Counterfactual Reasoning in Algorithmic Fairness
|
|
- Myron Ford
- 5 years ago
- Views:
Transcription
1 Counterfactual Reasoning in Algorithmic Fairness Ricardo Silva University College London and The Alan Turing Institute Joint work with Matt Kusner (Warwick/Turing), Chris Russell (Sussex/Turing), and Joshua Loftus (NYU)
2 Fairness and Machine Learning The dream: if we teach machines to perform sensitive decisions, they will not suffer from human biases. The reality: the GIGO principle still holds, regardless of whether we are talking of statistical models or software.
3 The Message There is only so much data alone can tell you about fairness. I m not talking about just value judgments. We should highlight the role that the datagenerating causal process has in shaping our notions of fairness.
4 Nobody is Saying This is Easy At no point I will suggest that building a causal model is easy. Some untested and untestable assumptions will be needed. The idea is to make your assumptions as explicit as possible, hopefully being less wrong in the end.
5 <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="hp+6lruf2d3tzaldqaqqvekmxyw=">aaab2xicbzdnsgmxfixv1l86vq1rn8eiucozbnqpuhfzwbzco5rm5k4bmskmyr2hdh0bf25efc93vo3pz0jbdwq+zknivscullqubn9ebwd3b/+gfugfnfzjk9nmo2fz0gjsilzl5jnmfpxu2cvjcp8lgzylffbj6f0i77+gstlxtzqrmmr4wmtuck7o6oyaraadlmw2ivxdc9yanb+gss7kddujxa0dhefbucunsafw7g9liwuxuz7ggupnm7rrtrxzzi6dk7a0n+5oykv394ukz9bostjdzdhn7ga2mp/lbiwlt1eldvesarh6kc0vo5wtdmajnchizrxwyasblykjn1yqa8z3hysbg29d77odbu3wmya6nmmfxeein3ahd9cblghi4bxevyn35n2suqp569lo4i+8zx84xio4</latexit> <latexit sha1_base64="h0vkymv1txiryc/6suot3b8szzo=">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</latexit> <latexit sha1_base64="h0vkymv1txiryc/6suot3b8szzo=">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</latexit> <latexit sha1_base64="jgc96zbtujoycxypdl+iax6pzeu=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="hp+6lruf2d3tzaldqaqqvekmxyw=">aaab2xicbzdnsgmxfixv1l86vq1rn8eiucozbnqpuhfzwbzco5rm5k4bmskmyr2hdh0bf25efc93vo3pz0jbdwq+zknivscullqubn9ebwd3b/+gfugfnfzjk9nmo2fz0gjsilzl5jnmfpxu2cvjcp8lgzylffbj6f0i77+gstlxtzqrmmr4wmtuck7o6oyaraadlmw2ivxdc9yanb+gss7kddujxa0dhefbucunsafw7g9liwuxuz7ggupnm7rrtrxzzi6dk7a0n+5oykv394ukz9bostjdzdhn7ga2mp/lbiwlt1eldvesarh6kc0vo5wtdmajnchizrxwyasblykjn1yqa8z3hysbg29d77odbu3wmya6nmmfxeein3ahd9cblghi4bxevyn35n2suqp569lo4i+8zx84xio4</latexit> <latexit sha1_base64="h0vkymv1txiryc/6suot3b8szzo=">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</latexit> <latexit sha1_base64="h0vkymv1txiryc/6suot3b8szzo=">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</latexit> <latexit sha1_base64="jgc96zbtujoycxypdl+iax6pzeu=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">aaac2hicdvfnj9mwehxc11k+chy5wfqgdqhkebki0wixjotedwtnvu2cswpvssn7sqwkineaia78ng78cv4ctjdc291ltk9vnp9n3qsvko6i6hcqxrp85eq1veudgzdv3b4zvhvv0jnacpwio4ydpubqsy0tkqrwwlmemlv4lk7edv2jy7rogv2bnhxos1hqmusb5knf8m/0fx/me8lpzmugj1hwrzmjug3rtu+5sryirexrqsdnzkfzqtn5llmavjskg9c7blrgb1jzqygis1mg7yuoh//jqgcxsh6idqhe18jxplyzv3kfnaz3e2zsuoeqdijdgx/qv0g2or9u2w4ww1e0jrbfz4o4bypw18fi+cvjjkhl1cquodelo4rmdvisqme7sgqhfygvlhhmoyys3bzzhqbljzyt8dyhmbtnfmueftfa6dymtl2ybcrc2v5hxtsb1zs/nddsvz5vlu4+ymvvxdjdmwfs+uzvxgmqvvpzusjagr+hdv0i8dmvz4pdz+m4gsfvn4/23/rx7leh7cf7wml2gu2zd+yatzgijketfa2+hz/cl+h38mejnaz6n/fztou//wjnmos+</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">aaac2hicdvfnj9mwehxc11k+chy5wfqgdqhkebki0wixjotedwtnvu2cswpvssn7sqwkineaia78ng78cv4ctjdc291ltk9vnp9n3qsvko6i6hcqxrp85eq1veudgzdv3b4zvhvv0jnacpwio4ydpubqsy0tkqrwwlmemlv4lk7edv2jy7rogv2bnhxos1hqmusb5knf8m/0fx/me8lpzmugj1hwrzmjug3rtu+5sryirexrqsdnzkfzqtn5llmavjskg9c7blrgb1jzqygis1mg7yuoh//jqgcxsh6idqhe18jxplyzv3kfnaz3e2zsuoeqdijdgx/qv0g2or9u2w4ww1e0jrbfz4o4bypw18fi+cvjjkhl1cquodelo4rmdvisqme7sgqhfygvlhhmoyys3bzzhqbljzyt8dyhmbtnfmueftfa6dymtl2ybcrc2v5hxtsb1zs/nddsvz5vlu4+ymvvxdjdmwfs+uzvxgmqvvpzusjagr+hdv0i8dmvz4pdz+m4gsfvn4/23/rx7leh7cf7wml2gu2zd+yatzgijketfa2+hz/cl+h38mejnaz6n/fztou//wjnmos+</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="hp+6lruf2d3tzaldqaqqvekmxyw=">aaab2xicbzdnsgmxfixv1l86vq1rn8eiucozbnqpuhfzwbzco5rm5k4bmskmyr2hdh0bf25efc93vo3pz0jbdwq+zknivscullqubn9ebwd3b/+gfugfnfzjk9nmo2fz0gjsilzl5jnmfpxu2cvjcp8lgzylffbj6f0i77+gstlxtzqrmmr4wmtuck7o6oyaraadlmw2ivxdc9yanb+gss7kddujxa0dhefbucunsafw7g9liwuxuz7ggupnm7rrtrxzzi6dk7a0n+5oykv394ukz9bostjdzdhn7ga2mp/lbiwlt1eldvesarh6kc0vo5wtdmajnchizrxwyasblykjn1yqa8z3hysbg29d77odbu3wmya6nmmfxeein3ahd9cblghi4bxevyn35n2suqp569lo4i+8zx84xio4</latexit> <latexit sha1_base64="h0vkymv1txiryc/6suot3b8szzo=">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</latexit> <latexit sha1_base64="h0vkymv1txiryc/6suot3b8szzo=">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</latexit> <latexit