3

X1

X2

X4

X5

X7

X8

X9

X11

X6

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

X14

12 / 22

Recall Modularity<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

Intervening on avariable only changes
the causal mechanism(structural equation) forthat variable. All othercausal mechanisms
remain unchanged.
Mechanism for Y:
P (y | x4 , x5 )<latexit sha1_base64="dC/8wg8jD5F6qfJ3T3WscN3315k=">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</latexit>

Brady Neal

X3

X1

X2

X4

X5

X7

X8

X9

X11

X6

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

X14

12 / 22

Recall Modularity<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

Intervening on avariable only changes
the causal mechanism(structural equation) forthat variable. All othercausal mechanisms
remain unchanged.
Mechanism for Y:
P (y | x4 , x5 )<latexit sha1_base64="dC/8wg8jD5F6qfJ3T3WscN3315k=">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</latexit>

Brady Neal

X3

X1

X2

X4

X5

X7

X8

X9

X11

X6

X10

Y

X12

X13

X15

X16

Causal Insights for Transfer Learning

X14

12 / 22

Recall Modularity<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

Intervening on avariable only changes
the causal mechanism(structural equation) forthat variable. All othercausal mechanisms
remain unchanged.
Mechanism for Y:
P (y | x4 , x5 )<latexit sha1_base64="dC/8wg8jD5F6qfJ3T3WscN3315k=">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</latexit>

Brady Neal

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X12

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X15

X16

Causal Insights for Transfer Learning

X14

12 / 22

Recall Modularity<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

Intervening on avariable only changes
the causal mechanism(structural equation) forthat variable. All othercausal mechanisms
remain unchanged.
Mechanism for Y:
P (y | x4 , x5 )<latexit sha1_base64="dC/8wg8jD5F6qfJ3T3WscN3315k=">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</latexit>

X3

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Y

X12

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X14

Non-causal conditional:
P (y | x4 , x5 , x12 )<latexit sha1_base64="OiHzrsS4Xj9onzM1voj6mt6meGw=">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</latexit>

Brady Neal

Causal Insights for Transfer Learning

12 / 22

Recall Modularity<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">AAAahnicrVhZb9tGEN6krZu4V5y+uS9s7AAJQKuiDjtGYCCAk6AomiIBnMSqbRgUtZII8SpJWXEI/ZW+tk/9P/03nZldXhK1lItIELmcmW9mdo5drvqBY0dxs/nvrdufff7Fxpd37m5+9fU33353b+v+u8ifhhZ/a/mOH572zYg7tsffxnbs8NMg5Kbbd/j7/uQY+e+veBjZvncSXwf8wjVHnj20LTMG0uXWxtZ5n49sL4ntycfAtuJpyOebmvw81H7zBzzKns89eDyLYjPmF9qjU+OxZsbao6befKwlu6eXxu786SrZzmPtrM8df6Yd+UPt1LggREeBOMgQDh/GAtbRP0RjexgfNVpdy9WvxYNhuULdwWp1vUxbaI/GqTqE9RQ+dAFl9v0rXkD1hK2uAvZkwZiw1k2d36v2/olCYytzRKgSiJYCYbSW4yddT4zWXIXsLuSqlcK6Kthh5mKerxR6qMC1K3CHAtZWeWlU1Mdh6qmhnGC7ohiy2LSV0P2F2LRT2L4S1qwopAK4qQR3qmrXaKbgjgq8X2lYYvcRWOj2F4NRsdsDMx6LNufAoCZ+usRtZdxuBbeTcnsVzK6K2c70Hqr0wjKhUAyduMzdz7iQlmX2Yc421OxWBRsznfLbNfyqcBp5PI2qgBp5XKAWl/g9pXe9sm/EPufeoLT6X97baTaa9NGWB4Yc7DD5ee1v3f2HnbMB85nFpsxlnHkshrHDTBbB94wZrMkCoF2wBGghjGziczZnm4CdghQHCROoE7iO4OlMUj14Rp0RoS2w4sAvBKTGHkqZAYyHRBV3tK8VZFfZSEg3+ngN977U6QI1ZmOg1uFSyfVxffi6NbOOYQ5PaLY2zCQgCsbBKs15CHcHnmOYIV6vQZLDaACoEEYW0BygCgraCOEuIo+xGVMmTJLjMEKvVX6rvEa+D98J6DJhHJGn6KvGXsq8eGSZk68o41BOV2v8ADMU/ql0DbM52JR3MQuUPQHKhH2EUVmzymaxulZLxVLzvMIWysck4YF0BBSfKt+G+NkgUa5VC3SiDZMyOaK5BFRBDan5F+Kn1RxAPvfIn4iyplG92WQnkJ2V+5366cC9T7pDwCfAG5MdjIUO/nDSHVL1plHUSTqEpxk9WeSjtUBvUGdiPnWYE1aXDnp94NiZLhEJEc/UI+FdAjRNfvfYr2SfQ951igrWtQ5YnzwUVRyRLU/m/gjWFYyLC1ekYmRTik42XPJjTNUyAVpuT2hCez/BLCLqH0dqTeDuUNQC0qKTdYzPjMYuzQkzOiXbMxgPyA6iNVjrGuxA+jEvWcXY2LQ6LVvVqP6wd8XKuGgVcR51qiar8ogsNVlHzhgzgH4Xo2bJ1VCsyRHNbr7gVZ/8z1cNrNGyf4jK81PO1YCqcUyosl6MYvU8q+bSoj7RKZcY+xFIHBFqKOkR9WYgLW0W+u9YdolP+bDo7sqaEP2O1mYLnAToroyUA3Kiz3FPQs4foNOkmjqn6BX5Ko0vJD7dDXDFT4iqxr2jzipjQxglGUeNt2nEKT5ihBkX+gKKdUBz+TH7aQXOuro/jT5vSSNWbCw1rat9QLVRPfMTtgNrwQ5UVTmemGmkhPR2EsB4N5PcBb0oOyErHlCww5OMP6/NwXOgzGXEhsRL+zD3waaVb0CS57LyhIToqHhBArtabRXrsqpuBF2NHWT1XsYKuhqLlCuqTVuuBxHpOq3BxZQHRLiFKj2pQfm0Jwi68Lm3hn8e7bVTWReh9PB9DdKlNUbMypdd+arWv5hWwVDGBDG//48IXtH6N6c3oJvGMcfGN4pmjrv+XzHN8bMbRjZHujeMb478WIPktDfYGQ0xb2p7CqVe10qJeqrbEzjsJun7i+g2na59iCLu2QO5Mmlyv8GqELvjEa0B+lrr8/K+g51s1PbxlFbfPq1DIVkeKHrn7SfTlmdwusb8Iqr3/Hy5Tn9V4davm2PKQ9XqmHLEG0j+FpKeHqLC/mST/vxkkdBb4EzuZOn74B5VgMjhz7AWvoJ3rDlZasP7nQHXl6Vdb12teM6ayvWyqLcFmg9gv3pO+9rN9QbyNLRXOjOVbbwAC+lvLk+XubQjzwCrNN3EbrVmUXWDpbPIoiXxtjkGKXECu67RmNdxtT51fqs1i39KfDoB5OewT6G5eMZcT19eM6v/Eaj7n2L16TmA+xCuDq3okZR9TlXn0MrOYY0RJ3zx9nS89F8F1m8LugJrN3lwf8FeWf/i6XdEp9cpSOQy4hQTE1rQihpXnaMDwsTybTGiM2d+sm6UTspC36JvUaGuXKnvSK5HXO4Wm5f3dozFf9+WB+9aDaPbaL5p7TxryX/m7rAf2AP2COJ0wJ5BZ76GuFobHzb+3Phr4+/tO9uN7e72gRC9fUtivmelz/az/wCB9Z1V</latexit>

Intervening on avariable only changes
the causal mechanism(structural equation) forthat variable. All othercausal mechanisms
remain unchanged.
Mechanism for Y:
P (y | x4 , x5 )<latexit sha1_base64="dC/8wg8jD5F6qfJ3T3WscN3315k=">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</latexit>

X3

X1

X2

X4

X5

X7

X8

X9

X11

X6

X10

Y

X12

X13

X15

X16

X14

Non-causal conditional:
P (y | x4 , x5 , x12 )<latexit sha1_base64="OiHzrsS4Xj9onzM1voj6mt6meGw=">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</latexit>

Brady Neal

Causal Insights for Transfer Learning

12 / 22

Causal Mechanism is Optimal in Robust Sense

Brady Neal

Causal Insights for Transfer Learning

13 / 22

Causal Mechanism is Optimal in Robust Sense<latexit sha1_base64="3eC5wtDurlJK1ozO10lMkGs/F0o=">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</latexit>

E[Y | pa(Y )] = arg min max E(X,Y )⇠Ptest (Y
f

Ptest

f (X))

2(see, e.g., Rojas-Carulla et al., (2018, Appendix A.1))

Brady Neal

Causal Insights for Transfer Learning

13 / 22

Causal Mechanism is Optimal in Robust Sense<latexit sha1_base64="3eC5wtDurlJK1ozO10lMkGs/F0o=">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</latexit>

