Counterfactuals
and Mediation
Brady Nealcausalcourse.com

Counterfactuals Basics

Important Application: Mediation

Brady Neal

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Counterfactuals Basics

Important Application: Mediation

Brady NealCounterfactuals Basics

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Fundamental Problem of Causal Inferencedo(T = 1)

do(T = 0)

Brady NealYi (1) = 1: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment

TYi

Yi (0) = 0

Causal effectYi (1) Yi (0) = 1

Counterfactuals Basics

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Fundamental Problem of Causal Inferencedo(T = 1)

Yi (1) = 1

Counterfactualdo(T = 0)

Yi (0) = 0

Causal effectYi (1) Yi (0) = 1

FactualBrady Neal

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment

T
Y
iCounterfactuals Basics

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Fundamental Problem of Causal Inferencedo(T = 1)

Yi (1) = 1

Factualdo(T = 0)

Yi (0) = 0

Causal effectYi (1) Yi (0) = 1

Counterfactual

Brady Neal

: observed treatment: observed outcome: used in subscript to denote a
specific unit/individual
Yi (1) : potential outcome under treatment
Yi (0) : potential outcome under no treatment

T
Y
iCounterfactuals Basics

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We can compute counterfactuals
using a parametric SCM.

Counterfactuals
Counterfactual: P (Y (t) | T = t0 , Y = y 0 )<latexit sha1_base64="4CMBJKtzM2gzxDZ61/z0DuX68ao=">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</latexit>

Brady NealCounterfactuals Basics

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Counterfactuals
observation

Counterfactual: P (Y (t) | T = t0 , Y = y 0 )<latexit sha1_base64="4CMBJKtzM2gzxDZ61/z0DuX68ao=">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</latexit>

Brady NealCounterfactuals Basics

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Counterfactuals
observation

Counterfactual: P (Y (t) | T = t0 , Y = y 0 )<latexit sha1_base64="4CMBJKtzM2gzxDZ61/z0DuX68ao=">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</latexit>

hypothetical condition

Brady NealCounterfactuals Basics

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Counterfactuals
observation

Counterfactual: P (Y (t) | T = t0 , Y = y 0 )<latexit sha1_base64="4CMBJKtzM2gzxDZ61/z0DuX68ao=">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</latexit>

hypothetical condition

Different from CATE: E[Y (t) | X = x] = E[Y | do(t), X = x]<latexit sha1_base64="fLfT15enJ6H5GVklPn8BN+RWbrE=">AAAXfHicrVjbbttGEN2kt9TpxUmB+iEvbOQCaUurkpvWeREQwHZQFA3qAL6llmFIJCUR4q0kZdkR9Bf9mr62P9GfKTpzdimSupCSYAsWV7MzZy47M8vdduDYUVyr/Xvv/nvvf/DhRw8+3nj4yaeffb756PFp5A9CwzoxfMcPz9utyHJszzqJ7dixzoPQarltxzpr9/d5/uzaCiPb947j28C6dFtdz+7YRism0tVmtXl48fZZ/I3WdG1TO9ca2s0lfTFVkpoHPk3raupqs1Kr1vCnzQ7qalAR6u/If/TwiWgKU/jCEAPhCkt4IqaxI1oios+FqIuaCIh2KUZEC2lkY94SY7FBsgPisoijRdQ+fXfp14WievSbMSNIG6TFof+QJDXxteIxadwBVT5Zv5bhXaRjBGy28ZaebYXpEjUWPaKWySWcy8u16eOWeB2TDy/grU2eBKBwHIyczx16OvQ7Jg/5+5Y4LRqZJBXSyCCaQ1RJYR0hPWXkOTY9rEQLfBaN2Ooiu4us5nmfPn3CatE4gqVsqyZeqXXxoNmCrczjYE0XI96Qh9K+IqzOxAcb6y69YN5jovTFOxrlkYt0ZrNrMVeskMdzdDF/DA6PuCOi+Mh8m+JnE0c+Vw3CZB0trGQXvgTIoKpC/gXzSTYHtJ47sCfCqmnINxt6AlVZqd2JnQ4928AOSX5Ecz3o4VjoZI8F7BDZm0RRB3dIv4b4ZcBGY4peRWXyeurkE2eXTrg+zdgTLBkJGc/EImndiGia+uyIX6HfonXXERXOa51kfVgosziCLk+tfYP6CsfFpW+mcmQTig4dLuzoIVv6REv1SSTW9z15EaF+HIU6oqeDqAVA0aGd4zPE2IVPvKID6B7S2IQeltao11XFnrJjnNPKsbHRnWa1asg/rl3ZGae1spyHStVUVjagqSaeK495BdjubNQM1Q1lT47g3XjKqjbsT7sG52jePpZK1ye/ViaysQepPC5Hcb6f83zZRZ3oWEuOfZc4GpDqKHqE2gyUpo1M/e2rKvGxHgaersoJWe+sbTg1MyK6qyLlEJ+sc96TeOYPwmwhp5qIXna+CPFQySe7AXf8EajFcqeorLxsSKPRZKZY3sbIQnzkiFdc4gWIdQBfvpr8a5mZZbHvBs+bQeSMjRXSsugmcmO+58eiQr2gQlmVjyevNFNCvJ0ENN6ecG4TLvP2ocUjClf4aDI/Ll2DA6KMVcQ6mEvqMLXBRuczwdlUmSc5ZEXFUxxc1cVaOS/n5Y2kF8uak3zPy0p6sSxTrpGbtuoHEbDOS+RirANLuJksPS6R8rEnSLq0+e0S9nnYawcqL0Jl4VmJpIseI73yVVW+LrUvRhcMVUxY5vc1IniN/jfGG9CqcUxl45WimcrdrhXTVH64YmRTSXfF+KaS70okLewN9oTGMm9Ka4q5jkq5ZD6V7QkW7SbJ+4usNh3fbYoi79mm6kya2m84K+Tu2EAP0Jfqz7P7DldyvbSOB+i+bfShEJrNgto5uTO0dAUHS/gXId/T8+Uy9TVPbvm82cc6zOuOyUzZLuerk8CifVlyaDP727r7agBfW9CZ7NRXmZ1llJsvx7IyGTuLlMxKnAO8PTqoTYu6pYzXb8rypN6bJDMgWoBziBwV25FwTXuW0Nf1L8XNerkIdR1fvwUi83Sh+QY65G+u6YS2LmISiyyipK12BqziXOurs1yI05J8t07fr5NzcZR587IR2/TMPML5ZqgyNTnp7MAj2Z1+pl3+NZ0exqihH+jkUqfvV7n3uWVR+QZhoN4Esri7hLxH8TzAG9vquIE65+/kbgPyOg5JQ/I/VvcmKbcj5Ol2EdIqeucjy35qzpyypzXJc1SPuOTdwm0JYtqh5+MVr+98ZHkH6ONsm94w3AVy9vZkObw0ZxbfdZXdwC2+Fwro2aFvB109mlvZJ0LeXclzwf7MLRzn7y5VBefu6OnjKX15/Oma7uJeRvavhEeez2NIS9oy3SGATKzOQRFuU7L9InsHJPGmbYsyeeUqvIbqxUkv3bjarNSn75VnB6e71fqP1dqb55WXL9Sd8wPxRDwVzyhOe+IlVeYRxdUQf4q/xN/iny//29re+m5rR7Lev6dkvhC5v62f/gfTXpQA</latexit>

Brady NealCounterfactuals Basics

6 / 25

Counterfactuals
observation

Counterfactual: P (Y (t) | T = t0 , Y = y 0 )<latexit sha1_base64="4CMBJKtzM2gzxDZ61/z0DuX68ao=">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</latexit>

hypothetical condition

Different from CATE: E[Y (t) | X = x] = E[Y | do(t), X = x]<latexit sha1_base64="fLfT15enJ6H5GVklPn8BN+RWbrE=">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</latexit>

Brady Neal<latexit sha1_base64="fLfT15enJ6H5GVklPn8BN+RWbrE=">AAAXfHicrVjbbttGEN2kt9TpxUmB+iEvbOQCaUurkpvWeREQwHZQFA3qAL6llmFIJCUR4q0kZdkR9Bf9mr62P9GfKTpzdimSupCSYAsWV7MzZy47M8vdduDYUVyr/Xvv/nvvf/DhRw8+3nj4yaeffb756PFp5A9CwzoxfMcPz9utyHJszzqJ7dixzoPQarltxzpr9/d5/uzaCiPb947j28C6dFtdz+7YRism0tVmtXl48fZZ/I3WdG1TO9ca2s0lfTFVkpoHPk3raupqs1Kr1vCnzQ7qalAR6u/If/TwiWgKU/jCEAPhCkt4IqaxI1oios+FqIuaCIh2KUZEC2lkY94SY7FBsgPisoijRdQ+fXfp14WievSbMSNIG6TFof+QJDXxteIxadwBVT5Zv5bhXaRjBGy28ZaebYXpEjUWPaKWySWcy8u16eOWeB2TDy/grU2eBKBwHIyczx16OvQ7Jg/5+5Y4LRqZJBXSyCCaQ1RJYR0hPWXkOTY9rEQLfBaN2Ooiu4us5nmfPn3CatE4gqVsqyZeqXXxoNmCrczjYE0XI96Qh9K+IqzOxAcb6y69YN5jovTFOxrlkYt0ZrNrMVeskMdzdDF/DA6PuCOi+Mh8m+JnE0c+Vw3CZB0trGQXvgTIoKpC/gXzSTYHtJ47sCfCqmnINxt6AlVZqd2JnQ4928AOSX5Ecz3o4VjoZI8F7BDZm0RRB3dIv4b4ZcBGY4peRWXyeurkE2eXTrg+zdgTLBkJGc/EImndiGia+uyIX6HfonXXERXOa51kfVgosziCLk+tfYP6CsfFpW+mcmQTig4dLuzoIVv6REv1SSTW9z15EaF+HIU6oqeDqAVA0aGd4zPE2IVPvKID6B7S2IQeltao11XFnrJjnNPKsbHRnWa1asg/rl3ZGae1spyHStVUVjagqSaeK495BdjubNQM1Q1lT47g3XjKqjbsT7sG52jePpZK1ye/ViaysQepPC5Hcb6f83zZRZ3oWEuOfZc4GpDqKHqE2gyUpo1M/e2rKvGxHgaersoJWe+sbTg1MyK6qyLlEJ+sc96TeOYPwmwhp5qIXna+CPFQySe7AXf8EajFcqeorLxsSKPRZKZY3sbIQnzkiFdc4gWIdQBfvpr8a5mZZbHvBs+bQeSMjRXSsugmcmO+58eiQr2gQlmVjyevNFNCvJ0ENN6ecG4TLvP2ocUjClf4aDI/Ll2DA6KMVcQ6mEvqMLXBRuczwdlUmSc5ZEXFUxxc1cVaOS/n5Y2kF8uak3zPy0p6sSxTrpGbtuoHEbDOS+RirANLuJksPS6R8rEnSLq0+e0S9nnYawcqL0Jl4VmJpIseI73yVVW+LrUvRhcMVUxY5vc1IniN/jfGG9CqcUxl45WimcrdrhXTVH64YmRTSXfF+KaS70okLewN9oTGMm9Ka4q5jkq5ZD6V7QkW7SbJ+4usNh3fbYoi79mm6kya2m84K+Tu2EAP0Jfqz7P7DldyvbSOB+i+bfShEJrNgto5uTO0dAUHS/gXId/T8+Uy9TVPbvm82cc6zOuOyUzZLuerk8CifVlyaDP727r7agBfW9CZ7NRXmZ1llJsvx7IyGTuLlMxKnAO8PTqoTYu6pYzXb8rypN6bJDMgWoBziBwV25FwTXuW0Nf1L8XNerkIdR1fvwUi83Sh+QY65G+u6YS2LmISiyyipK12BqziXOurs1yI05J8t07fr5NzcZR587IR2/TMPML5ZqgyNTnp7MAj2Z1+pl3+NZ0exqihH+jkUqfvV7n3uWVR+QZhoN4Esri7hLxH8TzAG9vquIE65+/kbgPyOg5JQ/I/VvcmKbcj5Ol2EdIqeucjy35qzpyypzXJc1SPuOTdwm0JYtqh5+MVr+98ZHkH6ONsm94w3AVy9vZkObw0ZxbfdZXdwC2+Fwro2aFvB109mlvZJ0LeXclzwf7MLRzn7y5VBefu6OnjKX15/Oma7uJeRvavhEeez2NIS9oy3SGATKzOQRFuU7L9InsHJPGmbYsyeeUqvIbqxUkv3bjarNSn75VnB6e71fqP1dqb55WXL9Sd8wPxRDwVzyhOe+IlVeYRxdUQf4q/xN/iny//29re+m5rR7Lev6dkvhC5v62f/gfTXpQA</latexit>