sha1_base64="jgc96zbtujoycxypdl+iax6pzeu=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">aaac2hicdvfnj9mwehxc11k+chy5wfqgdqhkebki0wixjotedwtnvu2cswpvssn7sqwkineaia78ng78cv4ctjdc291ltk9vnp9n3qsvko6i6hcqxrp85eq1veudgzdv3b4zvhvv0jnacpwio4ydpubqsy0tkqrwwlmemlv4lk7edv2jy7rogv2bnhxos1hqmusb5knf8m/0fx/me8lpzmugj1hwrzmjug3rtu+5sryirexrqsdnzkfzqtn5llmavjskg9c7blrgb1jzqygis1mg7yuoh//jqgcxsh6idqhe18jxplyzv3kfnaz3e2zsuoeqdijdgx/qv0g2or9u2w4ww1e0jrbfz4o4bypw18fi+cvjjkhl1cquodelo4rmdvisqme7sgqhfygvlhhmoyys3bzzhqbljzyt8dyhmbtnfmueftfa6dymtl2ybcrc2v5hxtsb1zs/nddsvz5vlu4+ymvvxdjdmwfs+uzvxgmqvvpzusjagr+hdv0i8dmvz4pdz+m4gsfvn4/23/rx7leh7cf7wml2gu2zd+yatzgijketfa2+hz/cl+h38mejnaz6n/fztou//wjnmos+</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="hp+6lruf2d3tzaldqaqqvekmxyw=">aaab2xicbzdnsgmxfixv1l86vq1rn8eiucozbnqpuhfzwbzco5rm5k4bmskmyr2hdh0bf25efc93vo3pz0jbdwq+zknivscullqubn9ebwd3b/+gfugfnfzjk9nmo2fz0gjsilzl5jnmfpxu2cvjcp8lgzylffbj6f0i77+gstlxtzqrmmr4wmtuck7o6oyaraadlmw2ivxdc9yanb+gss7kddujxa0dhefbucunsafw7g9liwuxuz7ggupnm7rrtrxzzi6dk7a0n+5oykv394ukz9bostjdzdhn7ga2mp/lbiwlt1eldvesarh6kc0vo5wtdmajnchizrxwyasblykjn1yqa8z3hysbg29d77odbu3wmya6nmmfxeein3ahd9cblghi4bxevyn35n2suqp569lo4i+8zx84xio4</latexit> <latexit sha1_base64="h0vkymv1txiryc/6suot3b8szzo=">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</latexit> <latexit sha1_base64="h0vkymv1txiryc/6suot3b8szzo=">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</latexit> <latexit sha1_base64="jgc96zbtujoycxypdl+iax6pzeu=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="hp+6lruf2d3tzaldqaqqvekmxyw=">aaab2xicbzdnsgmxfixv1l86vq1rn8eiucozbnqpuhfzwbzco5rm5k4bmskmyr2hdh0bf25efc93vo3pz0jbdwq+zknivscullqubn9ebwd3b/+gfugfnfzjk9nmo2fz0gjsilzl5jnmfpxu2cvjcp8lgzylffbj6f0i77+gstlxtzqrmmr4wmtuck7o6oyaraadlmw2ivxdc9yanb+gss7kddujxa0dhefbucunsafw7g9liwuxuz7ggupnm7rrtrxzzi6dk7a0n+5oykv394ukz9bostjdzdhn7ga2mp/lbiwlt1eldvesarh6kc0vo5wtdmajnchizrxwyasblykjn1yqa8z3hysbg29d77odbu3wmya6nmmfxeein3ahd9cblghi4bxevyn35n2suqp569lo4i+8zx84xio4</latexit> <latexit sha1_base64="h0vkymv1txiryc/6suot3b8szzo=">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</latexit> <latexit sha1_base64="h0vkymv1txiryc/6suot3b8szzo=">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</latexit> <latexit sha1_base64="jgc96zbtujoycxypdl+iax6pzeu=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> <latexit sha1_base64="gz8rxw10rufhocup5exuumqzf3s=">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</latexit> The Scope of this Talk We consider prediction and intervention problems (more of the former). In prediction problems, we will have: X : features, or attributes of an individual A :theprotected attributes of an individual Y : thetarget, what we would like to predict Ŷ : our prediction
6 Prediction Problems Prediction here means inferring a property Y that will be used for decision making. For example: Y = 1 means this person will default on a loan (for the decision, should I give this person a loan?) Y = 1 means this person will commit a crime in two years (for the decision, should I release this convict now? ) We would like to predict Y in a fair way, meaning that our predictions should not be biased against particular instances of A.
7 Ŷ<latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> Primitives Even if we take the choice of what goes in A as a primitive, it is still not obvious what we mean by being fair. A first idea: ensure that does not use A. This is known to be unsatisfactory.
8 Ŷ<latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> Ŷ<latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> Ŷ<latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> <latexit sha1_base64="m+zgxsf9wvacj2hahalwrb9eawm=">aaab7xicbvdlsgnbeoz1gemr6thlyba8hv0r9bj04jgceuiyhnnjbdjmdmaz6rvcyd948aciv//hm3/jjnmdjhy0ffxddhdfqrqwff/bw1ldw9/ylgwvt3d29/zlb4cnqzpdej1pqu0ropzloxgdburesg2nssr5mxretp3mezdwahwpo5shce0reqtg0umnzoaieeiwyn7fn4eskyanzchr65a+oj3nsoqrzjja2w78fmmxnsiy5jnij7m8pwxi+7ztqkijt+f4du2endqlr2jtxckkm/x3xjgm1o6syhumfad20zuk/3ntdoorccxumifxbl4ozirbtaavk54wnkecoukzee5wwgbuuiyuokilivh8ezk0ziubxwnulsrv6zyoahzdczxbajdqhvuoqr0ypmizvmkbp70x7937mleuepnmefyb9/kdefyoya==</latexit> Examples Equalized odds: given the outcome Y, attribute A provides no further information about my prediction. Calibration: given my prediction, attribute A provides no further information about the outcome Y. If A is on average informative of Y, we cannot reconcile the above. Remember, here we do not control Y (directly). We decide on predictor.
9 Putting It in the Context of a Causal Model A toy model: imagine A is race, X is owns a red car and Y is crashes car in one year. Let s (informally) draw a causal diagram showing cause-effect relationships among those. It will include a unobserved trait U measuring aggressiveness. We will get into more formal definitions later.
10 A Causal Diagram
11 Some Initial Conclusions A is not a cause of Y. If we build a predictor based on X, it tells us something both about A and about U. Hence, our predictor will be different for different values of A, which does not seem appropriate.
12 A Second Causal Diagram
13 Which Conclusions? A is now a cause of Y (indirectly). It is now impossible to satisfy both equalized odds and calibration simultaneously. Judgment call: is the pathway A è X èy fair?