E[Y | pa(Y )] = arg min max E(X,Y )⇠Ptest (Y
Ptest

f

f (X))

2(see, e.g., Rojas-Carulla et al., (2018, Appendix A.1))
Still requires common support: supptrain (pa(Y )) = supptest (pa(Y ))<latexit sha1_base64="5miKSkrm7JlEguXeUHc/4htv0tk=">AAAXMHicrVhLb9tGEN6kr1R9OSngSy9slBZOIauSm8K5CAhgJyiKBnUAP5JGgUFRlESIr5KUFYfQP+ivybX9Ie2p6LX/oLfOfLsUSYkiJcMWLK5mZ7557Mwsd3u+bYVRq/XXjZvvvPve+x/c+rD20ceffPrZ1u07p6E3CQzzxPBsL3je00PTtlzzJLIi23zuB6bu9GzzrDc+4PmzCzMILc89ji5985WjD11rYBl6RKTzra+74cT3o0C33J2ur++8uH9f62iSaIZRQjvfqreaLfxpy4O2GtSF+jvybtf+E13RF54wxEQ4whSuiGhsC12E9Hkp2qIlfKK9EjHRAhpZmDfFTNRIdkJcJnHoRB3T95B+vVRUl34zZghpg7TY9B+QpCa+Ujx9Gg9AlU/Wr2V4V+mIgc02XtKzpzAdokZiRNQquYRzfbkefZwKryPy4SG8tcgTHxSOg5HzeUBPm35H5CF/XxKnSaM+SQU0MohmE1VSWEdATxl5js0IK6GDz6QRW11md5nVPO/RZ0xYOo1DWMq2auKJWhcXmk3Yyjw21nQ14mvyUNpXhjWY+2Bh3aUXzHtMlLF4Q6M8cpnObHat5ooU8qxAF/NH4HCJOySKh8y3KH4WceRz1SBM1qFjJYfwxUcGNRXyj5hPstmn9dyFPSFWTUO+WdDjq8pK7U7stOnZA3ZA8jHNjaCHY9Ege0xgB8jeJIoNcAf0a4pfBmw0FuhNVCavZ4N84uxqEK5HM9YcS0ZCxjOxSFoXE01Tn13xE/SbtO4NRIXzukGyHiyUWRxCl6vWvkN9hePi0DdTObIJpQEdDuwYIVvGREv1SSTW9y15EaJ+bIUa09NG1HygNKCd4zPF2IFPvKIT6J7SuA89LK1Rr2uKfWXHLKeVY2OhOy1r1ZB/XLuyMy5qZTkXlaqprOxAU0s8UB7zCrDd2agZqhvKnhzCu9mCVT3Yn3YNztG8fSyVrk9+rfrIxhGk8rgcxWI/i3zZQ500sJYc+yFxdCA1UPQQtekrTbVM/R2oKvGwHgaejsoJWe+sbbowExPdUZGyiU/WOe9JPPMrYerIqS6il50vQ3ys5JPdgDt+DGq53CkqKy8b0Ciez5TLWxiZiI8c8YpLPB+x9uHLl/N/LTOzLvb14LlLiJyxkUJaF72P3Cj2/FjUqRfUKavy8eSVZkqAtxOfxvfmnPcIl3nH0OIShSs8ns/PKtfgkCgzFbEB5pI6TG2w0Pn64OyqzJMcsqKiBQ6u6nKtnJdFeSPp5bL9eb7nZSW9XJYpF8hNS/WDEFjPK+QirANLOJksPa6Q8rAnSLq0+cUa9rnYaycqLwJl4VmFpIMeI73yVFU+rbQvQhcMVExY5pcrRPAC/W+GN6BN45jKRhtFM5W7vFJMU/nphpFNJZ0N45tKvqmQNLE3WHMayzyrrCnmOqrkkvlUtSeYtJsk7y+y2hr47lEUec/uq86kqf2Gs0Lujh30gMZa/Xl53+FKblfW8QTdt4c+FEBzv6R2Tq4NLV3ByRr+hcj39Hy5Tn0Vya2fNwdYh6LumMxU7XKeOgms2pclh7a0v111X/Xhqw6dyU59ntlZ4tx8NZaZydhlpGRW4hzi7dFGbZrULWW8flaWJ/XeJZkJ0XycQ+So3I6Ea9GzhH5V/1LcrJerUK/i6zdAZJ4hNL+GDvmbazqhXRUxiUUWUdJq6iPfjZMzbZh5a7IQl/S8G+NsMlVZlpxSdmGN7Cw/0A79lN78Z8j/7+jU0abvJ7l3sXVR+fQ/Ubt4FnePkPcpFod429oc11dn9N3cST6v4zFpSP5n6s4j5ebzRqTenYqQNtFbjCx7YX/phLyoSZ6BRsQl7wUuKxDT7lqMV76+xcjy/s7DuTS9HbgO5OzNx3p4ac6svqequj1bfafj03NA3zY6clhYlSdC3jvJd/qDpRs0zt89qgrO3fjunQV9efzFO5kh7lRk70l45Nk6grSkZRFX3e74kInUGSbETUh639PM3d9IvEXbwkxeOQqvo/po0gdr51v19uKd8PLgdK/Z/r7Zevag/uihui++Jb4Qd8UOxWlfPKLKPKK4GuI38Vb8Lv7Yfrv95/bf2/9I1ps3lMznIve3/e//QjuJSQ==</latexit>

Brady Neal

Causal Insights for Transfer Learning

13 / 22

Causal Mechanism is Optimal in Robust Sense<latexit sha1_base64="3eC5wtDurlJK1ozO10lMkGs/F0o=">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</latexit>

E[Y | pa(Y )] = arg min max E(X,Y )⇠Ptest (Y
Ptest

f

f (X))

2(see, e.g., Rojas-Carulla et al., (2018, Appendix A.1))
Still requires common support: supptrain (pa(Y )) = supptest (pa(Y ))<latexit sha1_base64="5miKSkrm7JlEguXeUHc/4htv0tk=">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</latexit>

Or that we can extrapolate well from supptrain (pa(Y )) to supptest (pa(Y ))<latexit sha1_base64="tm3fMWtTHZ4cE0oBGLaoyOmuxZ0=">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</latexit>

Brady Neal

Causal Insights for Transfer Learning<latexit sha1_base64="9UTbqxc4nH+AIaULCjqLHF+NOTE=">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</latexit>

13 / 22

Relaxation of Covariate Shift

Brady Neal

Causal Insights for Transfer Learning

14 / 22

Relaxation of Covariate Shift
Covariate shift: Ptrain (y | x) = Ptest (y | x)<latexit sha1_base64="gbbbRx9WB2HZasKzwjO6UAhgDCI=">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</latexit>

Brady Neal

Causal Insights for Transfer Learning

14 / 22

Relaxation of Covariate Shift
Covariate shift: Ptrain (y | x) = Ptest (y | x)<latexit sha1_base64="gbbbRx9WB2HZasKzwjO6UAhgDCI=">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</latexit>