Counterfactuals Basics

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Counterfactuals
observation

Counterfactual: P (Y (t) | T = t0 , Y = y 0 )<latexit sha1_base64="4CMBJKtzM2gzxDZ61/z0DuX68ao=">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</latexit>

hypothetical condition

Different from CATE: E[Y (t) | X = x] = E[Y | do(t), X = x]<latexit sha1_base64="fLfT15enJ6H5GVklPn8BN+RWbrE=">AAAXfHicrVjbbttGEN2kt9TpxUmB+iEvbOQCaUurkpvWeREQwHZQFA3qAL6llmFIJCUR4q0kZdkR9Bf9mr62P9GfKTpzdimSupCSYAsWV7MzZy47M8vdduDYUVyr/Xvv/nvvf/DhRw8+3nj4yaeffb756PFp5A9CwzoxfMcPz9utyHJszzqJ7dixzoPQarltxzpr9/d5/uzaCiPb947j28C6dFtdz+7YRism0tVmtXl48fZZ/I3WdG1TO9ca2s0lfTFVkpoHPk3raupqs1Kr1vCnzQ7qalAR6u/If/TwiWgKU/jCEAPhCkt4IqaxI1oios+FqIuaCIh2KUZEC2lkY94SY7FBsgPisoijRdQ+fXfp14WievSbMSNIG6TFof+QJDXxteIxadwBVT5Zv5bhXaRjBGy28ZaebYXpEjUWPaKWySWcy8u16eOWeB2TDy/grU2eBKBwHIyczx16OvQ7Jg/5+5Y4LRqZJBXSyCCaQ1RJYR0hPWXkOTY9rEQLfBaN2Ooiu4us5nmfPn3CatE4gqVsqyZeqXXxoNmCrczjYE0XI96Qh9K+IqzOxAcb6y69YN5jovTFOxrlkYt0ZrNrMVeskMdzdDF/DA6PuCOi+Mh8m+JnE0c+Vw3CZB0trGQXvgTIoKpC/gXzSTYHtJ47sCfCqmnINxt6AlVZqd2JnQ4928AOSX5Ecz3o4VjoZI8F7BDZm0RRB3dIv4b4ZcBGY4peRWXyeurkE2eXTrg+zdgTLBkJGc/EImndiGia+uyIX6HfonXXERXOa51kfVgosziCLk+tfYP6CsfFpW+mcmQTig4dLuzoIVv6REv1SSTW9z15EaF+HIU6oqeDqAVA0aGd4zPE2IVPvKID6B7S2IQeltao11XFnrJjnNPKsbHRnWa1asg/rl3ZGae1spyHStVUVjagqSaeK495BdjubNQM1Q1lT47g3XjKqjbsT7sG52jePpZK1ye/ViaysQepPC5Hcb6f83zZRZ3oWEuOfZc4GpDqKHqE2gyUpo1M/e2rKvGxHgaersoJWe+sbTg1MyK6qyLlEJ+sc96TeOYPwmwhp5qIXna+CPFQySe7AXf8EajFcqeorLxsSKPRZKZY3sbIQnzkiFdc4gWIdQBfvpr8a5mZZbHvBs+bQeSMjRXSsugmcmO+58eiQr2gQlmVjyevNFNCvJ0ENN6ecG4TLvP2ocUjClf4aDI/Ll2DA6KMVcQ6mEvqMLXBRuczwdlUmSc5ZEXFUxxc1cVaOS/n5Y2kF8uak3zPy0p6sSxTrpGbtuoHEbDOS+RirANLuJksPS6R8rEnSLq0+e0S9nnYawcqL0Jl4VmJpIseI73yVVW+LrUvRhcMVUxY5vc1IniN/jfGG9CqcUxl45WimcrdrhXTVH64YmRTSXfF+KaS70okLewN9oTGMm9Ka4q5jkq5ZD6V7QkW7SbJ+4usNh3fbYoi79mm6kya2m84K+Tu2EAP0Jfqz7P7DldyvbSOB+i+bfShEJrNgto5uTO0dAUHS/gXId/T8+Uy9TVPbvm82cc6zOuOyUzZLuerk8CifVlyaDP727r7agBfW9CZ7NRXmZ1llJsvx7IyGTuLlMxKnAO8PTqoTYu6pYzXb8rypN6bJDMgWoBziBwV25FwTXuW0Nf1L8XNerkIdR1fvwUi83Sh+QY65G+u6YS2LmISiyyipK12BqziXOurs1yI05J8t07fr5NzcZR587IR2/TMPML5ZqgyNTnp7MAj2Z1+pl3+NZ0exqihH+jkUqfvV7n3uWVR+QZhoN4Esri7hLxH8TzAG9vquIE65+/kbgPyOg5JQ/I/VvcmKbcj5Ol2EdIqeucjy35qzpyypzXJc1SPuOTdwm0JYtqh5+MVr+98ZHkH6ONsm94w3AVy9vZkObw0ZxbfdZXdwC2+Fwro2aFvB109mlvZJ0LeXclzwf7MLRzn7y5VBefu6OnjKX15/Oma7uJeRvavhEeez2NIS9oy3SGATKzOQRFuU7L9InsHJPGmbYsyeeUqvIbqxUkv3bjarNSn75VnB6e71fqP1dqb55WXL9Sd8wPxRDwVzyhOe+IlVeYRxdUQf4q/xN/iny//29re+m5rR7Lev6dkvhC5v62f/gfTXpQA</latexit><latexit sha1_base64="fLfT15enJ6H5GVklPn8BN+RWbrE=">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</latexit>

Cannot express counterfactuals using do-notation

Brady NealCounterfactuals Basics

6 / 25

Roadmap for Computing Counterfactuals

Brady NealCounterfactuals Basics

7 / 25

Roadmap for Computing Counterfactuals
Given: Observation of (T, Y) (observation of potential outcome Y(t)where t is the observed value of T)

Brady NealCounterfactuals Basics

7 / 25

Roadmap for Computing Counterfactuals
Given: Observation of (T, Y) (observation of potential outcome Y(t)where t is the observed value of T)
Main ingredient necessary: correct parametric model for the structural
equation for Y

Brady NealCounterfactuals Basics

7 / 25

Roadmap for Computing Counterfactuals
Given: Observation of (T, Y) (observation of potential outcome Y(t)where t is the observed value of T)
Main ingredient necessary: correct parametric model for the structural
equation for Y
Result: access to counterfactuals Y(t’) at the unit-level

Brady NealCounterfactuals Basics

7 / 25

Computing Counterfactuals: Simple Example

Brady NealCounterfactuals Basics

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)

Brady NealCounterfactuals Basics

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Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)

Brady NealCounterfactuals Basics

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)

Brady NealCounterfactuals Basics

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

T := . . .
Y := U T + (1Brady Neal

U )(1

T)

Counterfactuals Basics

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Y := U T + (1Brady Neal

Observation: T = 0 and Y = 0

T := . . .
U )(1

T)

Counterfactuals Basics

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Y := U T + (1Brady Neal

Observation: T = 0 and Y = 0

T := . . .
U )(1<latexit sha1_base64="ctlkolEpWDuIut0+ty5E2I5TOAA=">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</latexit>(Yu (0) = 0)

T)Counterfactuals Basics

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Y := U T + (1Brady Neal

Observation: T = 0 and Y = 0

T := . . .
U )(1

T)

Counterfactuals Basics<latexit sha1_base64="ctlkolEpWDuIut0+ty5E2I5TOAA=">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</latexit>(Yu (0) = 0)<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Yu (1)?

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">AAAXd3icrVjbbttGEN2kt9TpxUmB+iEPZeM0sFtZldy0LgoICGAnKIoGdQDJdhoFhkRSEiHeSlJWHELf0K/pa/sd/ZS+debsUiR1ISUhFiyuZmfOXHZmlrtd37bCqFb798bNd9597/0Pbn24dfujjz/5dPvO3bPQGwW62dI92wsuup3QtC3XbEVWZJsXfmB2nK5tnneHxzx/fmUGoeW5zejaN185nb5r9Sy9ExHpcnu/qT38qaG1bcOLQq3d3nqB362m9o22V9cOtNY+Hs39y+3dWrWGP21+UFeDXaH+Tr07t++JtjCEJ3QxEo4whSsiGtuiI0L6vBR1URM+0V6JmGgBjSzMm2Iitkh2RFwmcXSIOqTvPv16qagu/WbMENI6abHpPyBJTXyleAwa90CVT9avZXiX6YiBzTZe07OrMB2iRmJA1DK5hHN1uS59nBKvI/LhR3hrkSc+KBwHPedzj542/Y7IQ/6+Jk6TRgZJBTTSiWYTVVJYR0BPGXmOzQAr0QGfSSO2usjuIqt53qPPkLA6NA5hKduqiadqXVxoNmEr89hY0+WIr8lDaV8RVm/qg4V1l14wb5MoQ/GGRnnkIp3Z7FrOFSnkyQJdzB+BwyXukCgeMt+i+FnEkc9VnTBZRwcr2YcvPjKoqpB/wXySzT6t5wHsCbFqGvLNgh5fVVZqd2KnTc8usAOSj2luAD0ciwrZYwI7QPYmUayAO6BfY/zSYaM+Q6+iMnk9K+QTZ1eFcD2asaZYMhIynolF0rqYaJr6HIhfod+kda8gKpzXFZL1YKHM4hC6XLX2DeorHBeHvpnKkU0oFehwYMcA2TIkWqpPIrG+b8mLEPVjK9SYnjai5gOlAu0cnzHGDnziFR1B95jGBvSwtEa9riqOlB2TnFaOjYXuNK9VQ/5x7crOOKuV5VxUqqaysgFNNfFIecwrwHZno6arbih7cgjvJjNWdWF/2jU4R/P2sVS6Pvm1MpCNA0jlcTmKi/1c5Msh6qSCteTY94mjAameooeoTV9p2srU37GqEg/roePpqJyQ9c7axjMzMdEdFSmb+GSd857EM38QZgc51Ub0svNFiE+UfLIbcMePQS2WO0Nl5WUDGsXTmWJ5CyMT8ZEjXnGJ5yPWPnz5cvqvZWZWxX47eO4cImdspJBWRTeQG4s9b4pd6gW7lFX5ePJKMyXA24lP4wdTzgeEy7xDaHGJwhUeT+cnpWtwQpSJilgPc0kdpjZY6HwGONsq8ySHrKhohoOrulgr5+WivJH0Ylljmu95WUkvlmXKFXLTUv0gBNZFiVyEdWAJJ5OlzRIpD3uCpEubX6xgn4u9dqTyIlAWnpdIOugx0itPVeWzUvsidMFAxYRlft8gglfofxO8Aa0bx1Q2Wiuaqdz1RjFN5cdrRjaVdNaMbyr5pkTSxN5gTWks87y0ppjrtJRL5lPZnmDSbpK8v8hqq+C7S1HkPdtQnUlT+w1nhdwdG+gBlZX68/y+w5VcL63jEbpvF30ogGajoHZabw0tXcHRCv6FyPf0fLlKfS2SWz1vjrEOi7pjMlO2y3nqJLBsX5Yc2tz+tum+6sPXDnQmO/VlZmeJc/PlWGYmY+eRklmJc4K3Rxu1aVK3lPH6TVme1HubZEZE83EOkaNiOxKuWc8S+qb+pbhZL5ehbuLr10Bknj40v4YO+ZtrOqFtipjEIosoaeudAas413rqLBfgtCTfrdP36+RcHGbevCzENj0zxzjfjFWmJiedA3gku9PPtMs/o9PDBDX0HZ1c6vT9NPc+tyoq3yCM1JtAFveQkI8onid4Y1sf11fn/IPcbUBexxPSkPxP1L1Jym0LebpdhrSO3sXIsp8ac6fsWU3yHDUgLnm3cF2CmHboxXjF67sYWd4BejjbpjcMbwM5e3uyGl6aM8vvuspu4JbfC/n07NG3ja4eLqzslpB3V/JccDx3C8f5e0hVwbkb3787oy+PP1vTfdzLyP6V8MjzeQRpSVulO/iQidQ5KMRtSrZfZO+AJN6sbWEmrxyF11C9OOmlW5fbu/XZe+X5wdlhtf59tfb80e7jH9Sd8y1xT9wXexSnI/GYKvOU4qqLP8Vf4m/xz+f/7Xyx83BnT7LevKFkPhO5v536/7aokCY=</latexit>

Y := U T + (1Brady Neal

Observation: T = 0 and Y = 0

T := . . .
U )(1

T)

Counterfactuals Basics<latexit sha1_base64="ctlkolEpWDuIut0+ty5E2I5TOAA=">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</latexit>(Yu (0) = 0)<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Yu (1)?

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Observation: T = 0 and Y = 0

T := . . .
Y := U T + (1

U )(1

T)

(Yu (0) = 0)<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Step 1: Solve for U

Brady Neal<latexit sha1_base64="ctlkolEpWDuIut0+ty5E2I5TOAA=">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</latexit>

Counterfactuals Basics

Yu (1)?

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Observation: T = 0 and Y = 0

T := . . .
Y := U T + (1

U )(1

T)