14 Zooming In, with Another Example A here stands for race, Y for loan default. Same idea, augmented with a mediator:
15 A Causal Primitive: Counterfactual Fairness If we have some protected attribute like race, and a decision such as length of sentence, then our decision satisfies counterfactual fairness if had the protected attributes (e.g., race) of the individual been different, other things being equal, the decision would have remained the same A causal model is necessary to infer such claims from data.
16 Workflow Regardless of the machine learning algorithm to be used, work with a domain expert to estimate a causal model of your data. It s a model of the world, not of your software. Choose any machine learning algorithm of interest, any black-box that takes as inputs observed and unobserved variables in your domain. Select a set of variables based on which sets respect counterfactual fairness. If necessary, infer unobserved variables from the observed ones.
17 Formalizing the Idea Formal notions of counterfactuals date back at least to Jerzy Neyman in the 1920s. I will follow mostly the Structural Causal Model (SCM) framework of Judea Pearl, which has close links to the work of James Robins, and that of Spirtes, Glymour and Scheines.
18 Structural Causal Models A directed acyclic graph (DAG) postulates direct cause-effect pairs. Each vertex in the graph is a random variable in a distribution. Each variable V is given an equation that deterministically defines the value of V as a function of its parents. Such equations are postulated to be structural, in the sense that it follows the cause-effect direction.
19 The Operational Meaning This DAG with equations is causal in the sense that it must encode the effects of a perfect intervention. U Rain U Barometer Rain = f R (U Rain ) Barometer = f B (Rain, U Barometer ) Rain Barometer
20 Interventions Another primitive. It is a overriding operator, sets a variable to a fixed value of interest. Lower case here represents constants. U Rain U Barometer Rain = r Barometer = f B (r, U Barometer ) r Barometer
21 Interventions It is the notion of intervention that leads to the asymmetric nature of causality. U Rain U Barometer Rain = f R (U Rain ) Barometer = b Rain b
22 Interventions It is the notion of intervention that explains why correlation is not causation. Surviving old age Surviving old age Moving to Florida Dying of old age yes Dying of old age
23 Notation: the do Operator We must express how Dying of old age varies with Moving to Florida in both cases. Traditionally, conditional probabilities can be used for that. But notice that, in our example, P(Dying of old age = True Moving to Florida = True) P(Dying of old age = True Moving to Florida = False) is true in the observational case (no intervention), but false in the interventional case.
24 Notation: the do Operator In Pearl s calculus, this is distinguished by using the do operator to indicate an intervention as opposed to an observation. P(Dying of old age = True Moving to Florida = True) P(Dying of old age = True Moving to Florida = False) P(Dying of old age = True do(moving to Florida = True)) = P(Dying of old age = True do(moving to Florida = False))
25 Averages Vs. Individuals This type of notation can be used to express whether a drug is effective or not, averaging over a population, using a randomized controlled trial: P(Healthy = True do(treatment = Drug))?= P(Healthy = True do(treatment = Placebo)) It does not make any claims, however, on whether there is a balance of positive/ negative cases that cancel out.
26 Notation: Counterfactual Indices Meant to capture individual-level variability. For V j a variable in the system, and V i a variable being intervened at value v, we use V j (v) as the counterfactual value of V j, had V i being set to v. Context will tell us which variable the value v refers to.
27 Example Notice: it is common to represent V j (v) as just V j if V i is not a (direct or indirect) cause of V j. U rain (r) U barometer (r) U rain U barometer r Barometer(r) r Barometer(r)
28 Other Things Being Equal That is, A counterfactual value replaces the cause of interest The counterfactual value propagates downstream the causal graph via the structural equations Everything else remains the same ( other things being equal ), i.e., the non-descendants of the manipulated variable.
29 Multiple Worlds A counterfactual is just a different version of the same individual. All versions co-exist in one big joint distribution. U Rain U Barometer Rain Barometer r Barometer(r) r Barometer(r )
30 Workflow Regardless of the machine learning algorithm to be used, work with a domain expert to estimate a causal model of your data. It s a model of the world, not of your software. Choose any machine learning algorithm of interest, any black-box that takes as inputs observed and unobserved variables in your domain. Select a set of variables based on which sets respect counterfactual fairness. If necessary, infer unobserved variables from the observed ones.
31 Back to Counterfactual Fairness The law of counterfactual propagation means that, if we want to ensure P (Ŷ (a) =y A = a, X = x) =P (Ŷ (a0 )=y A = a, X = x) <latexit sha1_base64="ika4uwrxxdg1jbktp7zsue5mo/g=">aaacl3icdvdlsgmxfm34rpu16tjnsigvpmyiobuhkojlcvyhnahcsdm2npmgyyhd7r+58ve6evherx9hpi2orr4iozxzlsk9xsszvjb1yszmzs0vlgawsssrq2vr5szmryaxilrmqh6kmgeschbqsmkk01okkpgep1wve5h61tsqjaudg5ve1pwhhbawi6c01davs3mnawrf5mefn+lewq8optmmdnbnx/ep+p3z+yfumhnwwroctxn7thjojfldhdjnkmq+drthigxdtill9kaorjjtz51y0ghif9q0rmkappvub7hvh+9qpylbodanuhio/pzogs9l4ns66ypqyekvff/y6rfqnbg9fksxogezpdskovyhtsvdtsyoutzrbihg+q+ydeaaubrirc7bnlx5mlqoc7zvsk+pcsxzcr0zti12ub7z6bgv0ruqotii6ben0ct6m56mz+pd+bhfz4zxzbb6bepzc/vyork=</latexit> <latexit sha1_base64="ika4uwrxxdg1jbktp7zsue5mo/g=">aaacl3icdvdlsgmxfm34rpu16tjnsigvpmyiobuhkojlcvyhnahcsdm2npmgyyhd7r+58ve6evherx9hpi2orr4iozxzlsk9xsszvjb1yszmzs0vlgawsssrq2vr5szmryaxilrmqh6kmgeschbqsmkk01okkpgep1wve5h61tsqjaudg5ve1pwhhbawi6c01davs3mnawrf5mefn+lewq8optmmdnbnx/ep+p3z+yfumhnwwroctxn7thjojfldhdjnkmq+drthigxdtill9kaorjjtz51y0ghif9q0rmkappvub7hvh+9qpylbodanuhio/pzogs9l4ns66ypqyekvff/y6rfqnbg9fksxogezpdskovyhtsvdtsyoutzrbihg+q+ydeaaubrirc7bnlx5mlqoc7zvsk+pcsxzcr0zti12ub7z6bgv0ruqotii6ben0ct6m56mz+pd+bhfz4zxzbb6bepzc/vyork=</latexit> <latexit sha1_base64="ika4uwrxxdg1jbktp7zsue5mo/g=">aaacl3icdvdlsgmxfm34rpu16tjnsigvpmyiobuhkojlcvyhnahcsdm2npmgyyhd7r+58ve6evherx9hpi2orr4iozxzlsk9xsszvjb1yszmzs0vlgawsssrq2vr5szmryaxilrmqh6kmgeschbqsmkk01okkpgep1wve5h61tsqjaudg5ve1pwhhbawi6c01davs3mnawrf5mefn+lewq8optmmdnbnx/ep+p3z+yfumhnwwroctxn7thjojfldhdjnkmq+drthigxdtill9kaorjjtz51y0ghif9q0rmkappvub7hvh+9qpylbodanuhio/pzogs9l4ns66ypqyekvff/y6rfqnbg9fksxogezpdskovyhtsvdtsyoutzrbihg+q+ydeaaubrirc7bnlx5mlqoc7zvsk+pcsxzcr0zti12ub7z6bgv0ruqotii6ben0ct6m56mz+pd+bhfz4zxzbb6bepzc/vyork=</latexit> <latexit sha1_base64="ika4uwrxxdg1jbktp7zsue5mo/g=">aaacl3icdvdlsgmxfm34rpu16tjnsigvpmyiobuhkojlcvyhnahcsdm2npmgyyhd7r+58ve6evherx9hpi2orr4iozxzlsk9xsszvjb1yszmzs0vlgawsssrq2vr5szmryaxilrmqh6kmgeschbqsmkk01okkpgep1wve5h61tsqjaudg5ve1pwhhbawi6c01davs3mnawrf5mefn+lewq8optmmdnbnx/ep+p3z+yfumhnwwroctxn7thjojfldhdjnkmq+drthigxdtill9kaorjjtz51y0ghif9q0rmkappvub7hvh+9qpylbodanuhio/pzogs9l4ns66ypqyekvff/y6rfqnbg9fksxogezpdskovyhtsvdtsyoutzrbihg+q+ydeaaubrirc7bnlx5mlqoc7zvsk+pcsxzcr0zti12ub7z6bgv0ruqotii6ben0ct6m56mz+pd+bhfz4zxzbb6bepzc/vyork=</latexit> it is sufficient (and necessary, in general) to include only the non-descendants of A in the definition of the predictor.