Modularity: Ptrain (y | pa(Y )) = Ptest (y | pa(Y ))<latexit sha1_base64="67ZNHnTrg9YA0AXO7lNtuY2AepA=">AAAXOHicrVhLb9tGEN6kr1R9xEkBX3phoxRwClmV3BTORUAAO0FRNKgD+JVGgUFRlESIr5KUFUXQv+ivybX9E731VvTaW2+d+XYpkhJFSkIsWFzNznzz2JlZ7nZ82wqjRuPPGzffe/+DDz+69XHlk08/+/z2zp2756E3CgzzzPBsL7js6KFpW655FlmRbV76gak7Hdu86AyPeP7i2gxCy3NPo4lvvnL0vmv1LEOPiHS1U2/7UaBb7t5EaztWV2v7+t6LBw+0Fo0iM4wW6Fc71Ua9gT9tedBUg6pQfyfencp/oi26whOGGAlHmMIVEY1toYuQPi9FUzSET7RXYkq0gEYW5k0xExWSHRGXSRw6UYf03adfLxXVpd+MGULaIC02/QckqYmvFU+Xxj1Q5ZP1ayneVTqmwGYbJ/TsKEyHqJEYELVMLuZcX65DH6fE64h8eARvLfLEB4XjYGR87tHTpt8RecjfE+I0adQlqYBGBtFsokoK6wjoKSPPsRlgJXTwmTRiq4vsLrKa5z36DAlLp3EIS9lWTTxV6+JCswlbmcfGmq5GfE0eSvuKsHpzHyysu/SCeU+JMhRvaJRFLtKZzq7VXJFCnuXoYv4IHC5xh0TxkPkWxc8ijmyuGoTJOnSsZB+++MigukL+EfNxNvu0nvuwJ8Sqacg3C3p8VVmJ3bGdNj07wA5IfkpzA+jhWNTIHhPYAbI3jmIN3AH9GuOXARuNBXodlcnrWSOfOLtqhOvRjDXHkpGQ8YwtktZNiaapz774CfpNWvcaosJ5XSNZDxbKLA6hy1Vr36K+wnFx6JupHNmYUoMOB3YMkC1DoiX6JBLr+5a8CFE/tkKd0tNG1Hyg1KCd4zPG2IFPvKIj6B7TuAs9LK1Rr6uLQ2XHLKOVY2OhOy1r1ZB/XLuyMy5qZTkXlaqprGxBU0M8VB7zCrDd6agZqhvKnhzCu9mCVR3Yn3QNztGsfSyVrE92rbrIxgGksrgcxXw/83w5QJ3UsJYc+z5xtCDVU/QQtekrTZVU/R2pKvGwHgaejsoJWe+sbbwwMyW6oyJlE5+sc96TeOZXwtSRU21ELz1fhPhEyce7AXf8KajFcueorKxsQKPpfKZY3sLIRHzkiFdc4vmItQ9fvpr/a6mZdbHfDZ67hMgZGymkddG7yI18z09FlXpBlbIqG09eaaYEeDvxaXx/znmfcJl3CC0uUbjCp/P5WekaHBNlpiLWw1xch4kNFjpfF5xtlXmSQ1ZUtMDBVV2slfMyL28kvVi2O8/3rKykF8sy5Rq5aal+EALrskQuwjqwhJPK0tMSKQ97gqRLm1+sYZ+LvXak8iJQFl6USDroMdIrT1Xls1L7InTBQMWEZX7ZIoLX6H8zvAFtGsdENtoomoncZKuYJvLjDSObSDobxjeRfFMiaWJvsOY0lnleWlPMdVLKJfOpbE8waTeJ319ktdXw3aEo8p7dVZ1JU/sNZ4XcHVvoAbW1+vPyvsOV3Cyt4xG6bwd9KIDmbkHtnL0ztGQFR2v4FyLfk/PlOvWVJ7d+3hxhHfK6YzxTtst56iSwal+WHNrS/rbtvurDVx064536KrWzTDPz5VhmKmOXkeJZiXOMt0cbtWlSt5Tx+llZHtd7m2RGRPNxDpGjYjtirkXPYvq2/iW4aS9XoW7j6zdAZJ4+NL+GDvmbazqmbYsYxyKNKGkV9ZHvxvGZNky9NVmIS3LeneJsMlZZFp9S9mGN7Cw/0A79jN78Z8j/7+jU0aTvp5l3sXVR+fQ/Urt4GveAkA8pFsd429oc11dn9P3MST6r4wlpiP9n6s4j4ebzRqTenfKQNtGbjyx7YXfphLyoSZ6BBsQl7wUmJYhJd83HK17ffGR5f+fhXJrcDrwL5PTNx3p4Sc6svqcquz1bfafj07NH3zY6cphblWdC3jvJd/qjpRs0zt8DqgrO3em9uwv6sviLdzJ93KnI3hPzyLN1BGlJSyOuut3xIROpM0yIm5Dkvqeeub+ReIu2ham8chReS/XRuA9WrnaqzcU74eXB+UG9+X298fxh9fEjdV98S3wp7ok9itOheEyVeUJxNcRv4q34Xfyx+3b3r92/d/+RrDdvKJkvROZv99//ASwbi4c=</latexit>

Brady Neal

Causal Insights for Transfer Learning

14 / 22

Questions:
1. What is the Markov blanket of Y in this graph?
2. What task is the Markov blanket good for?
3. What input variables should we use for optimal robust prediction?
<latexit sha1_base64="wJ3A7SSt3Pe2wN4Dq3YazAQa2f8=">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</latexit>

X3

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Causal Insights for Transfer Learning

Transportability of Causal Effects Across Populations

Brady Neal

Transportability of Causal Effects Across Populations

16 / 22

Transportability Problem

Brady Neal

Transportability of Causal Effects Across Populations

17 / 22

Transportability Problem
Source Population ⇧<latexit sha1_base64="kAXR8YPQ7qfUfniLBphNkceNVKo=">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</latexit>

Brady Neal

Transportability of Causal Effects Across Populations

17 / 22

Transportability Problem
Source Population ⇧<latexit sha1_base64="kAXR8YPQ7qfUfniLBphNkceNVKo=">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</latexit>

Brady Neal

Target Population ⇧⇤Transportability of Causal Effects Across Populations<latexit sha1_base64="kB2IrOPaT/rS19xtMKqesRZVGKY=">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</latexit>

17 / 22

Transportability Problem
Source Population ⇧<latexit sha1_base64="kAXR8YPQ7qfUfniLBphNkceNVKo=">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</latexit>

Target Population ⇧⇤<latexit sha1_base64="kB2IrOPaT/rS19xtMKqesRZVGKY=">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</latexit><latexit sha1_base64="m7DSFZusbv5fj3/a7Nzsw+yFg7E=">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</latexit>

P (y, t, . . .)Brady Neal

Transportability of Causal Effects Across Populations

17 / 22

Transportability Problem
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Target Population ⇧⇤<latexit sha1_base64="kB2IrOPaT/rS19xtMKqesRZVGKY=">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</latexit>

P ⇤ (y, t, . . .)<latexit sha1_base64="vTXhwYX9nKBrAcwXU3orFduTrMk=">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</latexit><latexit sha1_base64="m7DSFZusbv5fj3/a7Nzsw+yFg7E=">AAAXGnicrVhLb9tGEN6kr1R9OSngSy9slAJpQauSm8K5GAhgJyiKBnUAv9ooMCSSkgiRIktSVhRC/yTX9of0VvTaS39Kb535dimSEkVKgi1YXM3OfPPYmVnudn3HDqNm899bt9959733P7jzYe2jjz/59LOdu/fOQ28cGNaZ4TlecNnthJZjj6yzyI4c69IPrI7bdayL7vCI5y+urSC0vdFpNPWtV26nP7J7ttGJiHS1s9P2H051LdK1tmN6Ufj11U692WjiT1setNSgLtTfiXe39p9oC1N4whBj4QpLjEREY0d0REifl6IlmsIn2isREy2gkY15S8xEjWTHxGURR4eoQ/ru06+Xijqi34wZQtogLQ79BySpia8Uj0njHqjyyfq1DO8qHTGw2cYpPbsK0yVqJAZErZJLONeX69LHrfA6Ih8ew1ubPPFB4TgYOZ979HTod0Qe8veUOC0amSQV0MggmkNUSWEdAT1l5Dk2A6xEB3wWjdjqMrvLrOZ5jz5DwurQOISlbKsmnql1GUGzBVuZx8GarkZ8TR5K+8qwenMfbKy79IJ5T4kyFG9olEcu05nNrtVckUKeFehi/ggcI+IOieIh822Kn00c+Vw1CJN1dLCSffjiI4MaCvlHzCfZ7NN67sGeEKumId9s6PFVZaV2J3Y69OwCOyD5mOYG0MOx0MkeC9gBsjeJog7ugH5N8MuAjcYCvYHK5PXUySfOLp1wPZqx51gyEjKeiUXSuphomvrsiZ+g36J11xEVzmudZD1YKLM4hK6RWvtD6iscF5e+mcqRTSg6dLiwY4BsGRIt1SeRWN+35EWI+nEUakxPB1HzgaJDO8dngrELn3hFx9A9obEJPSytUa9riANlxyynlWNjozsta9WQf1y7sjMuamW5ESpVU1l5CE1N8Uh5zCvAdmejZqhuKHtyCO9mC1Z1YX/aNThH8/axVLo++bUykY0DSOVxOYrFfhb5so860bGWHPs+cRxCqqfoIWrTV5pqmfo7UlXiYT0MPF2VE7LeWdtkYSYmuqsi5RCfrHPek3jmN8LsIKfaiF52vgzxqZJPdgPu+DGo5XLnqKy8bECjeD5TLm9jZCE+csQrLvF8xNqHL1/O/7XMzLrYN4M3WkLkjI0U0rroJnKj2PNTUadeUKesyseTV5opAd5OfBo/mHM+IFzmHULLiChc4fF8fla5BsdEmamI9TCX1GFqg43OZ4KzrTJPcsiKihY4uKrLtXJeFuWNpJfLmvN8z8tKerksU66Rm7bqByGwLivkIqwDS7iZLD2tkPKwJ0i6tPmXNewbYa8dq7wIlIUXFZIueoz0ylNV+bzSvghdMFAxYZlft4jgNfrfDG9Am8YxlY02imYqN90qpqn8ZMPIppLuhvFNJd9USFrYG+w5jWVeVNYUc51Ucsl8qtoTLNpNkvcXWW06vrsURd6zTdWZNLXfcFbI3fEQPUBfqz8v7ztcya3KOh6j+3bRhwJoNktq5+zG0NIVHK/hX4h8T8+X69RXkdz6eXOEdSjqjslM1S7nqZPAqn1ZcmhL+9u2+6oPXzvQmezUV5mdJc7NV2NZmYxdRkpmJc4x3h4d1KZF3VLG62dleVLvbZIZE83HOUSOyu1IuBY9S+jb+pfiZr1chbqNr98AkXn60PwaOuRvrumEti1iEossoqTV1Ee+Gydn2jDz1mQjLul5N8bZZKKyLDml7MEa2Vl+oB36Ob35z5D/39Gpo0Xfz3LvYuui8ul/rHbxLO4+IR9QLI7xtrU5rq/O6Hu5k3xex1PSkPzP1J1Hys3njUi9OxUhbaK3GFn2QnPphLyoSZ6BBsQl7wWmFYhpdy3GK1/fYmR5f+fhXJreDtwEcvbmYz28NGdW31NV3Z6tvtPx6dmjbwcdOSysyjMh753kO/3R0g0a5+8+VQXnbnz/3oK+PP7inUwfdyqy9yQ88mwdQVrSsoirbnd8yETqDBPiJiS972nk7m8k3qJtYSavXIV3qPpo0gdrVzv11uKd8PLgfL/R+r7RfPGo/uSxui++I74Q98VDitOBeEKVeUJxNWg3fSt+F3/svt39c/ev3b8l6+1bSuZzkfvb/ed/iYWAng==</latexit>