<latexit sha1_base64="ctlkolEpWDuIut0+ty5E2I5TOAA=">AAAXWHicrVjbbttGEN2klyROL44L1A95YeMUcApZld207ouBAHaComhQB/AlaRwYEkXJhEmRJSk7jqDf6Gv7W+3X9MzZpUjqQkqGLVhczc6cuezMLHdboefGSaPx763bH338yad37t5buv/Z5198ufxg5SgO+pHtHNqBF0SvW83Y8dyec5i4iee8DiOn6bc857h1vivzxxdOFLtB7yC5Cp13frPbczuu3UxAOll/c9pfbzyxdqzGk9PltUa9wT9rcrBpBmvK/O0HD+4/VCeqrQJlq77ylaN6KsHYU00V4/NWbaqGCkF7pwagRRi5nHfUUC1Btg8uBxxNUM/x3cWvt4baw2/BjCltQ4uH/wiSlvrW8LQx7pCqn6LfyvHO0jEgtth4hWfLYPqgJuoM1Cq5lHN+uRY+foXXCXz4md668CQkReJgF3zu4OnhdwIP5fsKnA5GbUhFGNmgeaBqiuiI8NSRl9iccSWa5HMwEqvL7C6zWuYDfM6B1cQ4pqViq6VemHXpUbNDW4XH45rORnwPD7V9ZVidkQ8u1117IbwHoJyrDxgVkct05rNrNldikIdTdAl/Qo4euGNQAma+i/i54Cjmqg1M0dHkSnbpS8gMqhvkXzmfZnOI9dygPTFXzWK+udQTmsrK7E7t9PBsETuC/ABzZ9QjsajBHofYEbM3jWKN3BF+XfKXTRvtMXqdlSnrWYNPkl014AaYcUdYOhI6nqlF2roBaJb5bKjfqN/ButcYFcnrGmQDWqizOKaunln7HfQViYuPb6FKZFNKjTp82nHGbDkHLdOnkUTf9/AiZv14BnWAp8eohUSpUbvE55Jjnz7Jivap+xLjNvWItIVeV1fbxo5hQavExmV3mtRqMf+kdnVnHNcqcj1WqmWycoeaGuqp8VhWQOzOR8023VD35JjeDcesatH+rGtIjhbtE6lsfYpr1WY2nlGqiCtRnO7nNF+2WCc1rqXEvguOHUp1DD1mbYZG01Ku/nZNlQRcD5tP3+SErnfRdjk2MwDdN5HywKfrXPYkmfkTmE3m1Amjl58vQ3xu5NPdQDr+gNRyuSNWVlE2wmgwmimXdzlyGB89khXXeCFjHdKXb0b/Vm5mXuybwetNIErGJgZpXvQ2c2O65wdqDb1gDVlVjKestFAivp2EGD8ecT4GrvCeU0sPFKnwwWh+WLkGe6AMTcQ6nEvrMLPBZedrk/PEZJ7m0BWVjHFIVZdrlbycljeaXi7bHuV7UVbTy2WFcsHcdE0/iIn1ukIu4TqIhJ/L0oMKqYB7gqZrm9/MYV+Pe23f5EVkLDyukPTZY7RXganKl5X2JeyCkYmJyPxxjQhesP8N+Qa0aBwz2WShaGZyV9eKaSZ/uWBkM0l/wfhmkh8qJB3uDe6IJjKvKmtKuPYruXQ+Ve0JDnaT9P1FV1uN3y1EUfbstulMltlvJCv07rjDHlCbqz9P7jtSyZuVddxn922xD0XU3C6pncMbQ8tWsD+HfzHzPTtfzlNf0+Tmz5tdrsO07pjOVO1ygTkJzNqXNYc1sb9dd18N6WuTOtOd+jS3swwK89VYTi5jJ5HSWY2zx7dHj7XpoFvqeP1uLE/r/QQyfdBCnkP0qNyOlGvcs5R+Xf8y3LyXs1Cv4+t3RBSeLjW/pw79W2o6pV0XMY1FHlHTFjsD1nmuDcxZLuJpSb9bZ+/X6bk4zr15uYxtdmYe8HxzaTI1Pels0CPdnX7BLv8Sp4cha+gHnFw28f2i8D43L6rcIPTNm0AedwvI24jnHt/YFscNzTl/o3AbUNTxHBrS/6G5N8m4PaVPt7OQFtE7HVn30/bEKXtckz5HnYFL3y1cVSBmHXo6Xvn6TkfWd4ABz7bZDcNNIOdvT+bDy3Jm9l1X1Q3c7HuhEM8Ovj129XhqZR8qfXelzwW7E7dwkr9bqArJ3cGjlTF9Rfzxmu7yXkb3r5RHn88TSmvaPN0hpExizkExb1Py/SJ/B6Txxm2Lc3nlG7wd04vTXrp0ury2OX6vPDk42qpv/lhvvHq69uwnc+d8Vz1Uj9Q64rStnqEy9xFXG1r+Un+rf77+b1Wt3lm9p1lv3zIyX6nC3+rK/yFUiBk=</latexit>(Yu (0) = 0)<latexit sha1_base64="byS3DFAjfaS2lB04sVJrbVA8Ro8=">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</latexit>

Y = U T + (1

U )(1

T)

0 = U (0) + (1

U )(1

0)

0=1

<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Yu (1)?

U

U =1

Brady NealCounterfactuals Basics

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Y := U T + (1Brady Neal

Observation: T = 0 and Y = 0

T := . . .
U )(1

T)

<latexit 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(0) = 0)<latexit sha1_base64="byS3DFAjfaS2lB04sVJrbVA8Ro8=">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</latexit><latexit sha1_base64="byS3DFAjfaS2lB04sVJrbVA8Ro8=">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</latexit>

Y = U T + (1 U )(1 T )0 = U (0) + (1 U )(1 0)
0=1 U
U =1

Counterfactuals Basics<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Yu (1)?

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Y := U T + (1Brady Neal

Observation: T = 0 and Y = 0

T := . . .
U )(1

T)

<latexit 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(0) = 0)<latexit 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Y = U T + (1 U )(1 T )0 = U (0) + (1 U )(1 0)
0=1 U
U =1

Counterfactuals Basics<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Yu (1)?

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Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Y := U T + (1Brady Neal

Observation: T = 0 and Y = 0

T := . . .
U )(1

T)

<latexit 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(0) = 0)<latexit 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Y = U T + (1 U )(1 T )0 = U (0) + (1 U )(1 0)
0=1 U
U =1

Counterfactuals Basics<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Yu (1)?

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Observation: T = 0 and Y = 0

T := . . .
Y := U T + (1U )(1Step 2: Individualized SCM

Brady Neal

T)<latexit 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(0) = 0)<latexit 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sha1_base64="byS3DFAjfaS2lB04sVJrbVA8Ro8=">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</latexit>

Y = U T + (1 U )(1 T )0 = U (0) + (1 U )(1 0)
0=1 U
U =1

Counterfactuals Basics<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">AAAXU3icrVhbbxtVED4tBUpKIS1S89CXpS5SQY6xQyG8RFRKWiFERSrlVpoqstdre+W9sbuOm1r+EbzC7+KB38ILM9+c9e76smtbiRXv8ZyZby5nZs6e0wocO4rr9X9v3Pzg1ocffXz7k407n9797PPNe/dPIn8Qmtax6Tt+eNZqRpZje9ZxbMeOdRaEVtNtOdZpq7/P86eXVhjZvncUXwXWW7fZ9eyObTZjIp2+vhg8aXz908VmpV6r48+YHTT0oKL036F/785Dda7aylemGihXWcpTMY0d1VQRfd6ohqqrgGhv1YhoIY1szFtqrDZIdkBcFnE0idqn7y79eqOpHv1mzAjSJmlx6D8kSUN9pXnaNO6AKk/Wb2R4F+kYAZttvKJnS2O6RI1Vj6hlcgnn8nIt+rglXsfkw4/w1iZPAlA4DmbO5w49Hfodk4f8fUWcFo3aJBXSyCSaQ1ShsI6QnhJ5jk0PK9EEn0UjtrrI7iKred6nT5+wmjSOYCnbaqgXel08aLZgK/M4WNPFiO/IQ7GvCKsz8cHGuosXzHtElL56T6M8cpHObHYt5oo18niOLuaPweERd0QUH5lvU/xs4sjnqkmYrKOJlezClwAZVNPIv2A+yeaA1nMb9kRYNQP5ZkNPoCsrtTux06FnC9ghyY9orgc9HIsq2WMBO0T2JlGsgjukX0P8MmGjOUWvoTJ5PavkE2dXlXB9mrEnWBIJiWdikVg3IpqhP9vqV+i3aN2riArndZVkfVgoWRxBl6fXfo/6CsfFpW+mcmQTShU6XNjRQ7b0iZbqEyTW9y15EaF+HI06oqeDqAVAqUI7x2eIsQufeEUH0D2kcRt6WNqgXldTu9qOcU4rx8ZGd5rVaiD/uHalM05rZTkPlWrorNyDprp6qj3mFWC7s1EzdTeUnhzBu/GUVS3Yn3YNztG8fSyVrk9+rdrIxh6k8rgcxfl+zvNlB3VSxVpy7LvEsQepjqZHqM1Aa9rI1N++rhIf62Hi6eqckHpnbcOpmRHRXR0ph/ikznlP4pk/CLOJnDpH9LLzRYjPtXyyG3DHH4FaLHeCysrLhjQaTWaK5W2MLMRHRrzighcg1gF8+XLyb2RmlsW+HjxvBpEzNtZIy6K3kRvzPT9SFeoFFcqqfDx5pZkS4u0koPHjCedjwmXePrR4ROEKH03mx6VrcECUsY5YB3NJHaY22Oh8bXCe68wTDqmoeIqDq7pYK+flvLwRerFse5LveVmhF8sy5RK5aet+EAHrrEQuxjqwhJvJ0qMSKR97gtDF5tdL2Odhrx3ovAi1haclki56jHjl66p8WWpfjC4Y6piwzO9rRPAS/W+MN6BV45jKxitFM5W7Wiumqfxwxcimku6K8U0l35dIWtgb7AmNZV6V1hRzHZZyST6V7QkW7SbJ+4tUWxXfLYoi79lt3ZkMvd9wVsjuuIceUF2qP8/uO1zJjdI6HqD7ttCHQmhuF9TO8bWhpSs4WMK/CPmeni+Xqa95csvnzT7WYV53TGbKdjlfnwQW7cvCYczsb+vuqwF8bUJnslNfZHaWUW6+HMvKZOwsUjIrOAd4e3RQmxZ1S4nXb9rypN7PSWZAtADnEBkV25FwTXuW0Nf1L8XNerkIdR1fvwEi83Sh+R10yG+u6YS2LmISiyyi0FY7A9ZwrvX1WS7EaUnerdP36+RcHGXevGzENj0zj3C+GepMTU462/BIutPPtMu/pNPDGDX0HZ1cGvT9Ivc+tywq3yAM9JtAFneHkHcpngd4Y1sdN9Dn/O3cbUBex3PSkPyP9b1Jyu0oOd0uQlpF73xk6aftmVP2tCY5R/WIS+4WrkoQ0w49H694fecjyx2gj7NtesNwHcjZ25Pl8NKcWXzXVXYDt/heKKBnh74ddPVobmUfK7m7knPB/swtHOfvDlUF5+7o0f0pfXn86Zru4l5G+lfCI+fzGNJCW6Y7BJCJ9Tkowm1Ktl9k74AEb9q2KJNXrsbb07046aUbF5uVxvS98uzgZKfW+L5Wf/W08uwHfed8Wz1Uj9QTitOuekaVeUhx5ZX4U/2l/n7wz4P/tm5u3RLWmze0zBcq97d1938WYIgo</latexit>

Yu (1)?

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Observation: T = 0 and Y = 0

T := . . .
Y := U T + (1

U )(1

T)

Step 2: Individualized SCM<latexit sha1_base64="NL0t4UNq9WfyHsPdvKMI3uQ19os=">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</latexit><latexit sha1_base64="NL0t4UNq9WfyHsPdvKMI3uQ19os=">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</latexit>

Brady Neal

T := . . .
Y := (1)T + (1

1)(1

T)

<latexit 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(0) = 0)<latexit 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Y = U T + (1 U )(1 T )0 = U (0) + (1 U )(1 0)
0=1 U
U =1

Counterfactuals Basics<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Yu (1)?

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Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Observation: T = 0 and Y = 0

T := . . .
Y := U T + (1

U )(1

T)

Step 2: Individualized SCM
T := . . .
TY := (1)T+ (1<latexit sha1_base64="kTHtR9LRMs91te/sq+RjhAmwoeM=">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</latexit><latexit sha1_base64="NL0t4UNq9WfyHsPdvKMI3uQ19os=">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</latexit>

Brady Neal

1)(1

T)

<latexit 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(0) = 0)<latexit 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Y = U T + (1 U )(1 T )0 = U (0) + (1 U )(1 0)
0=1 U
U =1

Counterfactuals Basics<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Yu (1)?

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Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Observation: T = 0 and Y = 0

T := . . .
Y := U T + (1U )(1Step 2: Individualized SCM
T := 1. . .
Y := T<latexit sha1_base64="+lM8jzLsVBYP1wCv3+XWf8a7ZEc=">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</latexit><latexit sha1_base64="kTHtR9LRMs91te/sq+RjhAmwoeM=">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</latexit>

Brady Neal

T)<latexit 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(0) = 0)<latexit 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Y = U T + (1 U )(1 T )0 = U (0) + (1 U )(1 0)
0=1 U
U =1

Counterfactuals Basics<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Yu (1)?

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Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Observation: T = 0 and Y = 0

T := . . .
Y := U T + (1U )(1Step 2: Individualized SCM
T := 1. . .
Y := T<latexit sha1_base64="+lM8jzLsVBYP1wCv3+XWf8a7ZEc=">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</latexit><latexit sha1_base64="kTHtR9LRMs91te/sq+RjhAmwoeM=">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</latexit>

T)<latexit sha1_base64="ctlkolEpWDuIut0+ty5E2I5TOAA=">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</latexit>(Yu (0) = 0)<latexit 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sha1_base64="byS3DFAjfaS2lB04sVJrbVA8Ro8=">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</latexit>