32 Examples A, X cannot be used. U can. A, Employed cannot be used. Prejudiced and Qualifications can. If it is judged that Prejudiced cannot be used, it should be labelled as a protected attributed.
33 How to Extract Unobserved Variables? Use the factual distribution to get a distribution over the unobserved variables by standard probabilistic conditioning. P(Unobserved Observed) Monte Carlo data augmentation approach: replace each data point in your training sample by a set of training points with the unobserved variables being filled by a Monte Carlo sample.
34 Workflow Regardless of the machine learning algorithm to be used, work with a domain expert to estimate a causal model of your data. It s a model of the world, not of your software. Choose any machine learning algorithm of interest, any black-box that takes as inputs observed and unobserved variables in your domain. Select a set of variables based on which sets respect counterfactual fairness. If necessary, infer unobserved variables from the observed ones.
35 Algorithm
36 Interpretation Extract causes of Y which are not mediators between A and Y. Find the best approximation to Y within the space of functions that exclude such mediators. Even if Y is unfair (A is a cause of it), by construction the predictor will be counterfactually fair.
37 Challenges Counterfactual fairness clarifies that algorithmic fairness in general is not explicitly modeling how the word becomes fairer with fair predictions. Even if our decision of giving a loan is fair, it doesn t mean that in aggregate the probability of a person of a particular demographic group won t have difficulties in repaying it (A still causes Y). The delayed impact of fair predictions is also a research topic see Liu et al.,
38 Workflow Regardless of the machine learning algorithm to be used, work with a domain expert to estimate a causal model of your data. It s a model of the world, not of your software. Choose any machine learning algorithm of interest, any black-box that takes as inputs observed and unobserved variables in your domain. Select a set of variables based on which sets respect counterfactual fairness. If necessary, infer unobserved variables from the observed ones.
39 Some Words of Caution Structural equations use unobserved variables. It is common that some of these variables are default choices based on some generic modeling assumption such as additive errors. Output = Signal + Noise Nature and society couldn t care less whether your mathematically convenient way of separating signal and noise is elegant or not.
40 Some Words of Caution That is, there are infinitely many structural equations V j = f j (V i, U j ) compatible with P(V i V j ) and P(V i do(v j )). Signal vs. noise must be determined by realworld assumptions ( simplicity assumptions, of the Ockham s razor type, can be sometimes adequate as long as caveats are advertised).
41 Some Words of Caution By now, there are several good papers on how to tackle fairness by generating unobserved variables which are independent of A, using assorted methods. U A Then off you go to plug-in U on a machine learning algorithm! X
42 However There are papers not causally motivated, which I find of difficult interpretation. Remember: there are infinitely many ways of extracting U. There are papers causally motivated, but which commit themselves to a domain-free family of structural equations. OK enough, but why would you do that? Counterfactual fairness emphasizes that the causal modeling step is separate from the prediction learning process. Finally, do beware of any paper that claims to do assumption-free extraction of causal latent factors. Those are selling you snake oil!
43 Interpretation of Counterfactuals But what does it mean to say had my race been different?? First, make sure to understand the difference between A and Perception of A : these can lead to conceptually different interpretations, even if the model stays the same. Without going in details, if those counterfactuals make you feel uneasy, just interpret them as comparing two different people who happen to match on the other things being equal factors. This is also related to fairness through awareness (Dwork et al., 2011,
44 Non-Counterfactual Causal Models Contrary to folk knowledge, causality does not require counterfactuals: the do operator is an way of comparing treatments without comparing individuals. However, if features X are affected by A, then in general there is no individual where P(Y = y X = x, do(a = a)) = P(Y = y X = x, do(a = a )) If features X are not affected by A, then we can show we do not need to explicitly model structural equations anyway!
45 The Upside Because structural causal models rely on unobserved variables, at least they can be partially falsified by eventually measuring some of those variables. Just keep in mind: while it is preposterous to say you have the causal model of a social process, you should (must?) be able to explain your assumptions to a regulator or a customer. Having passed testable implications, the remaining components of a counterfactual model should be understood as conjectures formulated according to the best of our knowledge. Such models should always be deemed provisional and prone to modifications.
46 Illustration The Law School Admission Council conducted a survey across 163 law schools in the United States It contains information on 21,790 law students such as their entrance exam scores (LSAT), their grade-point average (GPA) collected prior to law school, and their first year average grade (FYA). Task: predict if an applicant will have a high FYA Example of decision problem: make an offer
47 Setup I will present some simple causal models for this domain, which by no means I intend to sell as well-thought models. Their purpose is for illustration. We will fit real data to a model, then generate synthetic counterfactuals out of it. The point is to quantify to what extent a causally-oblivious method violates counterfactual fairness.
48 Two Models Fair K Fair Add
49 Predictive Error (Real Data) Comparison against Full (linear model with all variables) and Unaware (linear model without race and gender, but the other two predictors) Evaluation by root mean squared error
50 Fairness Violations (Simulated Counterfactuals)
51 Extension: Using Multiple Models We just saw two different counterfactual models that give different predictions despite being undistinguishable given the same data. This is OK assuming little difference between models, but we may have competing theories with some sizeable difference. We would like to be approximately counterfactually fair to all of them.