P (y, t, . . .)Brady Neal

Transportability of Causal Effects Across Populations

17 / 22

Transportability Problem
Source Population ⇧<latexit sha1_base64="kAXR8YPQ7qfUfniLBphNkceNVKo=">AAAXC3icrVjbbttGEN2kt9S9xEkBv/RFjVOgKGRVclM4LwYC2AmKokEd1Lc2DgyJomTCpMiSlBVH0Cfktf2QvhV97Uf0U/rWM2eXIqkLKQm2YHE1O3PmsjOz3G0FrhPF9fq/t26/8+57739w58O1jz7+5NO76/fuH0d+P7TsI8t3/fC01Yxs1+nZR7ETu/ZpENpNr+XaJ63LPZk/ubLDyPF7h/F1YL/ymt2e03GsZgzSz2cHzvn6Zr1W519letAwg01l/g78e2v/qTPVVr6yVF95ylY9FWPsqqaK8HmpGqquAtBeqSFoIUYO5201UmuQ7YPLBkcT1Et8d/HrpaH28FswI0pb0OLiP4RkRX1peNoYd0jVT9FfyfDO0zEktth4jWfLYHqgxuoC1DK5hHNxuRY+XonXMXx4TG8deBKQInGwcj538HTxO4aH8n0NThujNqRCjCzQXFA1RXSEeOrIS2wuuBJN8tkYidVFdhdZLfM+PpfAamIc0VKxtaKemXXpUbNNW4XH5ZrOR3wND7V9RVidsQ8O1117IbyHoFyqNxjlkYt0ZrNrPldskEczdAl/TI4euCNQfGa+g/g54MjnqgVM0dHkSnbpS8AMqhnkHzifZHOA9dyiPRFXrcJ8c6gnMJWV2p3Y6eLZInYI+SHmLqhHYlGFPTaxQ2ZvEsUquUP8GvCXRRutCXqNlSnrWYVPkl1V4PqYccZYOhI6nolF2rohaBXz2VI/Ur+Nda8yKpLXVcj6tFBncURdPbP2u+grEhcP30KVyCaUKnV4tOOC2XIJWqpPI4m+b+BFxPpxDeoQT5dRC4hSpXaJz4Bjjz7Jivape4Bxm3pEuoJeV1M7xo5RTqvExmF3mtZaYf5J7erOOKlV5Hqs1IrJyl1qqqtHxmNZAbE7GzXLdEPdkyN6N5qwqkX7064hOZq3T6TS9cmvVZvZeEGpPK5Ecbafs3zZZp1UuZYS+y44dinVMfSItRkYTWuZ+tszVeJzPSw+PZMTut5F22BiZgi6ZyLlgk/XuexJMvMbMJvMqTNGLztfhPjUyCe7gXT8IanFcsesrLxsiNFwPFMs73BkMz56JCuu8QLGOqAvX4z/K5mZRbFvBq83hSgZGxukRdHbzI3Znh+qTfSCTWRVPp6y0kIJ+XYSYPxwzPkQuMJ7SS09UKTCh+P5Ueka7IMyMhHrcC6pw9QGh52vTc4zk3maQ1dUPMEhVV2sVfJyVt5oerFse5zveVlNL5YVyhVz0zH9ICLWaYlczHUQCS+TpYclUj73BE3XNv+ygH097rV9kxehsfCkRNJjj9Fe+aYqn5faF7MLhiYmIvPrChG8Yv8b8Q1o2TimsvFS0UzlrleKaSo/WDKyqaS3ZHxTyTclkjb3BmdME5kXpTUlXAelXDqfyvYEG7tJ8v6iq63K7xaiKHt223SmitlvJCv07rjLHlBdqD9P7ztSyY3SOu6z+7bYh0JqbhfUztGNoaUr2F/Av4j5np4vF6mvWXKL580e12FWd0xmynY535wE5u3LmqMytb+tuq8G9LVJnclOfZ7ZWYa5+XIsO5Ox00jJrMbZ59ujy9q00S11vH4ylif1fgaZPmgBzyF6VGxHwjXpWUJf1b8UN+vlPNRVfP2aiMLTpebX1KF/S00ntFURk1hkETVtzXz0u3Fypo0yb00O45Ked4c8mwxMliWnlC1aozvL99ihn+PNf8T8/xanjga+n+XexRZFldN/3+ziWdxtIO8gFvt821oeNzBn9K3cST6v4yk0JP8jc+eRcst5IzbvTrOQltE7G1n3wvbUCXlSkz4DXYBL3wtclyCm3XU2XvH6zkbW93c+z6Xp7cBNIGdvPhbDS3Nm/j1V2e3Z/DudAM8Ovl125GhmVR4pfe+k3+n3pm7QJH+3URWSu8MH9yf05fEn72S6vFPRvSfh0WfrmNKalkWcd7sTUCY2Z5iINyHpfU8td3+j8SZtizJ55Rm8XdNHkz64dr6+2Zi8E54eHG/XGt/V6i8ebT55bO6L76jP1QP1FeK0o56gMg8QVwsWvFW/qz823m78ufHXxt+a9fYtI/OZyv1t/PM/fhR73A==</latexit>

Target Population ⇧⇤<latexit sha1_base64="kB2IrOPaT/rS19xtMKqesRZVGKY=">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</latexit>

P ⇤ (y, t, . . .)<latexit sha1_base64="vTXhwYX9nKBrAcwXU3orFduTrMk=">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</latexit><latexit sha1_base64="m7DSFZusbv5fj3/a7Nzsw+yFg7E=">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</latexit>

P (y, t, . . .)Given P (y | do(t), x)<latexit sha1_base64="K78GVsWHwf2plCHOX8pNPOGsB0I=">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</latexit>