Y = U T + (1 U )(1 T )0 = U (0) + (1 U )(1 0)
0=1 U
U =1

<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">AAAXU3icrVhbbxtVED4tBUpKIS1S89CXpS5SQY6xQyG8RFRKWiFERSrlVpoqstdre+W9sbuOm1r+EbzC7+KB38ILM9+c9e76smtbiRXv8ZyZby5nZs6e0wocO4rr9X9v3Pzg1ocffXz7k407n9797PPNe/dPIn8Qmtax6Tt+eNZqRpZje9ZxbMeOdRaEVtNtOdZpq7/P86eXVhjZvncUXwXWW7fZ9eyObTZjIp2+vhg8aXz908VmpV6r48+YHTT0oKL036F/785Dda7aylemGihXWcpTMY0d1VQRfd6ohqqrgGhv1YhoIY1szFtqrDZIdkBcFnE0idqn7y79eqOpHv1mzAjSJmlx6D8kSUN9pXnaNO6AKk/Wb2R4F+kYAZttvKJnS2O6RI1Vj6hlcgnn8nIt+rglXsfkw4/w1iZPAlA4DmbO5w49Hfodk4f8fUWcFo3aJBXSyCSaQ1ShsI6QnhJ5jk0PK9EEn0UjtrrI7iKred6nT5+wmjSOYCnbaqgXel08aLZgK/M4WNPFiO/IQ7GvCKsz8cHGuosXzHtElL56T6M8cpHObHYt5oo18niOLuaPweERd0QUH5lvU/xs4sjnqkmYrKOJlezClwAZVNPIv2A+yeaA1nMb9kRYNQP5ZkNPoCsrtTux06FnC9ghyY9orgc9HIsq2WMBO0T2JlGsgjukX0P8MmGjOUWvoTJ5PavkE2dXlXB9mrEnWBIJiWdikVg3IpqhP9vqV+i3aN2riArndZVkfVgoWRxBl6fXfo/6CsfFpW+mcmQTShU6XNjRQ7b0iZbqEyTW9y15EaF+HI06oqeDqAVAqUI7x2eIsQufeEUH0D2kcRt6WNqgXldTu9qOcU4rx8ZGd5rVaiD/uHalM05rZTkPlWrorNyDprp6qj3mFWC7s1EzdTeUnhzBu/GUVS3Yn3YNztG8fSyVrk9+rdrIxh6k8rgcxfl+zvNlB3VSxVpy7LvEsQepjqZHqM1Aa9rI1N++rhIf62Hi6eqckHpnbcOpmRHRXR0ph/ikznlP4pk/CLOJnDpH9LLzRYjPtXyyG3DHH4FaLHeCysrLhjQaTWaK5W2MLMRHRrzighcg1gF8+XLyb2RmlsW+HjxvBpEzNtZIy6K3kRvzPT9SFeoFFcqqfDx5pZkS4u0koPHjCedjwmXePrR4ROEKH03mx6VrcECUsY5YB3NJHaY22Oh8bXCe68wTDqmoeIqDq7pYK+flvLwRerFse5LveVmhF8sy5RK5aet+EAHrrEQuxjqwhJvJ0qMSKR97gtDF5tdL2Odhrx3ovAi1haclki56jHjl66p8WWpfjC4Y6piwzO9rRPAS/W+MN6BV45jKxitFM5W7Wiumqfxwxcimku6K8U0l35dIWtgb7AmNZV6V1hRzHZZyST6V7QkW7SbJ+4tUWxXfLYoi79lt3ZkMvd9wVsjuuIceUF2qP8/uO1zJjdI6HqD7ttCHQmhuF9TO8bWhpSs4WMK/CPmeni+Xqa95csvnzT7WYV53TGbKdjlfnwQW7cvCYczsb+vuqwF8bUJnslNfZHaWUW6+HMvKZOwsUjIrOAd4e3RQmxZ1S4nXb9rypN7PSWZAtADnEBkV25FwTXuW0Nf1L8XNerkIdR1fvwEi83Sh+R10yG+u6YS2LmISiyyi0FY7A9ZwrvX1WS7EaUnerdP36+RcHGXevGzENj0zj3C+GepMTU462/BIutPPtMu/pNPDGDX0HZ1cGvT9Ivc+tywq3yAM9JtAFneHkHcpngd4Y1sdN9Dn/O3cbUBex3PSkPyP9b1Jyu0oOd0uQlpF73xk6aftmVP2tCY5R/WIS+4WrkoQ0w49H694fecjyx2gj7NtesNwHcjZ25Pl8NKcWXzXVXYDt/heKKBnh74ddPVobmUfK7m7knPB/swtHOfvDlUF5+7o0f0pfXn86Zru4l5G+lfCI+fzGNJCW6Y7BJCJ9Tkowm1Ktl9k74AEb9q2KJNXrsbb07046aUbF5uVxvS98uzgZKfW+L5Wf/W08uwHfed8Wz1Uj9QTitOuekaVeUhx5ZX4U/2l/n7wz4P/tm5u3RLWmze0zBcq97d1938WYIgo</latexit>

Yu (1)?

<latexit sha1_base64="rExaXPO3SNMVs6ZP3NJvKxwpZXY=">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</latexit>

Yu (1) = 1
Brady NealCounterfactuals Basics

8 / 25

Computing Counterfactuals: Simple Example
Y: happy or unhappy (1 or 0)
T: get a dog or don’t (1 or 0)
U: unobserved variable describing the individual (1 if dog person; 0 if anti-dog person)
SCM:<latexit sha1_base64="sP+W5Ab4hhTmllKufSmtQwKPrZo=">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</latexit>

Observation: T = 0 and Y = 0

T := . . .
Y := U T + (1U )(1Step 2: Individualized SCM
T := 1. . .
Y := T<latexit sha1_base64="+lM8jzLsVBYP1wCv3+XWf8a7ZEc=">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</latexit><latexit sha1_base64="kTHtR9LRMs91te/sq+RjhAmwoeM=">AAAXZXicrVjrjttEFJ6WW0kpbAvq/ugPTFMqhLIhWQqLkCJV2m2FEBVbKXspTbVKbCex4hu2s+nWytPwFx6IJ+A1OOebcWznYifRbrTx5Mw537nMOWc80/NtK4wajX9v3Hzv/Q8+/OjWx5Xbn9z59LOdu/dOQ28c6OaJ7tlecN7rhqZtueZJZEW2ee4HZtfp2eZZb3TI82eXZhBantuOrnzzjdMduFbf0rsRkS52dtva459bWsc2vCjUOp3KK/xuX+xUG/UG/rTFQVMNqkL9HXt3bz8QHWEIT+hiLBxhCldENLZFV4T0eS2aoiF8or0RMdECGlmYN8VUVEh2TFwmcXSJOqLvAf16ragu/WbMENI6abHpPyBJTXyteAwa90GVT9avZXhX6YiBzTZe0bOnMB2iRmJI1DK5hHN9uR59nBKvI/LhJ3hrkSc+KBwHPedzn542/Y7IQ/6+Ik6TRgZJBTTSiWYTVVJYR0BPGXmOzRAr0QWfSSO2usjuIqt53qPPiLC6NA5hKduqiedqXVxoNmEr89hY09WIb8lDaV8RVn/mg4V1l14wb5soI/GORnnkIp3Z7FrNFSnk6RJdzB+BwyXukCgeMt+i+FnEkc9VnTBZRxcrOYAvPjKorpB/xXySzT6t5x7sCbFqGvLNgh5fVVZqd2KnTc8esAOSj2luCD0cixrZYwI7QPYmUayBO6BfE/zSYaM+R6+jMnk9a+QTZ1eNcD2asWZYMhIynolF0rqYaJr67InfoN+kda8hKpzXNZL1YKHM4hC6XLX2LeorHBeHvpnKkU0oNehwYMcQ2TIiWqpPIrG+78iLEPVjK9SYnjai5gOlBu0cnwnGDnziFR1D94TGBvSwtEa9ri4OlB3TnFaOjYXutKhVQ/5x7crOOK+V5VxUqqaysgVNDfFEecwrwHZno6arbih7cgjvpnNW9WB/2jU4R/P2sVS6Pvm1MpCNQ0jlcTmKy/1c5ss+6qSGteTYD4ijBam+ooeoTV9pqmTq71BViYf10PF0VE7Iemdtk7mZmOiOipRNfLLOeU/imT8Js4uc6iB62fkixGdKPtkNuOPHoBbLnaKy8rIBjeLZTLG8hZGJ+MgRr7jE8xFrH758NfvXMjPrYl8PnruAyBkbKaR10Q3kxnLP26JKvaBKWZWPJ680UwK8nfg0fjTjfES4zDuCFpcoXOHxbH5augZHRJmqiPUxl9RhaoOFzmeAs6MyT3LIiormOLiqi7VyXi7LG0kvljVm+Z6XlfRiWaZcIjct1Q9CYJ2XyEVYB5ZwMlnaLpHysCdIurT51Rr2udhrxyovAmXhWYmkgx4jvfJUVb4otS9CFwxUTFjmjy0ieIn+N8Ub0KZxTGWjjaKZyl1tFdNUfrJhZFNJZ8P4ppLvSiRN7A3WjMYyL0trirmOS7lkPpXtCSbtJsn7i6y2Gr57FEXesw3VmTS133BWyN2xhR5QW6s/L+47XMnN0joeo/v20IcCaDYKaufk2tDSFRyv4V+IfE/Pl+vU1zK59fPmEOuwrDsmM2W7nKdOAqv2ZcmhLexv2+6rPnztQmeyU19kdpY4N1+OZWYydhEpmZU4R3h7tFGbJnVLGa/fleVJvXdIZkw0H+cQOSq2I+Ga9yyhb+tfipv1chXqNr5+C0TmGUDzW+iQv7mmE9q2iEkssoiSttkZsI5zrafOcgFOS/LdOn2/Ts7FYebNy0Js0zNzjPPNRGVqctLZg0eyO/1Cu/wLOj1MUUPf08mlSd/Pc+9z66LyDcJYvQlkcfcJ+YDieYQ3ts1xfXXO38vdBuR1PCMNyf9U3Zuk3LaQp9tVSJvoXY4s+6mxcMqe1yTPUUPikncLVyWIaYdejle8vsuR5R2gh7NtesNwHcjZ25P18NKcWX3XVXYDt/peyKdnn75tdPVwaWWfCHl3Jc8Fhwu3cJy/+1QVnLvxw3tz+vL48zU9wL2M7F8JjzyfR5CWtHW6gw+ZSJ2DQtymZPtF9g5I4s3bFmbyylF4LdWLk15audipNufvlRcHp/v15g/1xssn1ac/qjvnW+KBeCi+oTgdiKdUmccUV16bv8Tf4p/7/+3e2f1i975kvXlDyXwucn+7X/4P8smMKw==</latexit>

T)<latexit 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(0) = 0)<latexit 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Y = U T + (1 U )(1 T )0 = U (0) + (1 U )(1 0)
0=1 U
U =1

ITE: Yu (1)<latexit sha1_base64="EoDGqPLKLd9OzAQhAuCi2ak7+hM=">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</latexit>

Yu (1)?

<latexit sha1_base64="MJ67+mtdvPDB4QMZCUmz+AqCqZ0=">AAAXanicrVjbbttGEN2kt9TpxXGBGoX7wMYpkBSyKrlp3RcDAZwERdGgCWDn6sCQKMoiRIosSdlxBL30a/ra/k3/oR/RM2eXIqkLKRmxYHE1O3PmsjOz3G2Hnhsnjca/V66+9/4HH3507eO16598+tnn6zc2nsbBMLKdIzvwguh5uxU7njtwjhI38ZznYeS0/LbnPGv3D2T+2ZkTxW4wOEwuQue13zoduF3XbiUgnax//eJkeLt5x9qxZNC4Y+1bTfxo8Hmyvt2oN/hnzQ6aZrCtzN/j4Mb1LXWsOipQthoqXzlqoBKMPdVSMT6vVFM1VAjaazUCLcLI5byjxmoNskNwOeBogdrH9yl+vTLUAX4LZkxpG1o8/EeQtNS3hqeDcZdU/RT9Vo53kY4RscXGCzzbBtMHNVE9UKvkUs7l5dr4+BVeJ/DhZ3rrwpOQFImDXfC5i6eH3wk8lO8LcDoYdSAVYWSD5oGqKaIjwlNHXmLT40q0yOdgJFaX2V1mtcwH+PSB1cI4pqViq6UemnUZULNDW4XH45ouRnwDD7V9ZVjdiQ8u1117IbyHoPTVW4yKyGU689m1mCsxyOM5uoQ/IccA3DEoATPfRfxccBRz1Qam6GhxJU/pS8gMqhvkXzmfZnOI9dyhPTFXzWK+udQTmsrK7E7t9PBsEzuC/AhzPeqRWNRgj0PsiNmbRrFG7gi/zvnLpo32FL3OypT1rMEnya4acAPMuBMsHQkdz9Qibd0INMt8dtRv1O9g3WuMiuR1DbIBLdRZHFPXwKz9PvqKxMXHt1AlsimlRh0+7egxW/qgZfo0kuj7Hl7ErB/PoI7w9Bi1kCg1apf4nHPs0ydZ0SF1n2PcoR6RttDr6mrP2DEuaJXYuOxOs1ot5p/Uru6M01pFbsBKtUxW7lNTQ901HssKiN35qNmmG+qeHNO78ZRVbdqfdQ3J0aJ9IpWtT3GtOszGHqWKuBLF+X7O82WXdVLjWkrsT8GxT6muoceszdBoWsvV34GpkoDrYfPpm5zQ9S7azqdmRqD7JlIe+HSdy54kM38As8WcOmb08vNliA+MfLobSMcfkVou95SVVZSNMBpNZsrlXY4cxkePZMU1XshYh/Tlm8m/lZtZFvvd4A1mECVjE4O0LHqHuTHf80O1jV6wjawqxlNWWigR305CjG9NOG8BV3j71DIARSp8NJkfV67BfVDGJmJdzqV1mNngsvN1yHlsMk9z6IpKpjikqsu1Sl7OyxtNL5ftTPK9KKvp5bJCOWNuuqYfxMR6XiGXcB1Ews9l6WGFVMA9QdO1zS+WsG/AvXZo8iIyFj6rkPTZY7RXganKR5X2JeyCkYmJyLy8RATP2P/GfANaNY6ZbLJSNDO5i0vFNJM/XzGymaS/YnwzybcVkg73BndCE5knlTUlXI8ruXQ+Ve0JDnaT9P1FV1uN321EUfbsjulMltlvJCv07rjPHlBbqj/P7jtSyc3KOh6y+7bZhyJq7pTUztE7Q8tWcLiEfzHzPTtfLlNf8+SWz5sDrsO87pjOVO1ygTkJLNqXNYc1s79ddl8N6WuLOtOd+iS3s4wK89VYTi5jZ5HSWY1zn2+PHmvTQbfU8frdWJ7W+zFkhqCFPIfoUbkdKde0Zyn9sv5luHkvF6FextfviCg8p9T8hjr0b6nplHZZxDQWeURNW+0MWOe5NjBnuYinJf1unb1fp+fiOPfm5TK22Zl5xPPNucnU9KSzQ490d/oFu/wjnB7GrKEfcHJp4vth4X1uWVS5QRiaN4E87i6Q9xDP+3xjWx03NOf8ncJtQFHHA2hI/8fm3iTj9pQ+3S5CWkXvfGTdTzszp+xpTfoc1QOXvlu4qEDMOvR8vPL1nY+s7wADnm2zG4Z3gZy/PVkOL8uZxXddVTdwi++FQjy7+PbY1eO5lX2k9N2VPhcczNzCSf7uoiokd0c3N6b0FfGna/qU9zK6f6U8+nyeUFrTlukOIWUScw6KeZuS7xf5OyCNN21bnMsr3+Dtm16c9tK1k/Xt5vS98uzg6W69+WO98eTu9r2fzJ3zNbWlbqrbiNOeuofKfIy42upP9Zf6W/3z5X+bG5tfbW5p1qtXjMwXqvC3eet/zdmMIQ==</latexit><latexit sha1_base64="rExaXPO3SNMVs6ZP3NJvKxwpZXY=">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</latexit>

Yu (1) = 1
Brady NealCounterfactuals Basics

Yu (0) = 1

0=1
8 / 25

General Steps for Deterministic Counterfactuals

Brady NealCounterfactuals Basics

9 / 25

General Steps for Deterministic Counterfactuals
From Chapter 4 of Pearl et al. (2016)’s Primer:

Brady NealCounterfactuals Basics

9 / 25

General Steps for Deterministic Counterfactuals
From Chapter 4 of Pearl et al. (2016)’s Primer:
1. Abduction: Use an observation to determine the value of U

Brady NealCounterfactuals Basics

9 / 25

General Steps for Deterministic Counterfactuals
From Chapter 4 of Pearl et al. (2016)’s Primer:
1. Abduction: Use an observation to determine the value of U
2. Action: Modify the SCM, by replacing the structural equation for T
with T := t

Brady NealCounterfactuals Basics

9 / 25

General Steps for Deterministic Counterfactuals
From Chapter 4 of Pearl et al. (2016)’s Primer:
1. Abduction: Use an observation to determine the value of U
2. Action: Modify the SCM, by replacing the structural equation for T
with T := t
3. Prediction: Use the value of U from step 1 and the modified SCM
from step 2 to compute the value of Y(t)Brady NealCounterfactuals Basics

9 / 25

Question:

Given the observation T = 1 and Y = 0, computeY(0) for this individual given the following SCM:
<latexit sha1_base64="Cu67tLdXINAxNYEboNZOA81sdrY=">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</latexit>

T := . . .