52 <latexit sha1_base64="6ftp5coo2gomwvzhxhd+kdhdw4c=">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</latexit> <latexit sha1_base64="6ftp5coo2gomwvzhxhd+kdhdw4c=">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</latexit> <latexit sha1_base64="6ftp5coo2gomwvzhxhd+kdhdw4c=">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</latexit> <latexit sha1_base64="6ftp5coo2gomwvzhxhd+kdhdw4c=">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</latexit> (ε, δ)- Counterfactual Fairness The following constraint provides a relaxation of counterfactual fairness: P ( Ŷ (a) Ŷ (a0 ) apple A = a, X = a) 1 The idea is to simultaneously satisfy such constraints according to different counterfactual models. It is not hard to show that this problem in general has no solution if ε = 0, hence the need for an approximate version.
53 Law School Revisited
54 COMPAS
55 A Different Direction: Interventions So far, we have solely discussed the creation of predictors. Ideally, we would like to destroy the pathways between A and Y, the outcome of interest. This is in general not possible. But let s assume we have an intervention with the ability of changing the contribution of A to Y. How is related to counterfactual fairness?
56 Imperfect Interventions and Interference We will assume two generalizations of the concept of intervention used so far. An intervention is represented generically as a set of (action) variables, which here I will denote as Z. We can define Z = 0 as the no action choice! Z 0 just means that one or more structural equations will change, not necessarily to a constant ( imperfect, or soft intervention).
57 Ideally Having available some Z = z which completely overrides the structural equation for Y to not depend on anything that starts on A. Z A Y
58 In Reality No such an intervention is typically available. And this is not a prediction problem anymore. What happens to Y? Setup: Assume Y is encoded so that high values are good. Model allows for interference: that is, treatment Z i given to person i might affect person j. How is this related to counterfactual fairness?
59 <latexit sha1_base64="xwnizrs8+qu/6klsoqwwnz6cx5y=">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</latexit> <latexit sha1_base64="xwnizrs8+qu/6klsoqwwnz6cx5y=">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</latexit> <latexit sha1_base64="xwnizrs8+qu/6klsoqwwnz6cx5y=">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</latexit> <latexit sha1_base64="xwnizrs8+qu/6klsoqwwnz6cx5y=">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</latexit> <latexit sha1_base64="4c9mlclgudbw9ia5ysx9ikequku=">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</latexit> <latexit sha1_base64="4c9mlclgudbw9ia5ysx9ikequku=">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</latexit> <latexit sha1_base64="4c9mlclgudbw9ia5ysx9ikequku=">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</latexit> <latexit sha1_base64="4c9mlclgudbw9ia5ysx9ikequku=">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</latexit> <latexit sha1_base64="we8pnaxqrv0xhrakialrilrid8q=">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</latexit> <latexit sha1_base64="we8pnaxqrv0xhrakialrilrid8q=">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</latexit> <latexit sha1_base64="we8pnaxqrv0xhrakialrilrid8q=">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</latexit> <latexit sha1_base64="hp+6lruf2d3tzaldqaqqvekmxyw=">aaab2xicbzdnsgmxfixv1l86vq1rn8eiucozbnqpuhfzwbzco5rm5k4bmskmyr2hdh0bf25efc93vo3pz0jbdwq+zknivscullqubn9ebwd3b/+gfugfnfzjk9nmo2fz0gjsilzl5jnmfpxu2cvjcp8lgzylffbj6f0i77+gstlxtzqrmmr4wmtuck7o6oyaraadlmw2ivxdc9yanb+gss7kddujxa0dhefbucunsafw7g9liwuxuz7ggupnm7rrtrxzzi6dk7a0n+5oykv394ukz9bostjdzdhn7ga2mp/lbiwlt1eldvesarh6kc0vo5wtdmajnchizrxwyasblykjn1yqa8z3hysbg29d77odbu3wmya6nmmfxeein3ahd9cblghi4bxevyn35n2suqp569lo4i+8zx84xio4</latexit> <latexit sha1_base64="z/0gulhd2yytxjooab1lgkjlzsa=">aaac0hicbvllbtqwfhxcq4qca1s2v1tqikwjpbsqo0ouhgbzjkydna7wjcdprtpogjuomyfswiaqx8eo3+alccyjavouzon4npu201jjy6pop+dfunzl6rwn68gnzzu3bg/ubb6aoq64gpncfdukrsou1gjspvviulyc81sjo/t0racffrcvkyv+a+elshi81jkthk2j2oaxzfgcnqswh0dvrlamhaxtlvbt56yre3h7xj+ki3b+9irn4ex0hzm7/s2dxsmkhynsmwl7geygmfnccyytlw9torpx9rqobuspdczqdl2d0xryhquispzbc6c60hweigqcf6e9co643fy6tvh38aouo6youcnababs9w1yvlafglwycbiasid/0gybg2xfw2hpcbhek7bfvnbabj/orob1lrtlco2zxlfpkwyrk7ksbubri0rkp3gspg5qzivjmuxbtpdamtnwrbqjlszzvymazi2z56nz7ayw61ph/k+b1jz7kjrsl7uvmvefslqblab7ajjjsncr5g4gr6trffgjvsit+w7deul1ks+cw91hha3jnxhzipfifbjdyvkypcovyqeze+5nve/ef++rn/if/w/9unxvtbe75b/zv/8gccnccq==</latexit> <latexit sha1_base64="z/0gulhd2yytxjooab1lgkjlzsa=">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</latexit> <latexit sha1_base64="0tqt7/ceh+4oxoofio+lz2mzclc=">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</latexit> <latexit sha1_base64="we8pnaxqrv0xhrakialrilrid8q=">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</latexit> <latexit sha1_base64="we8pnaxqrv0xhrakialrilrid8q=">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</latexit> <latexit sha1_base64="we8pnaxqrv0xhrakialrilrid8q=">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</latexit> <latexit sha1_base64="we8pnaxqrv0xhrakialrilrid8q=">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</latexit> <latexit sha1_base64="we8pnaxqrv0xhrakialrilrid8q=">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</latexit> <latexit sha1_base64="we8pnaxqrv0xhrakialrilrid8q=">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</latexit> Optimization Problem and Constraints Main family of constraints: E[Y i (a i, z) A i = a i,x i = x i ] E[Y i (a 0, z) A i = a i,x i = x i ] {z } G ia 0 < (as opposed to E[Y i (a i, z) A i = a i,x i = x i ] E[Y i (a 0, z) A i = a i,x i = x i ] < ) max z 1,...,z n s.t., nx i=1 nx i=1 E[Y i (z) A i = a i,x i = x i ] z i apple B G ia 0 apple 8a 0 2 A,i2 1,...,n,
60 Intuitive Toy Example Protected attribute A is such that A in {b, w}, X is some quantitative measure of professional competence, and Y is a measure of wealth in 5 years time. Z i = 1 means individual i gets a subsidy to move to a neighborhood with better transport links.