Brady Neal

Transportability of Causal Effects Across Populations

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Transportability Problem
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P ⇤ (y, t, . . .)<latexit sha1_base64="vTXhwYX9nKBrAcwXU3orFduTrMk=">AAAXHHicrVhLb9tGEN6kr1R9OS7gSy9slAJpQKuSm8K5GAhgJyiKBnUAv9ooNSSSkgjxVZKyohD6K7m2P6S3otcC/Sm9debbpUjqQcqCLVhczc5889iZWe52A8eO4mbz31u333n3vfc/uPNh7aOPP/n0s62722eRPwoN69TwHT+86HYiy7E96zS2Y8e6CEKr43Yd67w7POT58ysrjGzfO4kngfXK7fQ9u2cbnZhIl1vb7eDXhw8muhbrWtsx/Tj6+nKr3mw08actDlpqUBfq79i/W/tPtIUpfGGIkXCFJTwR09gRHRHR56VoiaYIiPZKJEQLaWRj3hJTUSPZEXFZxNEh6pC++/TrpaJ69JsxI0gbpMWh/5AkNfGV4jFp3ANVPlm/luNdpSMBNts4oWdXYbpEjcWAqFVyKef6cl36uBVex+TDY3hrkycBKBwHo+Bzj54O/Y7JQ/6eEKdFI5OkQhoZRHOIKimsI6SnjDzHZoCV6IDPohFbXWZ3mdU879NnSFgdGkewlG3VxDO1Lh40W7CVeRys6WrE1+ShtK8Mqzfzwca6Sy+Y94QoQ/GGRkXkMp357FrNFSvk6RJdzB+DwyPuiCg+Mt+m+NnEUcxVgzBZRwcr2YcvATKooZB/wHyazQGt5y7sibBqGvLNhp5AVVZmd2qnQ88usEOST2huAD0cC53ssYAdInvTKOrgDunXGL8M2GjM0RuoTF5PnXzi7NIJ16cZe4YlIyHjmVokrUuIpqnPrvgR+i1adx1R4bzWSdaHhTKLI+jy1NofUF/huLj0zVSObErRocOFHQNky5BomT6JxPq+IS8i1I+jUBN6OohaABQd2jk+Y4xd+MQrOoLuMY1N6GFpjXpdQ+wrO6YFrRwbG91pUauG/OPalZ1xXivLeahUTWXlATQ1xSPlMa8A252PmqG6oezJEbybzlnVhf1Z1+AcLdrHUtn6FNfKRDYOIFXE5Sgu93OZL3uoEx1rybHvE8cBpHqKHqE2A6Wplqu/Q1UlPtbDwNNVOSHrnbWN52YSorsqUg7xyTrnPYlnfiPMDnKqjejl58sQnyr5dDfgjp+AWi53hsoqyoY0SmYz5fI2RhbiI0e84hIvQKwD+PLl7F/LzayLfTN43gIiZ2yskNZFN5Ebyz0/EXXqBXXKqmI8eaWZEuLtJKDx/RnnfcJl3iG0eEThCk9m89PKNTgiylRFrIe5tA4zG2x0PhOcbZV5kkNWVDzHwVVdrpXzclneSHq5rDnL96KspJfLMuUKuWmrfhAB66JCLsY6sISby9KTCikfe4KkS5t/XsM+D3vtSOVFqCw8r5B00WOkV76qyueV9sXogqGKCcv8skEEr9D/pngDum4cM9n4WtHM5CYbxTSTH18zspmke834ZpJvKiQt7A32jMYyLyprirmOK7lkPlXtCRbtJun7i6w2Hd9diiLv2abqTJrabzgr5O54gB6gr9WfF/cdruRWZR2P0H276EMhNJsltXN6Y2jZCo7W8C9Cvmfny3Xqa5nc+nlziHVY1h3TmapdzlcngVX7suTQFva3TffVAL52oDPdqS9zO0tSmK/GsnIZu4iUzkqcI7w9OqhNi7qljNdPyvK03tskMyJagHOIHJXbkXLNe5bSN/Uvw817uQp1E18fApF5+tD8Gjrkb67plLYpYhqLPKKk1dRHvhunZ9oo99ZkIy7ZeTfB2WSssiw9pezCGtlZvqcd+jm9+U+R/9/SqaNF388K72LrovLpf6R28TzuHiHvUyyO8LZ1fdxAndF3Cyf5oo6npCH9n6o7j4ybzxuxendahnQdvcuRZS80F07I85rkGWhAXPJeYFKBmHXX5Xjl67scWd7f+TiXZrcDN4Gcv/lYDy/LmdX3VFW3Z6vvdAJ69ujbQUeOllblqZD3TvKd/nDhBo3zd4+qgnM3ubc9p6+IP38n08ediuw9KY88W8eQlrQ84qrbnQAysTrDRLgJye57GoX7G4k3b1uUyytX4R2oPpr2wdrlVr01fye8ODjba7S+azRfPKo/eazui++IL8Q98YDitC+eUGUeU1wN6oNvxe/ij523O3/u/LXzt2S9fUvJfC4Kfzv//A8P3IE6</latexit><latexit sha1_base64="m7DSFZusbv5fj3/a7Nzsw+yFg7E=">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</latexit>

P (y, t, . . .)<latexit sha1_base64="SeeqMYwOORjc4BMlbgYMBNhOsqE=">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</latexit>

Given P (y | do(t), x)<latexit sha1_base64="K78GVsWHwf2plCHOX8pNPOGsB0I=">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</latexit>

Brady Neal?

P (y | do(t),Goal:x) = P ⇤ (y | do(t), x)

Transportability of Causal Effects Across Populations

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Transportability Problem
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P ⇤ (y, t, . . .)<latexit sha1_base64="vTXhwYX9nKBrAcwXU3orFduTrMk=">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</latexit><latexit sha1_base64="m7DSFZusbv5fj3/a7Nzsw+yFg7E=">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</latexit>

P (y, t, . . .)<latexit sha1_base64="SeeqMYwOORjc4BMlbgYMBNhOsqE=">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</latexit><latexit sha1_base64="SeeqMYwOORjc4BMlbgYMBNhOsqE=">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</latexit>

Given P (y | do(t), x)<latexit sha1_base64="K78GVsWHwf2plCHOX8pNPOGsB0I=">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</latexit>

Brady Neal?

P (y | do(t), x) = P ⇤ (y | do(t), x)

Transportability of Causal Effects Across Populations

17 / 22

Selection Diagrams

Brady Neal

Transportability of Causal Effects Across Populations

18 / 22

Selection Diagrams
Allow for different causal mechanisms across the two distributions

Brady Neal

Transportability of Causal Effects Across Populations

18 / 22

Selection Diagrams
Allow for different causal mechanisms across the two distributions<latexit sha1_base64="UwmK6pZdJ3mC2AEnkYKpqS6dlas=">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</latexit>

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Brady Neal

Transportability of Causal Effects Across Populations

Y

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Selection Diagrams
Allow for different causal mechanisms across the two distributions
Mechanism that differs between ⇧ and ⇧⇤<latexit 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Brady Neal

Transportability of Causal Effects Across Populations

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Selection Diagrams
Allow for different causal mechanisms across the two distributions
Mechanism that differs between ⇧ and ⇧⇤<latexit 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S

ST

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Brady Neal

Transportability of Causal Effects Across Populations

XY

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Selection Diagrams
Allow for different causal mechanisms across the two distributions
Mechanism that differs between ⇧ and ⇧⇤<latexit 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Brady Neal

Transportability of Causal Effects Across Populations

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Selection Diagrams
Allow for different causal mechanisms across the two distributions
Mechanism that differs between ⇧ and ⇧⇤<latexit 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Absence of S encodes invariance

S

X

T

Brady Neal

Transportability of Causal Effects Across Populations

Y

18 / 22

Selection Diagrams
Allow for different causal mechanisms across the two distributions
Mechanism that differs between ⇧ and ⇧⇤<latexit 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Absence of S encodes invarianceSX

P ⇤ (y | do(t), x) , P (y | do(t), x, s⇤ )<latexit sha1_base64="Fcw0l/vhuTTJq3B4DX7A+X/eW9k=">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</latexit>

T

Brady Neal

Transportability of Causal Effects Across Populations

Y

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Direct Transportability (External Validity)<latexit sha1_base64="Dq0c8IUwofikAUgrPhU3BOaIfBw=">AAAZk3icrVjbbttGEN24t9TpJUnRByMvbOwASUErspsiBQIBQe0ERdG0CSI5dm3DoMiVRJgiGZKy4xD6on5NH9v+TGfOLkVSF1I2YsHScnbmzGVnZrnbDT03TprNf66tfPTxJ59+dv3z1RtffPnV1zdv3d6Lg1Fky44deEG037Vi6bm+7CRu4sn9MJLWsOvJN93THZ5/cyaj2A38dnIRyuOh1ffdnmtbCZFObq3sHnVl3/XTxD19H7p2Mork+DB+O7Ii+bARJxeebKWR7I88KzLCwLvoB7459WzEriPj1qPx8aqh/+4ZvwdEmzwf+fR4GCdWIs2e63ktO/Adl01w/f4m3Dg27u8/MKzEuN80mw+MdGN/Y/xkHgAxth8Yh13pBeeGJ3uJ0Qp6xv4xibQXixxMRCK3PyjIHCyQMVUQSPQ1iVrd4EzmQq8XCTlWPJAOCXXm6GuzaIdFC4F65vSLgQqtZIBISKKzp08Wzh3MzrUr5l5nc/uzc50KfZ0iJiaPpO+UEubk5nqz0cSfMTvY0oN1of9eBrdu3BFHwhGBsMVIDIUUvkho7AlLxPQ5FFuiKUKiHYuUaBGNXMxLMRarJDsiLkkcFlFP6btPT4ea6tMzY8aQtkmLR/8RSRrinuZxaNwDVf2yfqPAu0hHCmy28YJ+uxpzSNREDIhaJ5dxLi/Xpc+wxuuEfPgJ3rrkSQgKx8Eu+dyjX4+eE/KQvy+IU9LIIamIRjbRPKIqCuuI6FdFnmMzwEpY4JM0Yqur7K6ymucD+pwSlkXjGJayrYZ4rtfFh2YJW5nHw5ouRnxHHir7qrB6Ex9crLvygnnbRDkV72lURq7SWcyuxVyJRh7P0cX8CTh84o6JEiDzXYqfSxzlXLUJk3VYWMk+fAmRQQ2N/Cvms2wOaT03YU+MVTOQby70hLqycrszOz367QI7IvmU5gbQw7EwyR4J7AjZm0XRBHdET+d4smGjPUVvoDJ5PU3yibPLJNyAZtwJloqEimdmkbIuJZqhP5viN+iXtO4mosJ5bZJsAAtVFsfQ5eu1b1Ff4bgM6ZupHNmMYkLHEHYMkC2nRMv1KSTW95C8iFE/nkZN6ddD1EKgmNDO8TnHeAifeEVH0H1OYwd6WNqgXtcQj7Ud45JWjo2L7jSr1UD+ce2qzjitleV8VKqhs7IFTU3xSHvMK8B2F6Nm626oenIM78ZTVnVhf941OEfL9rFUvj7ltXKQjQNIlXE5ivP9nOfLNurExFpy7PvE0YJUT9Nj1GaoNa0W6m9HV0mA9bDxO9Q5oeqdtZ1PzaREH+pIecSn6pz3JJ55S5gWcuoI0SvOVyE+0/LZbsAdPwW1Wm4PlVWWjWiUTmaq5V2MJOKjRrziCi9ErEP48t3k3yjMLIv9YfD8GUTO2EQjLYvuIDfme94W69QL1imryvHklWZKhLeTkMYbE84NwmXeU2jxicIVnk7mx7VrsEuUsY5YD3NZHeY2uOh8DjiPdOYpDlVRyRQHV3W1Vs7LeXmj6NWyziTfy7KKXi3LlDPkpqv7QQys/Rq5BOvAEsNClrZrpALsCYqubD5Ywj4fe+1I50WkLXxTIzlEj1FeBboqX9Tal6ALRjomLPPnFSJ4hv43xhvQZeOYyyaXimYud3GlmOby55eMbC45vGR8c8n3NZISe4M7obHMq9qaYq6XtVwqn+r2BEm7Sfb+oqrNxHeXosh7tqM7k6H3G84KtTu20APMpfrz7L7DlbxVW8cjdN8u+lAEzU5F7XQ+GFq+gqMl/IuR7/n5cpn6mie3fN7sYB3mdcdspm6XC/RJYNG+rDiMmf3tqvtqCF8t6Mx26pPCzpKW5uuxZCFjZ5GyWYWzi7dHD7UpqVuqeP2hLc/q/YhkRkQLcQ5Ro2o7Mq5pzzL6Vf3LcYteLkK9iq/fA5F5+tD8DjrUM9d0RrsqYhaLIqKiXe4M2MC5NtBnuQinJfVunb9fZ+fiuPDm5SK2+Zk5xfnmXGdqdtLZhEeqO/1Cu/wLOj2MUUM/0Mlli76fl97nlkXlG4SRfhMo4m4T8mOK5y7e2C6PG+pz/mbpNqCs4xlpyP7H+t4k5/aEOt0uQrqM3vnIqp86M6fsaU3qHDUgLnW3cFGDmHfo+XjV6zsfWd0BBjjb5jcMHwK5eHuyHF6eM4vvuupu4BbfC4X026NvD109nlvZHaHurtS5YGfmFo7zd5uqgnM3vXt7Sl8Zf7qm+7iXUf0r41Hn8wTSirZMdwghk+hzUIzblGK/KN4BKbxp2+JCXg01Xkv34qyXrp7cXN+avleeHextN7Z+bDRfba8/3dZ3ztfFHXFX3Kc4PRZPqTJfUlztlb9W/l75d+W/tW/Xnqz9vLarWFeuaZlvROlv7cX/hjsvSQ==</latexit>