Y := U T + (1

U )(1

T)

Can’t Always Determine Counterfactual
Even when we have the structural equation for Y, we can’t always
determine counterfactuals with probability 1

Brady NealCounterfactuals Basics

11 / 25

Can’t Always Determine Counterfactual
Even when we have the structural equation for Y, we can’t always
determine counterfactuals with probability 1
What if we can’t solve for U (function that maps U to Y for a fixed valueof T isn’t invertible)?

Brady NealCounterfactuals Basics

11 / 25

Can’t Always Determine Counterfactual
Even when we have the structural equation for Y, we can’t always
determine counterfactuals with probability 1
What if we can’t solve for U (function that maps U to Y for a fixed valueof T isn’t invertible)?

Example:Structural equation for Y:
8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

Brady NealCounterfactuals Basics

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater
11 / 25

Can’t Always Determine Counterfactual
Even when we have the structural equation for Y, we can’t always
determine counterfactuals with probability 1
What if we can’t solve for U (function that maps U to Y for a fixed valueof T isn’t invertible)?
Structural equation for Y:

Example:

8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

Observation: T = 1 and Y = 0

Brady NealCounterfactuals Basics

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater
11 / 25

Can’t Always Determine Counterfactual
Even when we have the structural equation for Y, we can’t always
determine counterfactuals with probability 1
What if we can’t solve for U (function that maps U to Y for a fixed valueof T isn’t invertible)?
Structural equation for Y:

Example:

8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

Observation: T = 1 and Y = 0

Brady NealCounterfactuals Basics

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater
11 / 25

Non-Invertible Example
Structural equation for Y:
8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater

Observation: T = 1 and Y = 0

Brady NealCounterfactuals Basics

12 / 25

Non-Invertible Example
Structural equation for Y:
8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater

Observation: T = 1 and Y = 0<latexit sha1_base64="bR0l3IB3lA3Osve21SW2hBv7OWE=">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</latexit>(Yu (1) = 0)

Brady NealCounterfactuals Basics

12 / 25

Non-Invertible Example
Structural equation for Y:
8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater

Observation: T = 1 and Y = 0<latexit sha1_base64="bR0l3IB3lA3Osve21SW2hBv7OWE=">AAAXWnicrVhZb9tGEN64V2L3cNKifsgLG6VAUsiq5KZwXgQEsBMURYM6gK8kTg2KoiRCvEpSVhxC/6Ov7b8q0B/TmW+X4iGJlARbsLianfnm2JlZ7nZ82wqjZvPfWxsfffzJp5/dvrO59fkXX361fffeaeiNAsM8MTzbC847emjalmueRFZkm+d+YOpOxzbPOsMDnj+7MoPQ8tzj6No33zl637V6lqFHRPrjkfb6cvSo9Vhra03t8eV2rdlo4k+bHbTUoCbU35F3d+u+uBBd4QlDjIQjTOGKiMa20EVIn7eiJZrCJ9o7ERMtoJGFeVNMxCbJjojLJA6dqEP67tOvt4rq0m/GDCFtkBab/gOS1MT3iqdL4x6o8sn6tQzvIh0xsNnGa3p2FKZD1EgMiFoll3AuL9ehj1PhdUQ+PIW3Fnnig8JxMHI+9+hp0++IPOTva+I0adQlqYBGBtFsokoK6wjoKSPPsRlgJXTwmTRiq8vsLrOa5z36DAlLp3EIS9lWTbxQ6+JCswlbmcfGmi5GfE8eSvvKsHpTHyysu/SCeY+JMhQfaJRHLtOZza7FXJFCnszRxfwROFziDoniIfMtip9FHPlcNQiTdehYyT588ZFBDYX8K+aTbPZpPXdhT4hV05BvFvT4qrJSuxM7bXp2gB2QfExzA+jhWNTJHhPYAbI3iWId3AH9GuOXARuNAr2ByuT1rJNPnF11wvVoxppiyUjIeCYWSetiomnqsyt+g36T1r2OqHBe10nWg4Uyi0PoctXat6mvcFwc+mYqRzah1KHDgR0DZMuQaKk+icT6fiQvQtSPrVBjetqImg+UOrRzfMYYO/CJV3QE3WMad6GHpTXqdQ2xr+yY5LRybCx0p1mtGvKPa1d2xqJWlnNRqZrKyjY0NcUT5TGvANudjZqhuqHsySG8mxSs6sD+tGtwjubtY6l0ffJr1UU2DiCVx+Uozvdzni97qJM61pJj3yeONqR6ih6iNn2laTNTfweqSjysh4Gno3JC1jtrGxdmYqI7KlI28ck65z2JZ/4kTB05dYHoZefLEJ8r+WQ34I4fg1oud4rKyssGNIqnM+XyFkYm4iNHvOISz0esffjy3fRfy8wsi30zeO4MImdspJCWRe8iN+Z7fixq1AtqlFX5ePJKMyXA24lP44dTzoeEy7xDaHGJwhUeT+cnlWtwSJSJilgPc0kdpjZY6HxdcF6ozJMcsqKiAgdXdblWzst5eSPp5bLdab7nZSW9XJYpV8hNS/WDEFjnFXIR1oElnEyWHldIedgTJF3a/HoJ+1zstSOVF4Gy8KxC0kGPkV55qipfVtoXoQsGKiYs82aNCF6h/03wBrRqHFPZaKVopnLXa8U0lR+vGNlU0lkxvqnkhwpJE3uDNaWxzKvKmmKuo0oumU9Ve4JJu0ny/iKrrY7vDkWR9+yu6kya2m84K+Tu2EYPqC/Vn2f3Ha7kVmUdj9B9O+hDATR3S2rn5MbQ0hUcLeFfiHxPz5fL1Nc8ueXz5gDrMK87JjNVu5ynTgKL9mXJoc3sb+vuqz581aEz2akvMztLnJuvxjIzGTuLlMxKnEO8PdqoTZO6pYzX78rypN4vSGZENB/nEDkqtyPhKnqW0Nf1L8XNerkIdR1ffwAi8/Sh+T10yN9c0wltXcQkFllESVvtDNjAudZTZ7kApyX5bp2+Xyfn4jDz5mUhtumZOcb5ZqwyNTnp7MIj2Z1+oV3+JZ0eJqihn+jk0qLvF7n3uWVR+QZhpN4Esrh7hLxP8TzEG9vquL465+/mbgPyOp6ThuR/ou5NUm5byNPtIqRV9M5Hlv20O3PKLmqS56gBccm7hesKxLRDz8crX9/5yPIO0MPZNr1huAnk7O3Jcnhpziy+66q6gVt8L+TTs0ffNrp6OLeyT4S8u5LngoOZWzjO3z2qCs7d+MG9gr48frGm+7iXkf0r4ZHn8wjSkrZMd/AhE6lzUIjblGy/yN4BSbyibWEmrxyF11a9OOmlm5fbtVbxXnl2cLrXaP3caL56Unv2VN053xb3xQPxiOK0L55RZR5RXPl+8y/xt/jn2/92Nnbu7GxJ1o1bSuZrkfvb+eZ/bIOIcA==</latexit>(Yu (1) = 0)<latexit sha1_base64="b/onWLPfGBGr5T4FFNmxglk1eC4=">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</latexit>

Yu (0) =?
Brady NealCounterfactuals Basics

12 / 25

Non-Invertible Example
Structural equation for Y:
8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater<latexit sha1_base64="FoSdOXE5EMTzGrTNysavbmC+YPw=">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</latexit>

P (U = always happy) = 0.3
P (U = never happy) = 0.2
P (U = dog-needer) = 0.4
P (U = dog-hater) = 0.1

Observation: T = 1 and Y = 0<latexit sha1_base64="bR0l3IB3lA3Osve21SW2hBv7OWE=">AAAXWnicrVhZb9tGEN64V2L3cNKifsgLG6VAUsiq5KZwXgQEsBMURYM6gK8kTg2KoiRCvEpSVhxC/6Ov7b8q0B/TmW+X4iGJlARbsLianfnm2JlZ7nZ82wqjZvPfWxsfffzJp5/dvrO59fkXX361fffeaeiNAsM8MTzbC847emjalmueRFZkm+d+YOpOxzbPOsMDnj+7MoPQ8tzj6No33zl637V6lqFHRPrjkfb6cvSo9Vhra03t8eV2rdlo4k+bHbTUoCbU35F3d+u+uBBd4QlDjIQjTOGKiMa20EVIn7eiJZrCJ9o7ERMtoJGFeVNMxCbJjojLJA6dqEP67tOvt4rq0m/GDCFtkBab/gOS1MT3iqdL4x6o8sn6tQzvIh0xsNnGa3p2FKZD1EgMiFoll3AuL9ehj1PhdUQ+PIW3Fnnig8JxMHI+9+hp0++IPOTva+I0adQlqYBGBtFsokoK6wjoKSPPsRlgJXTwmTRiq8vsLrOa5z36DAlLp3EIS9lWTbxQ6+JCswlbmcfGmi5GfE8eSvvKsHpTHyysu/SCeY+JMhQfaJRHLtOZza7FXJFCnszRxfwROFziDoniIfMtip9FHPlcNQiTdehYyT588ZFBDYX8K+aTbPZpPXdhT4hV05BvFvT4qrJSuxM7bXp2gB2QfExzA+jhWNTJHhPYAbI3iWId3AH9GuOXARuNAr2ByuT1rJNPnF11wvVoxppiyUjIeCYWSetiomnqsyt+g36T1r2OqHBe10nWg4Uyi0PoctXat6mvcFwc+mYqRzah1KHDgR0DZMuQaKk+icT6fiQvQtSPrVBjetqImg+UOrRzfMYYO/CJV3QE3WMad6GHpTXqdQ2xr+yY5LRybCx0p1mtGvKPa1d2xqJWlnNRqZrKyjY0NcUT5TGvANudjZqhuqHsySG8mxSs6sD+tGtwjubtY6l0ffJr1UU2DiCVx+Uozvdzni97qJM61pJj3yeONqR6ih6iNn2laTNTfweqSjysh4Gno3JC1jtrGxdmYqI7KlI28ck65z2JZ/4kTB05dYHoZefLEJ8r+WQ34I4fg1oud4rKyssGNIqnM+XyFkYm4iNHvOISz0esffjy3fRfy8wsi30zeO4MImdspJCWRe8iN+Z7fixq1AtqlFX5ePJKMyXA24lP44dTzoeEy7xDaHGJwhUeT+cnlWtwSJSJilgPc0kdpjZY6HxdcF6ozJMcsqKiAgdXdblWzst5eSPp5bLdab7nZSW9XJYpV8hNS/WDEFjnFXIR1oElnEyWHldIedgTJF3a/HoJ+1zstSOVF4Gy8KxC0kGPkV55qipfVtoXoQsGKiYs82aNCF6h/03wBrRqHFPZaKVopnLXa8U0lR+vGNlU0lkxvqnkhwpJE3uDNaWxzKvKmmKuo0oumU9Ve4JJu0ny/iKrrY7vDkWR9+yu6kya2m84K+Tu2EYPqC/Vn2f3Ha7kVmUdj9B9O+hDATR3S2rn5MbQ0hUcLeFfiHxPz5fL1Nc8ueXz5gDrMK87JjNVu5ynTgKL9mXJoc3sb+vuqz581aEz2akvMztLnJuvxjIzGTuLlMxKnEO8PdqoTZO6pYzX78rypN4vSGZENB/nEDkqtyPhKnqW0Nf1L8XNerkIdR1ffwAi8/Sh+T10yN9c0wltXcQkFllESVvtDNjAudZTZ7kApyX5bp2+Xyfn4jDz5mUhtumZOcb5ZqwyNTnp7MIj2Z1+oV3+JZ0eJqihn+jk0qLvF7n3uWVR+QZhpN4Esrh7hLxP8TzEG9vquL465+/mbgPyOp6ThuR/ou5NUm5byNPtIqRV9M5Hlv20O3PKLmqS56gBccm7hesKxLRDz8crX9/5yPIO0MPZNr1huAnk7O3Jcnhpziy+66q6gVt8L+TTs0ffNrp6OLeyT4S8u5LngoOZWzjO3z2qCs7d+MG9gr48frGm+7iXkf0r4ZHn8wjSkrZMd/AhE6lzUIjblGy/yN4BSbyibWEmrxyF11a9OOmlm5fbtVbxXnl2cLrXaP3caL56Unv2VN053xb3xQPxiOK0L55RZR5RXPl+8y/xt/jn2/92Nnbu7GxJ1o1bSuZrkfvb+eZ/bIOIcA==</latexit>(Yu (1) = 0)<latexit sha1_base64="b/onWLPfGBGr5T4FFNmxglk1eC4=">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</latexit>

Yu (0) =?
Brady NealCounterfactuals Basics

12 / 25

Non-Invertible Example
Structural equation for Y:
8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater<latexit sha1_base64="FoSdOXE5EMTzGrTNysavbmC+YPw=">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</latexit>

P (U = always happy) = 0.3
P (U = never happy) = 0.2
P (U = dog-needer) = 0.4
P (U = dog-hater) = 0.1