61 <latexit sha1_base64="g6o0fzji/vpizil8s528m+cdgfg=">aaacjxicbvbns8naen3ur1q/oh69lbahiprefd0ovl3oryjpq00im+22xbr5yhejlna/48w/4swdrqrp/hu3aq7aora7jzdvmjnnrywkarhfwmfufmfxqbhcwlldw9/qn7caiow5jhyowchbhhke0ybykkpgwhenypcyaxqdq7tefcrc0dc4k8oiod7qbbrlmzkkcvwze5fcc9hs/we0demha8dztix1iyc2j2tf85kbueuiez/tk4nl0pkrl42qkqwcbwyoyicpuqup7u6iy58eejmkrns0iukkieukgrmv7fiqcoeb6pg2ggfs850ku3ie9xttgd2qqxdimlg/oxlkczh0pavmnxbttzt8r9aozffuswgqxzieedkogzmoq5habjuueyzzuageovw7qtxhhggpje1nmkdpngwnw6ppvm3bo3ltmrejchbalqgae5yagrggdwabdj7bk3ghy+1fe9m+tm+jtkdlpdvgt2jfp7p3oho=</latexit> <latexit sha1_base64="g6o0fzji/vpizil8s528m+cdgfg=">aaacjxicbvbns8naen3ur1q/oh69lbahiprefd0ovl3oryjpq00im+22xbr5yhejlna/48w/4swdrqrp/hu3aq7aora7jzdvmjnnrywkarhfwmfufmfxqbhcwlldw9/qn7caiow5jhyowchbhhke0ybykkpgwhenypcyaxqdq7tefcrc0dc4k8oiod7qbbrlmzkkcvwze5fcc9hs/we0demha8dztix1iyc2j2tf85kbueuiez/tk4nl0pkrl42qkqwcbwyoyicpuqup7u6iy58eejmkrns0iukkieukgrmv7fiqcoeb6pg2ggfs850ku3ie9xttgd2qqxdimlg/oxlkczh0pavmnxbttzt8r9aozffuswgqxzieedkogzmoq5habjuueyzzuageovw7qtxhhggpje1nmkdpngwnw6ppvm3bo3ltmrejchbalqgae5yagrggdwabdj7bk3ghy+1fe9m+tm+jtkdlpdvgt2jfp7p3oho=</latexit> <latexit sha1_base64="g6o0fzji/vpizil8s528m+cdgfg=">aaacjxicbvbns8naen3ur1q/oh69lbahiprefd0ovl3oryjpq00im+22xbr5yhejlna/48w/4swdrqrp/hu3aq7aora7jzdvmjnnrywkarhfwmfufmfxqbhcwlldw9/qn7caiow5jhyowchbhhke0ybykkpgwhenypcyaxqdq7tefcrc0dc4k8oiod7qbbrlmzkkcvwze5fcc9hs/we0demha8dztix1iyc2j2tf85kbueuiez/tk4nl0pkrl42qkqwcbwyoyicpuqup7u6iy58eejmkrns0iukkieukgrmv7fiqcoeb6pg2ggfs850ku3ie9xttgd2qqxdimlg/oxlkczh0pavmnxbttzt8r9aozffuswgqxzieedkogzmoq5habjuueyzzuageovw7qtxhhggpje1nmkdpngwnw6ppvm3bo3ltmrejchbalqgae5yagrggdwabdj7bk3ghy+1fe9m+tm+jtkdlpdvgt2jfp7p3oho=</latexit> <latexit sha1_base64="hp+6lruf2d3tzaldqaqqvekmxyw=">aaab2xicbzdnsgmxfixv1l86vq1rn8eiucozbnqpuhfzwbzco5rm5k4bmskmyr2hdh0bf25efc93vo3pz0jbdwq+zknivscullqubn9ebwd3b/+gfugfnfzjk9nmo2fz0gjsilzl5jnmfpxu2cvjcp8lgzylffbj6f0i77+gstlxtzqrmmr4wmtuck7o6oyaraadlmw2ivxdc9yanb+gss7kddujxa0dhefbucunsafw7g9liwuxuz7ggupnm7rrtrxzzi6dk7a0n+5oykv394ukz9bostjdzdhn7ga2mp/lbiwlt1eldvesarh6kc0vo5wtdmajnchizrxwyasblykjn1yqa8z3hysbg29d77odbu3wmya6nmmfxeein3ahd9cblghi4bxevyn35n2suqp569lo4i+8zx84xio4</latexit> <latexit sha1_base64="5cglw4oc6hfwrkutbeluh14wmoc=">aaacgnicbvdlsgmxfl1tx7vwhd26crahipsmilpqunzoroj9afugtjq2ozkhsuypq3/gjb/ixovfbp/gzlqlbt2q5hducffe40wck43xt5vbwl5zxcuvfzakm1vb9k6xrsjyulajoqhl0yokcr6wmuzasgykgfe9wrre8dqtn56yvdwm7vuoyh2f9ape45roi7n2+ypl0qvqmvsiorg/zuqke9ua+0yhtk/0wpos23h5kjm/hxplzeuf1y7hcs6afokziywyoerak3y3plhpak0fuarl4eh3eii1p4knc+1ysyjqiemzlqebmf07sbblgb0ypyt6otqn0chtf/9iik/uypemm51yzdds8b9ak9a9s07cgyjwlkdtrr1yib2indlu5zjrluagecq5mrxrazgeahnsgoizv/iiqr9xhfxx7jdkyq/2oqwonmil3eavakdhbd7gaybwq/vufu7jylmz3hbhd6yvh9ponuo=</latexit> <latexit sha1_base64="5cglw4oc6hfwrkutbeluh14wmoc=">aaacgnicbvdlsgmxfl1tx7vwhd26crahipsmilpqunzoroj9afugtjq2ozkhsuypq3/gjb/ixovfbp/gzlqlbt2q5hducffe40wck43xt5vbwl5zxcuvfzakm1vb9k6xrsjyulajoqhl0yokcr6wmuzasgykgfe9wrre8dqtn56yvdwm7vuoyh2f9ape45roi7n2+ypl0qvqmvsiorg/zuqke9ua+0yhtk/0wpos23h5kjm/hxplzeuf1y7hcs6afokziywyoerak3y3plhpak0fuarl4eh3eii1p4knc+1ysyjqiemzlqebmf07sbblgb0ypyt6otqn0chtf/9iik/uypemm51yzdds8b9ak9a9s07cgyjwlkdtrr1yib2indlu5zjrluagecq5mrxrazgeahnsgoizv/iiqr9xhfxx7jdkyq/2oqwonmil3eavakdhbd7gaybwq/vufu7jylmz3hbhd6yvh9ponuo=</latexit> <latexit sha1_base64="6vdkphhjm2k4gyz6mkhn0+38qme=">aaacjxicbvdlsgmxfm34rpvvdekmwiskudkc6ekh6kz3fexdo8oqstntaozbklhk0j9x46+4cwerwzw/ymy6c229knzduedy7z1uxjluch0zc/mli0vlhzxi6tr6xmzpa7spw1gq2iahd0xbxzjyftcgyortdiqo9l1ow+7gkq23hqmqlazu1dcito97afmywupttuns3mhwhlb1fwhnhb4ycjxlszgfsmj5wpvdn7kzvs4y8dobljqcvnrkzvrfwcbzyoagdpkoo6wx1q1j7nnaey6l7jgounachwke01hriiwnmbnghu1ogga9306yk0dwxznd6ivcv0dbjp3dkwbfyqhvamw6szyuper/tu6svfm7yueukxqqysav5lcfmlumdpmgrpghbpgipnefpi8fjkobm5pgtp88c5phvrnvzvturl3mdhtaltgdfwcce1ad16aogocaz/ak3shyedheja/jcykdm/kehfanjo8fsregdg==</latexit> <latexit