S<latexit sha1_base64="SeeqMYwOORjc4BMlbgYMBNhOsqE=">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</latexit>?

P (y | do(t), x) = P ⇤ (y | do(t), x)<latexit sha1_base64="SY08VlP+gHZXrXuVKc6/BP753Lg=">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</latexit>

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Brady Neal

Transportability of Causal Effects Across Populations

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Direct Transportability (External Validity)<latexit sha1_base64="Dq0c8IUwofikAUgrPhU3BOaIfBw=">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</latexit>S

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Brady Neal

Transportability of Causal Effects Across Populations

19 / 22

Direct Transportability (External Validity)<latexit sha1_base64="Dq0c8IUwofikAUgrPhU3BOaIfBw=">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</latexit>S

P (y | do(t), x) = P ⇤ (y | do(t), x)<latexit sha1_base64="SY08VlP+gHZXrXuVKc6/BP753Lg=">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</latexit><latexit sha1_base64="k4z/nsa9yUyNzgzgheJimr85g9s=">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</latexit>

if Y ??GT S | T, X

Proof:X

T

Y

U

Brady Neal

Transportability of Causal Effects Across Populations

19 / 22

Direct Transportability (External Validity)<latexit sha1_base64="Dq0c8IUwofikAUgrPhU3BOaIfBw=">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</latexit>S

P (y | do(t), x) = P ⇤ (y | do(t), x)<latexit sha1_base64="SY08VlP+gHZXrXuVKc6/BP753Lg=">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</latexit><latexit sha1_base64="k4z/nsa9yUyNzgzgheJimr85g9s=">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</latexit>

if Y ??GT S | T, X

X

TProof:P ⇤ (y | do(t), x) = P (y | do(t), x, s⇤ )Y<latexit sha1_base64="bck3sOyyKTDPw/+/myBlmvrTJB0=">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</latexit>= P (y | do(t), x)

Brady Neal

U

Transportability of Causal Effects Across Populations

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Direct Transportability (External Validity)<latexit sha1_base64="Dq0c8IUwofikAUgrPhU3BOaIfBw=">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</latexit>S

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if Y ??GT S | T, X

X

TProof:P ⇤ (y | do(t), x) = P (y | do(t), x, s⇤ )= P (y | do(t), x)Y<latexit sha1_base64="bck3sOyyKTDPw/+/myBlmvrTJB0=">AAAXWHicrVjbbttGEN2klyTqzXGB+qEvbJQWtkGrkpvCeREQwE5QFE3rAL61UWJIFCURIkWWpCwrgv6qP9O8tj/Rt86cXYqkRJGyYAsWV7MzZy47M8vdlmdbQVit/n3n7gcffvTxvfsPSp98+tnnX2w83DwL3KFvmKeGa7v+RasZmLY1ME9DK7TNC883m07LNs9b/UOeP78y/cByByfh2DPfOM3uwOpYRjMk0uXGrw3v7e72WGs4VltrHLnb4Y6uXe+UvqtrDW+BrmvB290drdFYMr9zuVGuVqr40xYHNTUoC/V37D4s/Scaoi1cYYihcIQpBiKksS2aIqDPa1ETVeER7Y2YEM2nkYV5U0xFiWSHxGUSR5Ooffru0q/Xijqg34wZQNogLTb9+ySpiW8VT5vGHVDlk/VrCd5lOibAZhvH9GwpTIeooegRtUgu4lxdrkUfp8DrkHx4Cm8t8sQDheNgpHzu0NOm3yF5yN9j4jRp1CYpn0YG0WyiSgrr8OkpI8+x6WElmuAzacRW59mdZzXPu/TpE1aTxgEsZVs18UKtywCaTdjKPDbWdDniNXko7cvD6sx8sLDu0gvmPSFKX7yjURo5T2cyu5ZzhQp5mqGL+UNwDIg7IIqLzLcofhZxpHPVIEzW0cRKduGLhwyqKOSfMR9ls0fruQd7AqyahnyzoMdTlRXbHdlp07MFbJ/kJzTXgx6OhU72mMD2kb1RFHVw+/RrhF8GbDTm6BVUJq+nTj5xdumE69KMNcOSkZDxjCyS1k2IpqnPnvgF+k1adx1R4bzWSdaFhTKLA+gaqLWvU1/huDj0zVSObETRocOBHT1kS59osT6JxPq+Jy8C1I+tUCf0tBE1Dyg6tHN8Rhg78IlXdAjdIxq3oYelNep1FXGg7JimtHJsLHSnRa0a8o9rV3bGea0sN0Claior69BUFU+Ux7wCbHcyaobqhrInB/BuOmdVC/bHXYNzNG0fS8Xrk16rNrKxB6k0Lkcx288sX/ZRJzrWkmPfJY46pDqKHqA2PaWplKi/Q1UlLtbDwNNROSHrnbWN5mYmRHdUpGzik3XOexLP/EmYTeRUA9FLzuchPlfy0W7AHX8Car7cGSorLevTaDKbyZe3MDIRHzniFZd4HmLtwZdvZv9aYmZV7NvBGywgcsaGCmlV9DZyI9vzE1GmXlCmrErHk1eaKT7eTjwaP55xPiZc5u1Dy4AoXOGT2fy0cA2OiDJVEetgLqrD2AYLna8NzobKPMkhKyqc4+CqztfKeZmVN5KeL9ue5XtaVtLzZZlyhdy0VD8IgHVRIBdiHVjCSWTpSYGUiz1B0qXNv69g3wB77VDlha8sPC+QdNBjpFeuqsqXhfaF6IK+ignL/LFGBK/Q/6Z4A7ppHGPZ8EbRjOXGa8U0lh/dMLKxpHPD+MaS7wokTewN1ozGMq8Ka4q5jgu5ZD4V7Qkm7SbR+4usNh3fLYoi79lt1Zk0td9wVsjdsY4eoK/Unxf3Ha7kWmEdD9F9W+hDPjS3c2rn9NbQ4hUcruBfgHyPz5er1FeW3Op5c4h1yOqO0UzRLueqk8CyfVlyaAv727r7qgdfm9AZ7dSXiZ1lkpovxjITGbuIFM1KnCO8PdqoTZO6pYzXb8ryqN4bJDMkmodziBzl2xFxzXsW0df1L8ZNerkMdR1fd4HIPF1ovoYO+ZtrOqKtixjFIokoaSX1ke/G0Zk2SLw1WYhLfN6d4GwyUlkWnVL2YI3sLD/RDv2S3vynyP8f6NRRo+8XqXexVVH59D9Uu3gSd5+QDygWR3jbujmup87oe6mTfFrHc9IQ/U/VnUfMzeeNUL07ZSHdRG82suyF7YUT8rwmeQbqEZe8FxgXIMbdNRsvf32zkeX9nYtzaXw7cBvIyZuP1fDinFl+T1V0e7b8TsejZ4e+bXTkILMqT4W8d5Lv9IcLN2icv/tUFZy7k0ebc/rS+PN3Ml3cqcjeE/HIs3UIaUlLIi673fEgE6ozTICbkPi+p5K6v5F487YFibxyFF5d9dGoD5YuN8q1+TvhxcHZfqX2Y6X66kn52VN1X3xffC0eiW2K04F4RpV5THE1xF/ivfhH/PvV+y2xdW/rgWS9e0fJfClSf1ub/wPVOY+3</latexit><latexit sha1_base64="bck3sOyyKTDPw/+/myBlmvrTJB0=">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</latexit>