Observation: T = 1 and Y = 0<latexit sha1_base64="bR0l3IB3lA3Osve21SW2hBv7OWE=">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</latexit>(Yu (1) = 0)<latexit sha1_base64="b/onWLPfGBGr5T4FFNmxglk1eC4=">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</latexit>

Yu (0) =?
Brady NealCounterfactuals Basics

12 / 25

Non-Invertible Example
Structural equation for Y:
8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater<latexit sha1_base64="FoSdOXE5EMTzGrTNysavbmC+YPw=">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</latexit>

P (U = always happy) = 0.3
P (U = never happy) = 0.2
P (U = dog-needer) = 0.4
P (U = dog-hater) = 0.1<latexit sha1_base64="GWoKkW+XMg+Au+dSy6kfc8QMuqg=">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</latexit>

Observation: T = 1 and Y = 0<latexit sha1_base64="bR0l3IB3lA3Osve21SW2hBv7OWE=">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</latexit>(Yu (1) = 0)0.2
2=0.2 + 0.1
3
0.1
1
P (U = dog-hater | T = 1, Y = 0) ==0.2 + 0.1
3

P (U = never happy | T = 1, Y = 0) =<latexit sha1_base64="b/onWLPfGBGr5T4FFNmxglk1eC4=">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</latexit>

Yu (0) =?
Brady NealCounterfactuals Basics

12 / 25

Non-Invertible Example
Structural equation for Y:
8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater<latexit sha1_base64="FoSdOXE5EMTzGrTNysavbmC+YPw=">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</latexit>

P (U = always happy) = 0.3
P (U = never happy) = 0.2
P (U = dog-needer) = 0.4
P (U = dog-hater) = 0.1<latexit sha1_base64="GWoKkW+XMg+Au+dSy6kfc8QMuqg=">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</latexit>

Observation: T = 1 and Y = 0<latexit sha1_base64="bR0l3IB3lA3Osve21SW2hBv7OWE=">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</latexit>(Yu (1) = 0)0.2
2=0.2 + 0.1
3
0.1
1
P (U = dog-hater | T = 1, Y = 0) ==0.2 + 0.1
3

P (U = never happy | T = 1, Y = 0) =1
P (Yu (0) = 1) =3<latexit sha1_base64="sq9JiVvdcSPYjIabMw0nGcwnfSk=">AAAXanicrVjbbttGEN2kt9TpxXGBGoX7wEYu4BayKjlp3RcBAewERdGgDuBbEgUGRVESId5KUlYcQi/9mr62f9N/6Ed05uxSJHUhJcEWLK5mZ85cdmaWu23ftsKoXv/3zt333v/gw4/ufbxx/5NPP/t888HWeegNA8M8MzzbCy7bemjalmueRVZkm5d+YOpO2zYv2oMjnr+4NoPQ8tzT6MY33zh6z7W6lqFHRLra/Lrl7728Gu7Vv9OaWoO/Wt1AN+LGOH40vtqs1Gt1/Gmzg4YaVIT6O/Ee3N8RLdERnjDEUDjCFK6IaGwLXYT0eS0aoi58or0RMdECGlmYN8VYbJDskLhM4tCJOqDvHv16ragu/WbMENIGabHpPyBJTXyreDo07oIqn6xfy/Au0hEDm228oWdbYTpEjUSfqGVyCefycm36OCVeR+TDz/DWIk98UDgORs7nLj1t+h2Rh/x9Q5wmjTokFdDIIJpNVElhHQE9ZeQ5Nn2shA4+k0ZsdZHdRVbzvEefAWHpNA5hKduqiWdqXVxoNmEr89hY08WIb8lDaV8RVnfig4V1l14w7ylRBuIdjfLIRTqz2bWYK1LI4zm6mD8Ch0vcIVE8ZL5F8bOII5+rBmGyDh0r2YMvPjKoppB/xXySzT6t5z7sCbFqGvLNgh5fVVZqd2KnTc82sAOSj2muDz0ciyrZYwI7QPYmUayCO6BfI/wyYKMxRa+hMnk9q+QTZ1eVcD2asSZYMhIynolF0rqYaJr67IvfoN+kda8iKpzXVZL1YKHM4hC6XLX2TeorHBeHvpnKkU0oVehwYEcf2TIgWqpPIrG+H8iLEPVjK9SYnjai5gOlCu0cnxHGDnziFR1C94jGHehhaY16XU0cKjvGOa0cGwvdaVarhvzj2pWdcVory7moVE1lZROa6uKx8phXgO3ORs1Q3VD25BDejaesasP+tGtwjubtY6l0ffJr1UE29iGVx+Uozvdzni8HqJMq1pJj3yOOJqS6ih6iNn2laSNTf0eqSjysh4Gno3JC1jtrG03NxER3VKRs4pN1znsSz/xBmDpyqoXoZeeLEJ8q+WQ34I4fg1osd47KyssGNIonM8XyFkYm4iNHvOISz0esffjyzeRfy8wsi307eO4MImdspJCWRe8gN+Z7fioq1AsqlFX5ePJKMyXA24lP490J5y7hMu8AWlyicIXHk/lx6RocE2WsItbFXFKHqQ0WOl8HnC2VeZJDVlQ0xcFVXayV83Je3kh6sWxnku95WUkvlmXKNXLTUv0gBNZliVyEdWAJJ5OlpyVSHvYESZc2v1zCPhd77VDlRaAsvCiRdNBjpFeeqsrnpfZF6IKBignLvFojgtfof2O8Aa0ax1Q2WimaqdzNWjFN5UcrRjaVdFaMbyr5rkTSxN5gTWgs86K0ppjrpJRL5lPZnmDSbpK8v8hqq+K7TVHkPbujOpOm9hvOCrk7NtEDqkv159l9hyu5UVrHQ3TfNvpQAM2dgto5uzW0dAWHS/gXIt/T8+Uy9TVPbvm8OcI6zOuOyUzZLuepk8CifVlyaDP727r7qg9fdehMduqrzM4S5+bLscxMxs4iJbMS5xhvjzZq06RuKeP1u7I8qfcWyQyJ5uMcIkfFdiRc054l9HX9S3GzXi5CXcfX74HIPD1ofgsd8jfXdEJbFzGJRRZR0lY7A9ZwrvXUWS7AaUm+W6fv18m5OMy8eVmIbXpmjnG+GalMTU46+/BIdqdfaJd/TqeHMWroEZ1cGvT9LPc+tywq3yAM1ZtAFveAkA8pnsd4Y1sd11fn/P3cbUBex1PSkPyP1b1Jym0LebpdhLSK3vnIsp92Zk7Z05rkOapPXPJu4aYEMe3Q8/GK13c+srwD9HC2TW8YbgM5e3uyHF6aM4vvuspu4BbfC/n07NK3ja4ezq3sMyHvruS54GjmFo7z94CqgnM3frg1pS+PP13TPdzLyP6V8MjzeQRpSVumO/iQidQ5KMRtSrZfZO+AJN60bWEmrxyF11S9OOmlG1eblcb0vfLs4Pyg1vixVn/xuPLkJ3XnfE/siIdij+J0KJ5QZZ5QXA3xp/hL/C3++fK/7a3tr7Z3JOvdO0rmC5H72979HzAPjoA=</latexit><latexit sha1_base64="b/onWLPfGBGr5T4FFNmxglk1eC4=">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</latexit>

Yu (0) =?
Brady NealCounterfactuals Basics

12 / 25

General Steps for Probabilistic Counterfactuals

Brady NealCounterfactuals Basics

13 / 25

General Steps for Probabilistic Counterfactuals
From Chapter 4 of Pearl et al. (2016)’s Primer:

Brady NealCounterfactuals Basics

13 / 25

General Steps for Probabilistic Counterfactuals
From Chapter 4 of Pearl et al. (2016)’s Primer:
1. Abduction: Use an observation Z to update the distribution of U:
P(U | Z)

Brady NealCounterfactuals Basics

13 / 25

General Steps for Probabilistic Counterfactuals
From Chapter 4 of Pearl et al. (2016)’s Primer:
1. Abduction: Use an observation Z to update the distribution of U:
P(U | Z)
2. Action: Modify the SCM, by replacing the structural equation for T
with T := t

Brady NealCounterfactuals Basics

13 / 25

General Steps for Probabilistic Counterfactuals
From Chapter 4 of Pearl et al. (2016)’s Primer:
1. Abduction: Use an observation Z to update the distribution of U:
P(U | Z)
2. Action: Modify the SCM, by replacing the structural equation for T
with T := t
3. Prediction: Use the the updated distribution of U step 1 and themodified SCM from step 2 to compute the distribution of Y(t)Brady NealCounterfactuals Basics

13 / 25

No Unit-Level Counterfactuals without
Parametric Model
Main ingredient necessary for computing counterfactuals:
parametric model for the structural equation for Y

Brady NealCounterfactuals Basics

14 / 25

No Unit-Level Counterfactuals without
Parametric Model
Main ingredient necessary for computing counterfactuals:
parametric model for the structural equation for Y
Strong assumption

Brady NealCounterfactuals Basics

14 / 25

No Unit-Level Counterfactuals without
Parametric Model
Main ingredient necessary for computing counterfactuals:
parametric model for the structural equation for Y
Strong assumption
Without it, we are stuck with the fundamental problem of causal
inference.

Brady NealCounterfactuals Basics

14 / 25

Question:

Given the observation T = 1 and Y = 1, computeY(0) for this individual given the following SCM
and prior:
8
1
>
>
>
<0
Y :=
>
T
>
>
:
1
<latexit sha1_base64="/SK5IIZdAjochaa7e3H0JdxeF7w=">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</latexit>

T

U = always happy
U = never happy
U = dog-needer
U = dog-hater<latexit sha1_base64="FoSdOXE5EMTzGrTNysavbmC+YPw=">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</latexit>

P (U = always happy) = 0.3
P (U = never happy) = 0.2
P (U = dog-needer) = 0.4
P (U = dog-hater) = 0.1

Population-Level Doesn’t Require a Parametric Model
Population-level counterfactual: E[Y (t) | T = t0 ]<latexit sha1_base64="SYYdSgdt3A7S0d/pOJUafMGhWKo=">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</latexit>

Brady NealCounterfactuals Basics

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Population-Level Doesn’t Require a Parametric Model
Population-level counterfactual: E[Y (t) | T = t0 ]<latexit sha1_base64="SYYdSgdt3A7S0d/pOJUafMGhWKo=">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</latexit>

Just like we were able to identify the ATE E[Y (1) Y (0)]nonparametrically (using just the causal graph), we can do the same withpopulation-level counterfactual quantities, if they are identifiable<latexit sha1_base64="5fuE181ORSz5GwzYYzptnmDHjxw=">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</latexit>

Brady NealCounterfactuals Basics

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Population-Level Doesn’t Require a Parametric Model
Population-level counterfactual: E[Y (t) | T = t0 ]<latexit sha1_base64="SYYdSgdt3A7S0d/pOJUafMGhWKo=">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</latexit>

Just like we were able to identify the ATE E[Y (1) Y (0)]nonparametrically (using just the causal graph), we can do the same withpopulation-level counterfactual quantities, if they are identifiable<latexit sha1_base64="5fuE181ORSz5GwzYYzptnmDHjxw=">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</latexit>

Same with CATEs: E[Y (1)<latexit sha1_base64="KKjy4VarCTJAVCtQTN+MKP1HfPY=">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</latexit>

Brady Neal

Y (0) | X = x]

Counterfactuals Basics

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Population-Level Doesn’t Require a Parametric Model
Population-level counterfactual: E[Y (t) | T = t0 ]<latexit sha1_base64="SYYdSgdt3A7S0d/pOJUafMGhWKo=">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</latexit>

Just like we were able to identify the ATE E[Y (1) Y (0)]nonparametrically (using just the causal graph), we can do the same withpopulation-level counterfactual quantities, if they are identifiable<latexit sha1_base64="5fuE181ORSz5GwzYYzptnmDHjxw=">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</latexit>