sha1_base64="g6o0fzji/vpizil8s528m+cdgfg=">aaacjxicbvbns8naen3ur1q/oh69lbahiprefd0ovl3oryjpq00im+22xbr5yhejlna/48w/4swdrqrp/hu3aq7aora7jzdvmjnnrywkarhfwmfufmfxqbhcwlldw9/qn7caiow5jhyowchbhhke0ybykkpgwhenypcyaxqdq7tefcrc0dc4k8oiod7qbbrlmzkkcvwze5fcc9hs/we0demha8dztix1iyc2j2tf85kbueuiez/tk4nl0pkrl42qkqwcbwyoyicpuqup7u6iy58eejmkrns0iukkieukgrmv7fiqcoeb6pg2ggfs850ku3ie9xttgd2qqxdimlg/oxlkczh0pavmnxbttzt8r9aozffuswgqxzieedkogzmoq5habjuueyzzuageovw7qtxhhggpje1nmkdpngwnw6ppvm3bo3ltmrejchbalqgae5yagrggdwabdj7bk3ghy+1fe9m+tm+jtkdlpdvgt2jfp7p3oho=</latexit> <latexit sha1_base64="g6o0fzji/vpizil8s528m+cdgfg=">aaacjxicbvbns8naen3ur1q/oh69lbahiprefd0ovl3oryjpq00im+22xbr5yhejlna/48w/4swdrqrp/hu3aq7aora7jzdvmjnnrywkarhfwmfufmfxqbhcwlldw9/qn7caiow5jhyowchbhhke0ybykkpgwhenypcyaxqdq7tefcrc0dc4k8oiod7qbbrlmzkkcvwze5fcc9hs/we0demha8dztix1iyc2j2tf85kbueuiez/tk4nl0pkrl42qkqwcbwyoyicpuqup7u6iy58eejmkrns0iukkieukgrmv7fiqcoeb6pg2ggfs850ku3ie9xttgd2qqxdimlg/oxlkczh0pavmnxbttzt8r9aozffuswgqxzieedkogzmoq5habjuueyzzuageovw7qtxhhggpje1nmkdpngwnw6ppvm3bo3ltmrejchbalqgae5yagrggdwabdj7bk3ghy+1fe9m+tm+jtkdlpdvgt2jfp7p3oho=</latexit> <latexit sha1_base64="g6o0fzji/vpizil8s528m+cdgfg=">aaacjxicbvbns8naen3ur1q/oh69lbahiprefd0ovl3oryjpq00im+22xbr5yhejlna/48w/4swdrqrp/hu3aq7aora7jzdvmjnnrywkarhfwmfufmfxqbhcwlldw9/qn7caiow5jhyowchbhhke0ybykkpgwhenypcyaxqdq7tefcrc0dc4k8oiod7qbbrlmzkkcvwze5fcc9hs/we0demha8dztix1iyc2j2tf85kbueuiez/tk4nl0pkrl42qkqwcbwyoyicpuqup7u6iy58eejmkrns0iukkieukgrmv7fiqcoeb6pg2ggfs850ku3ie9xttgd2qqxdimlg/oxlkczh0pavmnxbttzt8r9aozffuswgqxzieedkogzmoq5habjuueyzzuageovw7qtxhhggpje1nmkdpngwnw6ppvm3bo3ltmrejchbalqgae5yagrggdwabdj7bk3ghy+1fe9m+tm+jtkdlpdvgt2jfp7p3oho=</latexit> <latexit sha1_base64="g6o0fzji/vpizil8s528m+cdgfg=">aaacjxicbvbns8naen3ur1q/oh69lbahiprefd0ovl3oryjpq00im+22xbr5yhejlna/48w/4swdrqrp/hu3aq7aora7jzdvmjnnrywkarhfwmfufmfxqbhcwlldw9/qn7caiow5jhyowchbhhke0ybykkpgwhenypcyaxqdq7tefcrc0dc4k8oiod7qbbrlmzkkcvwze5fcc9hs/we0demha8dztix1iyc2j2tf85kbueuiez/tk4nl0pkrl42qkqwcbwyoyicpuqup7u6iy58eejmkrns0iukkieukgrmv7fiqcoeb6pg2ggfs850ku3ie9xttgd2qqxdimlg/oxlkczh0pavmnxbttzt8r9aozffuswgqxzieedkogzmoq5habjuueyzzuageovw7qtxhhggpje1nmkdpngwnw6ppvm3bo3ltmrejchbalqgae5yagrggdwabdj7bk3ghy+1fe9m+tm+jtkdlpdvgt2jfp7p3oho=</latexit> <latexit sha1_base64="g6o0fzji/vpizil8s528m+cdgfg=">aaacjxicbvbns8naen3ur1q/oh69lbahiprefd0ovl3oryjpq00im+22xbr5yhejlna/48w/4swdrqrp/hu3aq7aora7jzdvmjnnrywkarhfwmfufmfxqbhcwlldw9/qn7caiow5jhyowchbhhke0ybykkpgwhenypcyaxqdq7tefcrc0dc4k8oiod7qbbrlmzkkcvwze5fcc9hs/we0demha8dztix1iyc2j2tf85kbueuiez/tk4nl0pkrl42qkqwcbwyoyicpuqup7u6iy58eejmkrns0iukkieukgrmv7fiqcoeb6pg2ggfs850ku3ie9xttgd2qqxdimlg/oxlkczh0pavmnxbttzt8r9aozffuswgqxzieedkogzmoq5habjuueyzzuageovw7qtxhhggpje1nmkdpngwnw6ppvm3bo3ltmrejchbalqgae5yagrggdwabdj7bk3ghy+1fe9m+tm+jtkdlpdvgt2jfp7p3oho=</latexit> <latexit sha1_base64="g6o0fzji/vpizil8s528m+cdgfg=">aaacjxicbvbns8naen3ur1q/oh69lbahiprefd0ovl3oryjpq00im+22xbr5yhejlna/48w/4swdrqrp/hu3aq7aora7jzdvmjnnrywkarhfwmfufmfxqbhcwlldw9/qn7caiow5jhyowchbhhke0ybykkpgwhenypcyaxqdq7tefcrc0dc4k8oiod7qbbrlmzkkcvwze5fcc9hs/we0demha8dztix1iyc2j2tf85kbueuiez/tk4nl0pkrl42qkqwcbwyoyicpuqup7u6iy58eejmkrns0iukkieukgrmv7fiqcoeb6pg2ggfs850ku3ie9xttgd2qqxdimlg/oxlkczh0pavmnxbttzt8r9aozffuswgqxzieedkogzmoq5habjuueyzzuageovw7qtxhhggpje1nmkdpngwnw6ppvm3bo3ltmrejchbalqgae5yagrggdwabdj7bk3ghy+1fe9m+tm+jtkdlpdvgt2jfp7p3oho=</latexit> Intuitive Toy Example Suppose structural equation is Y i = X i + 100Z i + 50Z i I(A i = w)+u i So if there are two individuals, one of type w and one of type b, and Z 1 + Z 2 = 1. Without the fairness constraint, type w gets the subsidy even if type b has up to 50 more units of professional ability!