Brady Neal

U

Transportability of Causal Effects Across Populations

19 / 22

Direct Transportability (External Validity)
P (y | do(t), x) = P ⇤ (y | do(t), x)<latexit 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if Y ??GT S | T, X

X

TProof:SYP ⇤ (y | do(t), x) = P (y | do(t), x, s⇤ )= P (y | do(t), x)<latexit sha1_base64="bck3sOyyKTDPw/+/myBlmvrTJB0=">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</latexit><latexit sha1_base64="bck3sOyyKTDPw/+/myBlmvrTJB0=">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</latexit>

Brady Neal

Transportability of Causal Effects Across Populations

19 / 22

Direct Transportability (External Validity)⇤

P (y | do(t), x) = P (y | do(t), x)

P (y | do(t), x) 6= P ⇤ (y | do(t), x)

if Y ??GT S | T, X

X

<latexit 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T

Proof:SYP ⇤ (y | do(t), x) = P (y | do(t), x, s⇤ )= P (y | do(t), x)<latexit sha1_base64="bck3sOyyKTDPw/+/myBlmvrTJB0=">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</latexit><latexit sha1_base64="bck3sOyyKTDPw/+/myBlmvrTJB0=">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</latexit>

Brady Neal

Transportability of Causal Effects Across Populations

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Trivial Transportability• Don’t have direct transportability: P (y | do(t), x) 6= P ⇤ (y | do(t), x)<latexit sha1_base64="m8XTtNDHGrY30TDYnZZznBNPa2A=">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</latexit><latexit sha1_base64="sqZFh7skkopQaERygQifmV585DY=">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</latexit>

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Brady Neal

Transportability of Causal Effects Across Populations

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Trivial Transportability• Don’t have direct transportability: P (y | do(t), x) 6= P ⇤ (y | do(t), x)• Have access to observational data from target population: P ⇤ (y, t, x)<latexit 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Brady Neal

Transportability of Causal Effects Across Populations

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20 / 22

Trivial Transportability• Don’t have direct transportability: P (y | do(t), x) 6= P ⇤ (y | do(t), x)• Have access to observational data from target population: P ⇤ (y, t, x)• Can identify estimand using only target data: P ⇤ (y | do(t), x) = P ⇤ (y | t, x)<latexit 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Brady Neal

Transportability of Causal Effects Across Populations

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Y

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Trivial Transportability• Don’t have direct transportability: P (y | do(t), x) 6= P ⇤ (y | do(t), x)• Have access to observational data from target population: P ⇤ (y, t, x)• Can identify estimand using only target data: P ⇤ (y | do(t), x) = P ⇤ (y | t, x)<latexit 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Identify P (y | do(t)) ?
<latexit sha1_base64="r9ycMo1ZqJaOKxpV9rwnTG3MVL8=">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</latexit>

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Brady Neal

Transportability of Causal Effects Across Populations

20 / 22

Trivial Transportability• Don’t have direct transportability: P (y | do(t), x) 6= P ⇤ (y | do(t), x)• Have access to observational data from target population: P ⇤ (y, t, x)• Can identify estimand using only target data: P ⇤ (y | do(t), x) = P ⇤ (y | t, x)<latexit 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sha1_base64="61wu9cIVUQRcQsX10i4CM5Bjhac=">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</latexit>

Identify P (y | do(t)) ?
<latexit sha1_base64="r9ycMo1ZqJaOKxpV9rwnTG3MVL8=">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</latexit>⇤

Combine aspects oftrivial and directTtransportability

S

X

X

T

S

Y

Y

U

Brady Neal

Transportability of Causal Effects Across Populations

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S-Admissibility and Transport Formula<latexit sha1_base64="61wu9cIVUQRcQsX10i4CM5Bjhac=">AAAZe3icrVjbbttGEN24t9TpJUlfauSFjR0gCWhFdhOkQCAggJOgKJo2QaTYrm0YlLiSCFEkQ1JWHEJf0a/pa/sV/ZgCnTm7FEldSNmIBUvL2Zkzl52Z5W47cJ0ortf/vbL2yaefff7F1S/Xr3319TffXr9x823kj8KObHV81w8P2lYkXceTrdiJXXkQhNIatl253x7s8fz+mQwjx/ea8XkgT4ZWz3O6TseKiXR6Y808bsue4yWxM/gQOJ14FMrJUfRuZIXyQS2Kz13ZSELZG7lWaAS+e97zPXPm2YgcW0aNh5OTdUP/3TF+84k2fT726PEoiq1Ynhh3D+4ZVmzcrZv1e0aydbA1ebKEsXnPOGpL1x8bruzGRsPvGgcnJNJcLnI4FQmdXj8nc7hExlTOkugbErXa/pnMhN4sE7KtqC9tEmot0Ndk0RaL5gLy3O7lAxJYcR+RkERnT58snTucn2uWzL1J5w7m51ol+lp5TEweS88uJMbp9c16rY4/Y36wowebQv+98m9cuyWOhS180REjMRRSeCKmsSssEdHnSOyIugiIdiISooU0cjAvxUSsk+yIuCRxWEQd0HePno401aNnxowg3SEtLv2HJGmIO5rHpnEXVPXL+o0c7zIdCbDZxnP6bWvMIVFj0SdqlVzKubpcmz7DCq9j8uEneOuQJwEoHIdOwecu/br0HJOH/H1OnJJGNkmFNOoQzSWqorCOkH5V5Dk2fayEBT5JI7a6zO4yq3nep8+AsCwaR7CUbTXEC70uHjRL2Mo8LtZ0OeJ78lDZV4bVnfrgYN2VF8zbJMpAfKBREblMZz67lnPFGnmyQBfzx+DwiDsiio/Mdyh+DnEUc7VDmKzDwkr24EuADKpp5F8wn2ZzQOu5DXsirJqBfHOgJ9CVldmd2unSbxvYIcknNNeHHo6FSfZIYIfI3jSKJrhDehrjqQMbOzP0GiqT19Mknzi7TML1acaZYqlIqHimFinrEqIZ+rMtfoV+SetuIiqc1ybJ+rBQZXEEXZ5e+wb1FY7LkL6ZypFNKSZ0DGFHH9kyIFqmTyGxvgfkRYT6cTVqQr8uohYAxYR2js8Y4yF84hUdQfeYxjb0sLRBva4mHms7JgWtHBsH3Wleq4H849pVnXFWK8t5qFRDZ2UDmuriofaYV4Dtzketo7uh6skRvJvMWNWG/VnX4Bwt2sdS2foU18pGNvYhVcTlKC72c5Evu6gTE2vJse8RRwNSXU2PUJuB1rSeq789XSU+1qOD36HOCVXvrG08M5MQfagj5RKfqnPek3jmHWFayKljRC8/X4b4XMunuwF3/ATUcrm3qKyibEijZDpTLu9gJBEfNeIVV3gBYh3Alx+m/0ZuZlXsj4PnzSFyxsYaaVV0G7mx2POm2KResElZVYwnrzRTQrydBDTemnJuES7zDqDFIwpXeDKdn1SuwTOiTHTEuphL6zCzwUHns8F5rDNPcaiKimc4uKrLtXJeLsobRS+Xtaf5XpRV9HJZppwhNx3dDyJgHVTIxVgHlhjmsrRZIeVjT1B0ZfPhCvZ52GtHOi9CbeF+heQQPUZ55euqfFlpX4wuGOqYsMwfl4jgGfrfBG9AF41jJhtfKJqZ3PmlYprJjy8Y2UxyeMH4ZpIfKiQl9gZnSmOZ15U1xVyvKrlUPlXtCZJ2k/T9RVWbie82RZH3bFt3JkPvN5wVandsoAeYK/Xn+X2HK3mnso5H6L5t9KEQmu2S2ml9NLRsBUcr+Bch37Pz5Sr1tUhu9bzZwzos6o7pTNUu5+uTwLJ9WXEYc/vbZffVAL5a0Jnu1Ke5nSUpzFdjyVzGziOlswrnGd4eXdSmpG6p4vW7tjyt92OSGREtwDlEjcrtSLlmPUvpl/Uvw817uQz1Mr7eByLz9KD5PXSoZ67plHZZxDQWeURFu9gZsIZzra/PciFOS+rdOnu/Ts/FUe7Ny0FsszNzgvPNWGdqetLZhkeqO/1Mu/xLOj1MUEM/0sllh75fFN7nVkXlG4SRfhPI4+4S8mOK5zO8sV0cN9Dn/O3CbUBRx3PSkP5P9L1Jxu0KdbpdhnQRvYuRVT+1507Zs5rUOapPXOpu4bwCMevQi/HK13cxsroD9HG2zW4YPgZy/vZkNbwsZ5bfdVXdwC2/Fwrot0vfLrp6tLCyW0LdXalzwd7cLRzn7y5VBeducvvmjL4i/mxN93Avo/pXyqPO5zGkFW2V7hBAJtbnoAi3Kfl+kb8DUniztkW5vBpqvIbuxWkvXT+9vrkze688P3i7W9t5VKu/3t18uqvvnK+KW+K2uEtxeiyeUmW+orh21v5c+2vt77V/vv9vY3Pj/oapWNeuaJnvROFv49H/tt4l9w==</latexit>