Same with CATEs: E[Y (1)<latexit sha1_base64="KKjy4VarCTJAVCtQTN+MKP1HfPY=">AAAXaHicrVjbbttGEN2kt9TpxUmBGm1eWCsFnEJWJTeF8yIggO2gKBrUAWzHjmUYEklJhHgrSVl2CAH9mr62v9Nf6Fd05uxSJHUhJcMWLK5mZ85cdmaWux3ftsKoXv/33v0PPvzo408efLr28LPPv/hy/dHjk9AbBrp5rHu2F5x22qFpW655HFmRbZ76gdl2Orb5tjPY4/m3V2YQWp57FN345oXT7rlW19LbEZEu179tHZyfbTWeadva2Vb9mdZyLEM71Zra9cXleqVeq+NPmx001KAi1N+h9+jhE9EShvCELobCEaZwRURjW7RFSJ9z0RB14RPtQsREC2hkYd4UY7FGskPiMomjTdQBfffo17miuvSbMUNI66TFpv+AJDXxveIxaNwFVT5Zv5bhXaQjBjbbeEPPjsJ0iBqJPlHL5BLO5eU69HFKvI7Ihxfw1iJPfFA4DnrO5y49bfodkYf8fUOcJo0MkgpopBPNJqqksI6AnjLyHJs+VqINPpNGbHWR3UVW87xHnwFhtWkcwlK2VROv1Lq40GzCVuaxsaaLEa/JQ2lfEVZ34oOFdZdeMO8RUQbiPY3yyEU6s9m1mCtSyOM5upg/AodL3CFRPGS+RfGziCOfqzphso42VrIHX3xkUE0h/4r5JJt9Ws9t2BNi1TTkmwU9vqqs1O7ETpueHWAHJB/TXB96OBZVsscEdoDsTaJYBXdAv0b4pcNGfYpeQ2XyelbJJ86uKuF6NGNNsGQkZDwTi6R1MdE09dkWv0G/SeteRVQ4r6sk68FCmcUhdLlq7ZvUVzguDn0zlSObUKrQ4cCOPrJlQLRUn0RifT+SFyHqx1aoMT1tRM0HShXaOT4jjB34xCs6hO4RjQ3oYWmNel1N7Co7xjmtHBsL3WlWq4b849qVnXFaK8u5qFRNZWUTmuriufKYV4DtzkZNV91Q9uQQ3o2nrOrA/rRrcI7m7WOpdH3ya2UgG/uQyuNyFOf7Oc+XHdRJFWvJse8RRxNSXUUPUZu+0rSWqb89VSUe1kPH01E5IeudtY2mZmKiOypSNvHJOuc9iWf+IMw2cqqF6GXnixAPlHyyG3DHj0EtljtBZeVlAxrFk5lieQsjE/GRI15xiecj1j58+W7yr2VmlsW+Gzx3BpEzNlJIy6IbyI35nh+JCvWCCmVVPp680kwJ8Hbi0/jphPMp4TLvAFpconCFx5P5ceka7BNlrCLWxVxSh6kNFjqfAc6WyjzJISsqmuLgqi7Wynk5L28kvVjWmOR7XlbSi2WZcoXctFQ/CIF1WiIXYR1Ywslk6VGJlIc9QdKlzWdL2Odirx2qvAiUhW9LJB30GOmVp6rydal9EbpgoGLCMu9uEcEr9L8x3oBWjWMqG60UzVTu5lYxTeVHK0Y2lXRWjG8q+b5E0sTeYE1oLPOmtKaY67CUS+ZT2Z5g0m6SvL/Iaqviu0NR5D3bUJ1JU/sNZ4XcHZvoAdWl+vPsvsOV3Cit4yG6bwd9KIBmo6B2ju8MLV3B4RL+hcj39Hy5TH3Nk1s+b/awDvO6YzJTtst56iSwaF+WHNrM/nbbfdWHr23oTHbqy8zOEufmy7HMTMbOIiWzEmcfb482atOkbinj9buyPKn3FskMiebjHCJHxXYkXNOeJfTb+pfiZr1chHobX38AIvP0oPkaOuRvrumEdlvEJBZZRElb7QxYw7nWU2e5AKcl+W6dvl8n5+Iw8+ZlIbbpmTnG+WakMjU56WzDI9mdfqFd/jWdHsaooZ/o5NKg71e597llUfkGYajeBLK4O4S8S/Hcxxvb6ri+Oudv524D8joOSEPyP1b3Jim3LeTpdhHSKnrnI8t+asycsqc1yXNUn7jk3cJNCWLaoefjFa/vfGR5B+jhbJveMNwFcvb2ZDm8NGcW33WV3cAtvhfy6dmlbxtdPZxb2cdC3l3Jc8HezC0c5+8OVQXnbrz5eEpfHn+6pnu4l5H9K+GR5/MI0pK2THfwIROpc1CI25Rsv8jeAUm8advCTF45Cq+penHSS9cu1yuN6Xvl2cHJTq3xc63+5nnl5Qt15/xAPBGbYovitCteUmUeUlx18af4S/wt/vn6v431jY2NbyTr/XtK5iuR+9vY/B8L1YzK</latexit>

Y (0) | X = x]

See Malinsky et al. (2019)’s potential outcome calculus (generalization of
do-calculus) for general identification of counterfactual quantities
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Counterfactuals Basics

Important Application: Mediation

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Important Application: Mediation

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Mediation

Get dog

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Happiness

Important Application: Mediation

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Mediation
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Happiness

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Mediation
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Brady Neal

Direct effectImportant Application: Mediation

Happiness

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Mediation

Indirect effect
Walk
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Brady Neal

Direct effectImportant Application: Mediation

Happiness

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Mediation

Indirect effect
Walk
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Get dog

Brady Neal

“Direct effect”

Important Application: Mediation

Happiness

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Mediation

Indirect effect
Walk
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Get dogMorefriends

Happiness“Direct effect”
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Important Application: Mediation

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Mediation

Indirect effect
Walk
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Get dogMorefriends“Direct effect”
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Important Application: Mediation

Happiness

How can we measure
the direct effect?
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Controlled Direct Effect (CDE)<latexit sha1_base64="OfK0ns+KKjjc/m4Cn5/Ghal1nM4=">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</latexit>

M

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Important Application: Mediation

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Controlled Direct Effect (CDE)<latexit sha1_base64="sbpqejjz+mm29RE0sPZcAMtsExA=">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</latexit>

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Brady NealE[Y | do(T = 0), M = m]

Important Application: Mediation

19 / 25

Controlled Direct Effect (CDE)<latexit sha1_base64="LZYYgMU/dMOZROt6+Py+fmIFMoY=">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</latexit>

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Brady NealE[Y | do(T = 0), M = m]

Important Application: Mediation

19 / 25

Controlled Direct Effect (CDE)<latexit sha1_base64="aZC75sSxfdw/OqkThjTxyBGuPXo=">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</latexit>

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Brady NealE[Y | do(T = 0), M = m]

Important Application: Mediation

19 / 25

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M

T

Y

E[Y | do(T = 1, M = m)]

E[Y | do(T = 0, M = m)]

E[Y | do(T = 1), M = m]

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Brady Neal

Important Application: Mediation

19 / 25

Controlled Direct Effect (CDE)<latexit sha1_base64="Cxszk/1ritG68/0lXF/ObkjBeww=">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</latexit>

M

T

Y

E[Y | do(T = 1, M = m)]

E[Y | do(T = 0, M = m)]

E[Y | do(T = 1), M = m]

E[Y | do(T = 0), M = m]<latexit sha1_base64="SFYRU40GekXha36XoWISP1ckKNE=">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</latexit><latexit sha1_base64="fbGmkiykYAwbP5VirVVkqAFfS38=">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</latexit>

Brady Neal

Important Application: Mediation

19 / 25

Controlled Direct Effect (CDE)<latexit sha1_base64="Cxszk/1ritG68/0lXF/ObkjBeww=">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</latexit>

Problems:
M

TY

E[Y | do(T = 1, M = m)]

E[Y | do(T = 0, M = m)]

E[Y | do(T = 1), M = m]

E[Y | do(T = 0), M = m]<latexit sha1_base64="SFYRU40GekXha36XoWISP1ckKNE=">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</latexit><latexit sha1_base64="fbGmkiykYAwbP5VirVVkqAFfS38=">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</latexit>

Brady Neal

Important Application: Mediation

19 / 25

Controlled Direct Effect (CDE)<latexit sha1_base64="Cxszk/1ritG68/0lXF/ObkjBeww=">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</latexit>

Problems:• CDE is specific to thearbitrary choice of m

M

T

Y

E[Y | do(T = 1, M = m)]

E[Y | do(T = 0, M = m)]

E[Y | do(T = 1), M = m]

E[Y | do(T = 0), M = m]<latexit sha1_base64="SFYRU40GekXha36XoWISP1ckKNE=">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</latexit><latexit sha1_base64="fbGmkiykYAwbP5VirVVkqAFfS38=">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</latexit>

Brady Neal

Important Application: Mediation

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Controlled Direct Effect (CDE)<latexit sha1_base64="Cxszk/1ritG68/0lXF/ObkjBeww=">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</latexit>

M

T

Y

Problems:• CDE is specific to thearbitrary choice of m
• How do we get the indirect
effect? Can’t just subtract theCDE from the total effectE[Y | do(T = 1, M = m)]

E[Y | do(T = 0, M = m)]

E[Y | do(T = 1), M = m]

E[Y | do(T = 0), M = m]<latexit sha1_base64="SFYRU40GekXha36XoWISP1ckKNE=">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</latexit><latexit sha1_base64="fbGmkiykYAwbP5VirVVkqAFfS38=">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</latexit>

Brady Neal

Important Application: Mediation

19 / 25

Natural Direct and Indirect Effects

Brady Neal

Important Application: Mediation

20 / 25

Natural Direct and Indirect Effects
Subscript notation:
E[Yt,m ] , E[Y | do(T = t, M = m)]<latexit sha1_base64="1yAHxBcn3VnkWVDlUoO1Mh9n+Tc=">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</latexit>

Brady Neal

Important Application: Mediation

20 / 25

Natural Direct and Indirect Effects
Subscript notation:
E[Yt,m ] , E[Y | do(T = t, M = m)]<latexit sha1_base64="1yAHxBcn3VnkWVDlUoO1Mh9n+Tc=">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</latexit>

Brady Neal

E[Mt ] , E[M | do(T = t)]<latexit sha1_base64="Qma081lCp+csiqQ7ip2eWtLgFqg=">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</latexit>

Important Application: Mediation

20 / 25

Natural Direct and Indirect Effects
Subscript notation:
E[Yt,m ] , E[Y | do(T = t, M = m)]<latexit sha1_base64="1yAHxBcn3VnkWVDlUoO1Mh9n+Tc=">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</latexit>

CDE , E[Y1,mE[Mt ] , E[M | do(T = t)]<latexit sha1_base64="Qma081lCp+csiqQ7ip2eWtLgFqg=">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</latexit><latexit sha1_base64="qAUfZEou0p+PWvGIWkYYj8AI7T4=">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</latexit>

Brady Neal

Y0,m ]

Important Application: Mediation

20 / 25

Natural Direct and Indirect Effects
Subscript notation:
E[Yt,m ] , E[Y | do(T = t, M = m)]<latexit sha1_base64="1yAHxBcn3VnkWVDlUoO1Mh9n+Tc=">AAAUpnicrVjbbttGEN0kvSTuzUn91hc2SgEXoFXJSeG8CAhgOyiKunAA31LLcCWSkgjxFpKy6hD6j35FP6Gv7S/0bzpzdimSupBSYAkiV7MzZ2bntlx2A8eO4kbjv3v3H3z08SefPny08dnnX3z51ebjJ2eRPwoN69TwHT+86HYiy7E96zS2Y8e6CEKr43Yd67w73Of58xsrjGzfO4lvA+vK7fQ9u2cbnZhI15u77cPLt9dJrGvu5Eprx6Hd8fqO9U5jutZ2bVNrH/jbJy3iONJamvv91fVmrVFv4KPND5pqUBPqc+w/fvSXaAtT+MIQI+EKS3giprEjOiKi76VoioYIiHYlEqKFNLIxb4mJ2CDZEXFZxNEh6pCuffp3qage/WfMCNIGaXHoF5KkJr5TPCaNe6DKO+vXcrzLdCTAZhtv6d5VmC5RYzEgapVcyrm6XJe+bsWqY1rDS6zWppUEoLAfjMKae3R36H9MK+TrLXFaNDJJKqSRQTSHqJLCOkK6S8+zbwaIRAd8Fo3Y6jK7y6zmeZ++Q8Lq0DiCpWyrJl6ruHjQbMFW5nEQ0+WIf9AKpX1lWL3pGmzEXa6CeU+IMhTvaVRELtOZz67lXLFCnizQxfwxODzijojiI/Nt8p9NHMVcNQiTdXQQyT7WEiCD6gr5Z8yn2RxQPHdgT4Soacg3G3oCVVmZ3amdDt27wA5JPqG5AfSwL3SyxwJ2iOxNvaiDO6R/Y/wzYKMxQ6+jMjmeOq2Js0snXJ9m7CmW9IT0Z2qRtC4hmqa+O+IX6Lco7jq8wnmtk6wPC2UWR9Dlqdi3qK+wX1y6MpU9m1J06HBhxwDZMiRapk8isb4faBUR6sdRqAndHXgtAIoO7eyfMcYu1sQRHUH3mMYm9LC0Rr2uLvaUHZOCVvaNje40r1VD/nHtys44q5XlPFSqprKyBU0N8UKtmCPAdue9ZqhuKHtyhNVNZqzqwv6sa3COFu1jqSw+xViZyMYBpIq47MXF61y0ll3UiY5Ysu/7xNGCVE/RI9RmoDRt5OpvX1WJj3gYuLsqJ2S9s7bxzExCdFd5yiE+Wee8J/HMO8LsIKfa8F5+vgzxUMmnuwF3/ATUcrkzVFZRNqRRMp0pl7cxsuAfOeKIS7wAvg6wlm+nPy03syr23eB5c4icsbFCWhXdRG4sXvmJqFEvqFFWFf3JkWZKiKeTgMbPppzPCJd5h9DiEYUrPJnOTypjcECUifJYD3NpHWY22Oh8JjjbKvMkh6yoeIaDq7pcK+floryR9HJZc5rvRVlJL5dlyg1y01b9IALWRYVcjDiwhJvL0pMKKR97gqRLm9+uYJ+HvXak8iJUFp5XSLroMXJVvqrKo0r7YnTBUPmEZX77AA/eoP9N8AS0rh8z2Xgtb2Zytx/k00x+vKZnM0l3Tf9mku8rJC3sDfaUxjJvKmuKuY4ruWQ+Ve0JFu0m6fOLrDYd1y55kfdsU3UmTe03nBVyd2yhB+gr9ef5fYcruVlZxyN03y76UAjNZkntnN4ZWhbB0Qrri5Dv2flylfpaJLd63uwjDou6Yzojn0Cyp5D09BDl9icb+NnJIsFT4FjtZOnz4A4yQMbwJ+qFR/SMNYGm5/R816Tr68Kutyoqn7NGql/mcXcJeY/2qwPsa+vjBuo0tFM4MxV1HJKG9DdRp8uM21FngGVI6+hdjCyzzpw7i8xqkk+bA+KSJ7DbCsQsjxfjlcd3MbJ8U+LjBJCdw+4COX/GXA0vy5nlbwSq3lMsPz0HdO/R1UFHjxTvAbLOQWe3qMfIE758etqfe1fB+btLVcG5mzx9MqOviD97+u3j9DoijoxHnmJiSEtaHnHZOTqATKyeFiOcObOTdb1wUpZ4s7ZFubxyFV5L9SNL7RYb15u15uzbt/nB2W69+WO98eZF7dVL9WbuofhGPBXb5Kc98Yoq85j8aog/xd/iH/Hv1vbWr1unW+eS9f49JfO1KHy2fv8fMWgE3g==</latexit>

CDE , E[Y1,m

Y0,m ]NDE , E[Y1,M0

Y0,M0 ]

E[Mt ] , E[M | do(T = t)]<latexit sha1_base64="Qma081lCp+csiqQ7ip2eWtLgFqg=">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</latexit><latexit 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sha1_base64="w1RwfD13HIayF6BIr98IJNwPauc=">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</latexit>

Brady Neal

Important Application: Mediation

20 / 25

Natural Direct and Indirect Effects
Subscript notation:
E[Yt,m ] , E[Y | do(T = t, M = m)]<latexit sha1_base64="1yAHxBcn3VnkWVDlUoO1Mh9n+Tc=">AAAUpnicrVjbbttGEN0kvSTuzUn91hc2SgEXoFXJSeG8CAhgOyiKunAA31LLcCWSkgjxFpKy6hD6j35FP6Gv7S/0bzpzdimSupBSYAkiV7MzZ2bntlx2A8eO4kbjv3v3H3z08SefPny08dnnX3z51ebjJ2eRPwoN69TwHT+86HYiy7E96zS2Y8e6CEKr43Yd67w73Of58xsrjGzfO4lvA+vK7fQ9u2cbnZhI15u77cPLt9dJrGvu5Eprx6Hd8fqO9U5jutZ2bVNrH/jbJy3iONJamvv91fVmrVFv4KPND5pqUBPqc+w/fvSXaAtT+MIQI+EKS3giprEjOiKi76VoioYIiHYlEqKFNLIxb4mJ2CDZEXFZxNEh6pCuffp3qage/WfMCNIGaXHoF5KkJr5TPCaNe6DKO+vXcrzLdCTAZhtv6d5VmC5RYzEgapVcyrm6XJe+bsWqY1rDS6zWppUEoLAfjMKae3R36H9MK+TrLXFaNDJJKqSRQTSHqJLCOkK6S8+zbwaIRAd8Fo3Y6jK7y6zmeZ++Q8Lq0DiCpWyrJl6ruHjQbMFW5nEQ0+WIf9AKpX1lWL3pGmzEXa6CeU+IMhTvaVRELtOZz67lXLFCnizQxfwxODzijojiI/Nt8p9NHMVcNQiTdXQQyT7WEiCD6gr5Z8yn2RxQPHdgT4Soacg3G3oCVVmZ3amdDt27wA5JPqG5AfSwL3SyxwJ2iOxNvaiDO6R/Y/wzYKMxQ6+jMjmeOq2Js0snXJ9m7CmW9IT0Z2qRtC4hmqa+O+IX6Lco7jq8wnmtk6wPC2UWR9Dlqdi3qK+wX1y6MpU9m1J06HBhxwDZMiRapk8isb4faBUR6sdRqAndHXgtAIoO7eyfMcYu1sQRHUH3mMYm9LC0Rr2uLvaUHZOCVvaNje40r1VD/nHtys44q5XlPFSqprKyBU0N8UKtmCPAdue9ZqhuKHtyhNVNZqzqwv6sa3COFu1jqSw+xViZyMYBpIq47MXF61y0ll3UiY5Ysu/7xNGCVE/RI9RmoDRt5OpvX1WJj3gYuLsqJ2S9s7bxzExCdFd5yiE+Wee8J/HMO8LsIKfa8F5+vgzxUMmnuwF3/ATUcrkzVFZRNqRRMp0pl7cxsuAfOeKIS7wAvg6wlm+nPy03syr23eB5c4icsbFCWhXdRG4sXvmJqFEvqFFWFf3JkWZKiKeTgMbPppzPCJd5h9DiEYUrPJnOTypjcECUifJYD3NpHWY22Oh8JjjbKvMkh6yoeIaDq7pcK+floryR9HJZc5rvRVlJL5dlyg1y01b9IALWRYVcjDiwhJvL0pMKKR97gqRLm9+uYJ+HvXak8iJUFp5XSLroMXJVvqrKo0r7YnTBUPmEZX77AA/eoP9N8AS0rh8z2Xgtb2Zytx/k00x+vKZnM0l3Tf9mku8rJC3sDfaUxjJvKmuKuY4ruWQ+Ve0JFu0m6fOLrDYd1y55kfdsU3UmTe03nBVyd2yhB+gr9ef5fYcruVlZxyN03y76UAjNZkntnN4ZWhbB0Qrri5Dv2flylfpaJLd63uwjDou6Yzojn0Cyp5D09BDl9icb+NnJIsFT4FjtZOnz4A4yQMbwJ+qFR/SMNYGm5/R816Tr68Kutyoqn7NGql/mcXcJeY/2qwPsa+vjBuo0tFM4MxV1HJKG9DdRp8uM21FngGVI6+hdjCyzzpw7i8xqkk+bA+KSJ7DbCsQsjxfjlcd3MbJ8U+LjBJCdw+4COX/GXA0vy5nlbwSq3lMsPz0HdO/R1UFHjxTvAbLOQWe3qMfIE758etqfe1fB+btLVcG5mzx9MqOviD97+u3j9DoijoxHnmJiSEtaHnHZOTqATKyeFiOcObOTdb1wUpZ4s7ZFubxyFV5L9SNL7RYb15u15uzbt/nB2W69+WO98eZF7dVL9WbuofhGPBXb5Kc98Yoq85j8aog/xd/iH/Hv1vbWr1unW+eS9f49JfO1KHy2fv8fMWgE3g==</latexit>

CDE , E[Y1,m

Y0,m ]NDE , E[Y1,M0

Y0,M0 ]

NIE , E[Y0,M1

Y0,M0 ]

E[Mt ] , E[M | do(T = t)]<latexit 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Brady Neal

Important Application: Mediation

20 / 25

Natural Direct and Indirect Effects
Subscript notation:
E[Yt,m ] , E[Y | do(T = t, M = m)]<latexit sha1_base64="1yAHxBcn3VnkWVDlUoO1Mh9n+Tc=">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</latexit>

CDE , E[Y1,m

Y0,m ]NDE , E[Y1,M0

Y0,M0 ]

NIE , E[Y0,M1

Y0,M0 ]

E[Mt ] , E[M | do(T = t)]<latexit 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Brady Neal

Recall problems with CDE:
• CDE is specific to thearbitrary choice of m
• How do we get the indirect
effect? Can’t just subtract theCDE from the total effectImportant Application: Mediation

20 / 25

Natural Direct and Indirect Effects
Subscript notation:
E[Yt,m ] , E[Y | do(T = t, M = m)]<latexit sha1_base64="1yAHxBcn3VnkWVDlUoO1Mh9n+Tc=">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</latexit>

CDE , E[Y1,m

Y0,m ]NDE , E[Y1,M0

Y0,M0 ]

NIE , E[Y0,M1

Y0,M0 ]TE = NDE

NIEr

E[Mt ] , E[M | do(T = t)]<latexit 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Brady Neal

Recall problems with CDE:
• CDE is specific to thearbitrary choice of m
• How do we get the indirect
effect? Can’t just subtract theCDE from the total effectImportant Application: Mediation

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Natural Direct and Indirect Effects
Subscript notation:
E[Yt,m ] , E[Y | do(T = t, M = m)]<latexit sha1_base64="1yAHxBcn3VnkWVDlUoO1Mh9n+Tc=">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</latexit>

CDE , E[Y1,m

Y0,m ]NDE , E[Y1,M0

Y0,M0 ]

NIE , E[Y0,M1

Y0,M0 ]TE = NDE

NIEr

E[Mt ] , E[M | do(T = t)]<latexit 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Recall problems with CDE:
• CDE is specific to thearbitrary choice of m
• How do we get the indirect
effect? Can’t just subtract theCDE from the total effectFor example in linear setting, TE = NDE + NIE<latexit sha1_base64="x2L7ZphSWAMU7s4uuvb1QudD3ec=">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</latexit>

Brady Neal

Important Application: Mediation

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Question:Show that TE = NDE
where NIEr , E[Y1,M0<latexit sha1_base64="+j4o6PLpcjUkGfkOlFFuMCD9KkY=">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</latexit><latexit sha1_base64="l5/7TzpQ691/gUdnhipLCDHYIBs=">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</latexit>

NIEr ,
Y1,M1 ].

Comparison of Controlled vs. Natural Mediation

Brady Neal

Important Application: Mediation

22 / 25

Comparison of Controlled vs. Natural Mediation
CDE can always be measured via experiments (do-operator), but it has noclear undirect effect since there is no decomposition

Brady Neal

Important Application: Mediation

22 / 25

Comparison of Controlled vs. Natural Mediation
CDE can always be measured via experiments (do-operator), but it has noclear undirect effect since there is no decomposition
NDE cannot always be measured via experiments since it is counterfactual,
but it allows for the complete decomposition of the total effect into the
NDE and NIE, which is what we’d like in mediation analysis

Brady Neal

Important Application: Mediation

22 / 25

When We Can Measure NDE and NIE

Brady Neal

Important Application: Mediation

23 / 25

When We Can Measure NDE and NIE
Adjustment set W

Brady Neal

Important Application: Mediation

23 / 25

When We Can Measure NDE and NIE
Adjustment set W
Sufficient conditions for identifying NDE:

Brady Neal

Important Application: Mediation

23 / 25

When We Can Measure NDE and NIE
Adjustment set W
Sufficient conditions for identifying NDE:1.

No member of W is a descendant of T

Brady Neal

Important Application: Mediation

23 / 25

When We Can Measure NDE and NIE
Adjustment set W
Sufficient conditions for identifying NDE:1.

No member of W is a descendant of T2.

W blocks all backdoor paths from M to Y

Brady Neal

Important Application: Mediation

23 / 25

When We Can Measure NDE and NIE
Adjustment set W
Sufficient conditions for identifying NDE:1.

No member of W is a descendant of T2.

W blocks all backdoor paths from M to Y<latexit sha1_base64="Kb6uX7WEoUg7/jP319SRIpxZvx0=">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</latexit>

NDE =

XX
m

w

(E[Y | do(T = 1, M = m), W = w]

E[Y | do(T = 0, M = m), W = w])⇥ P (M = m | do(T = 0), W = w)P (W = w)
XX=
(E[Y | T = 1, M = m, W = w] E[Y | T = 0, M = m, W = w])
mw⇥ P (M = m | T = 0, W = w)P (W = w)

Brady Neal

Important Application: Mediation

23 / 25

When We Can Measure NDE and NIE
Adjustment set W
Sufficient conditions for identifying NDE:1.

No member of W is a descendant of T2.

W blocks all backdoor paths from M to Y

3.

P (M = m | do(T = 0), W = w)
is identifiable (e.g. no unblockable backdoor paths from T to M)

4.

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NDE =

XX
m

w

(E[Y | do(T = 1, M = m), W = w]

E[Y | do(T = 0, M = m), W = w])⇥ P (M = m | do(T = 0), W = w)P (W = w)
XX=
(E[Y | T = 1, M = m, W = w] E[Y | T = 0, M = m, W = w])
mw⇥ P (M = m | T = 0, W = w)P (W = w)

Brady Neal

Important Application: Mediation

23 / 25

When We Can Measure NDE and NIE
Adjustment set W
Sufficient conditions for identifying NDE:1.

No member of W is a descendant of T2.

W blocks all backdoor paths from M to Y

3.

P (M = m | do(T = 0), W = w)
is identifiable (e.g. no unblockable backdoor paths from T to M)

4.

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E[Y | do(T = t, M = m), W = w]
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NDE =

XX
m

w

(E[Y | do(T = 1, M = m), W = w]

E[Y | do(T = 0, M = m), W = w])⇥ P (M = m | do(T = 0), W = w)P (W = w)
XX=
(E[Y | T = 1, M = m, W = w] E[Y | T = 0, M = m, W = w])
mw⇥ P (M = m | T = 0, W = w)P (W = w)

Brady Neal

Important Application: Mediation

23 / 25

When We Can Measure NDE and NIE
Adjustment set W
Sufficient conditions for identifying NDE:1.

No member of W is a descendant of T2.

W blocks all backdoor paths from M to Y

3.

P (M = m | do(T = 0), W = w)
is identifiable (e.g. no unblockable backdoor paths from T to M)

4.

<latexit sha1_base64="vD3uqHWNg4ZEmqwjjMpAZ9qMZqk=">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</latexit>

E[Y | do(T = t, M = m), W = w]
is identifiable (e.g. no unblockable backdoors paths from T to Y)<latexit sha1_base64="KlR9BwAmfmQZVeSppeNMaipszf4=">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</latexit><latexit sha1_base64="Kb6uX7WEoUg7/jP319SRIpxZvx0=">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</latexit>

NDE =

XX
m

w

(E[Y | do(T = 1, M = m), W = w]

E[Y | do(T = 0, M = m), W = w])⇥ P (M = m | do(T = 0), W = w)P (W = w)
XX=
(E[Y | T = 1, M = m, W = w] E[Y | T = 0, M = m, W = w])
mw⇥ P (M = m | T = 0, W = w)P (W = w)

Brady Neal

Important Application: Mediation

23 / 25

When We Can Measure NDE and NIE
Adjustment set W
Sufficient conditions for identifying NDE:

TE = NDE<latexit sha1_base64="l56fD28Cp0nOxtUJXhZefnKJmBA=">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</latexit>

NIEr

1.

No member of W is a descendant of T2.

W blocks all backdoor paths from M to Y

3.

P (M = m | do(T = 0), W = w)
is identifiable (e.g. no unblockable backdoor paths from T to M)

4.

<latexit sha1_base64="vD3uqHWNg4ZEmqwjjMpAZ9qMZqk=">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</latexit>

E[Y | do(T = t, M = m), W = w]
is identifiable (e.g. no unblockable backdoors paths from T to Y)<latexit sha1_base64="KlR9BwAmfmQZVeSppeNMaipszf4=">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</latexit><latexit sha1_base64="Kb6uX7WEoUg7/jP319SRIpxZvx0=">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</latexit>

NDE =

XX
m

w

(E[Y | do(T = 1, M = m), W = w]

E[Y | do(T = 0, M = m), W = w])⇥ P (M = m | do(T = 0), W = w)P (W = w)
XX=
(E[Y | T = 1, M = m, W = w] E[Y | T = 0, M = m, W = w])
mw⇥ P (M = m | T = 0, W = w)P (W = w)

Brady Neal

Important Application: Mediation

23 / 25

Question:
Come up with your own example of mediationand the corresponding graph. Then, determine
whether you can identify the NDE and NIE
from observational data.

Path-Specific Effects
Measure causal effects along arbitrary path or set of paths in the causal
graph
See “Identifiability of Path-Specific Effects” (Avin et al., 2005)

Brady Neal

Important Application: Mediation

25 / 25