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Transportability of Causal Effects Across Populations

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S-Admissibility and Transport Formula
S-Admissibility: A set of variables W is S-admissible if Y ??GT S | T, W
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Transportability of Causal Effects Across Populations

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S-Admissibility and Transport Formula
S-Admissibility: A set of variables W is S-admissible if Y ??GT S | T, W
<latexit sha1_base64="R3+oJY3URjeUnoKscuY/hxkOiTU=">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</latexit><latexit sha1_base64="61wu9cIVUQRcQsX10i4CM5Bjhac=">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</latexit>

S

Transport Result: If W is S-admissible, then<latexit sha1_base64="maw+FKMxZSEHjVhxaEhbkHL4W1A=">AAAXRnicrVhLb9tGEN6kr1R9OSngHnphoxSwA1qV3BTOoQIC2AmKokEdwI+kUWpIJCURIkWWpKwohP5Mf02u7a1/orei6K0z3y5FUg9SMmzB4mp25pvHzsxyt+M7dhjV63/duPnOu++9/8GtDysfffzJp59t3b5zFnqjwLBODc/xgueddmg59tA6jezIsZ77gdV2O4513hkc8vz5pRWEtjc8iSa+9cpt94Z21zbaEZEutr5v+b/e35loLdc2tdaRtxPt7mpNrRWO3Iux1vLzU7o23tVaugah8e7FVrVeq+NPWxw01KAq1N+xd7vyn2gJU3jCECPhCksMRURjR7RFSJ+XoiHqwifaKxETLaCRjXlLTEWFZEfEZRFHm6gD+u7Rr5eKOqTfjBlC2iAtDv0HJKmJrxWPSeMuqPLJ+rUM7yodMbDZxgk9OwrTJWok+kQtk0s415fr0Mct8ToiHx7CW5s88UHhOBg5n7v0dOh3RB7y94Q4LRqZJBXQyCCaQ1RJYR0BPWXkOTZ9rEQbfBaN2Ooiu4us5nmPPgPCatM4hKVsqyaeqHUZQrMFW5nHwZquRnxNHkr7irC6Mx9srLv0gnlPiDIQb2iURy7Smc2u1VyRQp4u0cX8ETiGxB0SxUPm2xQ/mzjyuWoQJutoYyV78MVHBtUU8o+YT7LZp/Xcgz0hVk1DvtnQ46vKSu1O7HTo2QF2QPIxzfWhh2Ohkz0WsANkbxJFHdwB/RrjlwEbjTl6DZXJ66mTT5xdOuF6NGPPsGQkZDwTi6R1MdE09dkTP0G/ReuuIyqc1zrJerBQZnEIXUO19k3qKxwXl76ZypFNKDp0uLCjj2wZEC3VJ5FY3zfkRYj6cRRqTE8HUfOBokM7x2eMsQufeEVH0D2msQk9LK1Rr6uJA2XHNKeVY2OjOy1q1ZB/XLuyM85rZbkhKlVTWdmEprp4oDzmFWC7s1EzVDeUPTmEd9M5qzqwP+0anKN5+1gqXZ/8WpnIxj6k8rgcxeV+LvNlH3WiYy059j3iaEKqq+ghatNXmiqZ+jtUVeJhPQw8XZUTst5Z23huJia6qyLlEJ+sc96TeOY3wmwjp1qIXna+CPGxkk92A+74MajFcmeorLxsQKN4NlMsb2NkIT5yxCsu8XzE2ocvX83+tczMutjXgzdcQOSMjRTSuugmcmO55yeiSr2gSlmVjyevNFMCvJ34NL4347xHuMw7gJYhUbjC49n8tHQNjogyVRHrYi6pw9QGG53PBGdLZZ7kkBUVzXFwVRdr5bxcljeSXixrzvI9LyvpxbJMuURu2qofhMB6XiIXYR1Yws1k6UmJlIc9QdKlzS/WsG+IvXak8iJQFp6XSLroMdIrT1Xl01L7InTBQMWEZX65QgQv0f+meAPaNI6pbLRRNFO5yZVimsqPN4xsKuluGN9U8k2JpIW9wZ7RWOZZaU0x13Epl8ynsj3Bot0keX+R1abju0NR5D3bVJ1JU/sNZ4XcHZvoAfpa/Xlx3+FKbpTW8Qjdt4M+FECzWVA7p9eGlq7gaA3/QuR7er5cp76Wya2fN4dYh2XdMZkp2+U8dRJYtS9LDm1hf7vqvurD1zZ0Jjv1RWZniXPz5VhWJmMXkZJZiXOEt0cHtWlRt5Tx+llZntR7i2RGRPNxDpGjYjsSrnnPEvpV/Utxs16uQr2Kr/eByDw9aH4NHfI313RCuypiEossoqRV1Ee+Gydn2jDz1mQjLul5N8bZZKyyLDml7MEa2Vl+oB36Kb35T5H/39Kpo0HfT3LvYuui8ul/pHbxLO4+IR9QLI7wtrU5rq/O6Hu5k3xex2PSkPxP1Z1Hys3njUi9Oy1D2kTvcmTZC82FE/K8JnkG6hOXvBeYlCCm3XU5XvH6LkeW93cezqXp7cB1IGdvPtbDS3Nm9T1V2e3Z6jsdn55d+nbQkcOlVXkq5L2TfKc/XLhB4/zdp6rg3I3v3pnTl8efv5Pp4U5F9p6ER56tI0hLWhZx1e2OD5lInWFC3ISk9z213P2NxJu3LczklavwmqqPJn2wcrFVbczfCS8OzvZrje9q9WcPqo8eqvviW+JLcVfsUJwOxCOqzGOKqyF+F2/FH+LP7bfbf2//s/2vZL15Q8l8LnJ/X4j/AdVkjgY=</latexit>⇤

P (y | do(t)) =Xw

P (y | do(t), w) P ⇤ (w)

X

T

Y

U

Brady Neal

Transportability of Causal Effects Across Populations

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S-Admissibility and Transport Formula
S-Admissibility: A set of variables W is S-admissible if Y ??GT S | T, W
<latexit sha1_base64="R3+oJY3URjeUnoKscuY/hxkOiTU=">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</latexit><latexit sha1_base64="61wu9cIVUQRcQsX10i4CM5Bjhac=">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</latexit>

S

Transport Result: If W is S-admissible, then<latexit sha1_base64="maw+FKMxZSEHjVhxaEhbkHL4W1A=">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</latexit>⇤

P (y | do(t)) =Xw

P (y | do(t), w) P ⇤ (w)Note: Another word for “sufficient adjustment set”
from week 4 is “admissible set.”

X

T

Y

U

Brady Neal

Transportability of Causal Effects Across Populations

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S-Admissibility and Transport Formula
S-Admissibility: A set of variables W is S-admissible if Y ??GT S | T, W
<latexit sha1_base64="R3+oJY3URjeUnoKscuY/hxkOiTU=">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</latexit><latexit sha1_base64="61wu9cIVUQRcQsX10i4CM5Bjhac=">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</latexit>

S

Transport Result: If W is S-admissible, then<latexit sha1_base64="maw+FKMxZSEHjVhxaEhbkHL4W1A=">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</latexit>⇤

P (y | do(t)) =Xw

P (y | do(t), w) P ⇤ (w)Note: Another word for “sufficient adjustment set”
from week 4 is “admissible set.”
Main Paper: Pearl & Bareinboim (2014)
Brady Neal

Transportability of Causal Effects Across Populations

X

T

Y

U

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Questions:
1. Describe direct transportability in your own words.
2. Describe trivial transportability in your own words.
3. Prove the transport result on the previous slide.