62 Considerations There might be no feasible solution if τ is small enough. It might be the case that the counterfactual gap in each constraint remains constant regardless of Z. These are features of the intervention, not of the fairness framework. Again, a good intervention is a matter of real world design, not of algorithm design!
63 Illustration (Partially Synthetic Data) NYC Public Schools: intervention Z is to provide calculus classes in schools. Attribute A is whether school has a white majority. Outcome Y is proportion of students taking the SAT/ACT. Geographical interference is assumed.
64 Results
Graphical Representation of Causal Effects. November 10, 2016
Graphical Representation of Causal Effects November 10, 2016 Lord s Paradox: Observed Data Units: Students; Covariates: Sex, September Weight; Potential Outcomes: June Weight under Treatment and Control;
More informationNP-Completeness. Until now we have been designing algorithms for specific problems
NP-Completeness 1 Introduction Until now we have been designing algorithms for specific problems We have seen running times O(log n), O(n), O(n log n), O(n 2 ), O(n 3 )... We have also discussed lower
More informationLecture 34 Woodward on Manipulation and Causation
Lecture 34 Woodward on Manipulation and Causation Patrick Maher Philosophy 270 Spring 2010 The book This book defends what I call a manipulationist or interventionist account of explanation and causation.
More informationSingle World Intervention Graphs (SWIGs):
Single World Intervention Graphs (SWIGs): Unifying the Counterfactual and Graphical Approaches to Causality Thomas Richardson Department of Statistics University of Washington Joint work with James Robins
More informationAbstract. Three Methods and Their Limitations. N-1 Experiments Suffice to Determine the Causal Relations Among N Variables
N-1 Experiments Suffice to Determine the Causal Relations Among N Variables Frederick Eberhardt Clark Glymour 1 Richard Scheines Carnegie Mellon University Abstract By combining experimental interventions
More informationWhat Causality Is (stats for mathematicians)
What Causality Is (stats for mathematicians) Andrew Critch UC Berkeley August 31, 2011 Introduction Foreword: The value of examples With any hard question, it helps to start with simple, concrete versions
More informationThe Effects of Interventions
3 The Effects of Interventions 3.1 Interventions The ultimate aim of many statistical studies is to predict the effects of interventions. When we collect data on factors associated with wildfires in the
More informationThe decision theoretic approach to causal inference OR Rethinking the paradigms of causal modelling
The decision theoretic approach to causal inference OR Rethinking the paradigms of causal modelling A.P.Dawid 1 and S.Geneletti 2 1 University of Cambridge, Statistical Laboratory 2 Imperial College Department
More informationAutomatic Causal Discovery
Automatic Causal Discovery Richard Scheines Peter Spirtes, Clark Glymour Dept. of Philosophy & CALD Carnegie Mellon 1 Outline 1. Motivation 2. Representation 3. Discovery 4. Using Regression for Causal
More informationCausality. Pedro A. Ortega. 18th February Computational & Biological Learning Lab University of Cambridge
Causality Pedro A. Ortega Computational & Biological Learning Lab University of Cambridge 18th February 2010 Why is causality important? The future of machine learning is to control (the world). Examples
More informationCausality II: How does causal inference fit into public health and what it is the role of statistics?
Causality II: How does causal inference fit into public health and what it is the role of statistics? Statistics for Psychosocial Research II November 13, 2006 1 Outline Potential Outcomes / Counterfactual
More informationIntroduction to Causal Calculus
Introduction to Causal Calculus Sanna Tyrväinen University of British Columbia August 1, 2017 1 / 1 2 / 1 Bayesian network Bayesian networks are Directed Acyclic Graphs (DAGs) whose nodes represent random
More information1 Impact Evaluation: Randomized Controlled Trial (RCT)
Introductory Applied Econometrics EEP/IAS 118 Fall 2013 Daley Kutzman Section #12 11-20-13 Warm-Up Consider the two panel data regressions below, where i indexes individuals and t indexes time in months:
More informationBounding the Probability of Causation in Mediation Analysis
arxiv:1411.2636v1 [math.st] 10 Nov 2014 Bounding the Probability of Causation in Mediation Analysis A. P. Dawid R. Murtas M. Musio February 16, 2018 Abstract Given empirical evidence for the dependence
More informationStatistical Models for Causal Analysis
Statistical Models for Causal Analysis Teppei Yamamoto Keio University Introduction to Causal Inference Spring 2016 Three Modes of Statistical Inference 1. Descriptive Inference: summarizing and exploring
More informationProbabilistic Models
Bayes Nets 1 Probabilistic Models Models describe how (a portion of) the world works Models are always simplifications May not account for every variable May not account for all interactions between variables
More informationOUTLINE THE MATHEMATICS OF CAUSAL INFERENCE IN STATISTICS. Judea Pearl University of California Los Angeles (www.cs.ucla.
THE MATHEMATICS OF CAUSAL INFERENCE IN STATISTICS Judea Pearl University of California Los Angeles (www.cs.ucla.edu/~judea) OUTLINE Modeling: Statistical vs. Causal Causal Models and Identifiability to
More informationCausal Inference & Reasoning with Causal Bayesian Networks
Causal Inference & Reasoning with Causal Bayesian Networks Neyman-Rubin Framework Potential Outcome Framework: for each unit k and each treatment i, there is a potential outcome on an attribute U, U ik,
More informationSummary of the Bayes Net Formalism. David Danks Institute for Human & Machine Cognition
Summary of the Bayes Net Formalism David Danks Institute for Human & Machine Cognition Bayesian Networks Two components: 1. Directed Acyclic Graph (DAG) G: There is a node for every variable D: Some nodes
More informationOF CAUSAL INFERENCE THE MATHEMATICS IN STATISTICS. Department of Computer Science. Judea Pearl UCLA
THE MATHEMATICS OF CAUSAL INFERENCE IN STATISTICS Judea earl Department of Computer Science UCLA OUTLINE Statistical vs. Causal Modeling: distinction and mental barriers N-R vs. structural model: strengths
More informationA Distinction between Causal Effects in Structural and Rubin Causal Models
A istinction between Causal Effects in Structural and Rubin Causal Models ionissi Aliprantis April 28, 2017 Abstract: Unspecified mediators play different roles in the outcome equations of Structural Causal
More informationCausality in Econometrics (3)
Graphical Causal Models References Causality in Econometrics (3) Alessio Moneta Max Planck Institute of Economics Jena moneta@econ.mpg.de 26 April 2011 GSBC Lecture Friedrich-Schiller-Universität Jena
More informationCAUSAL INFERENCE IN THE EMPIRICAL SCIENCES. Judea Pearl University of California Los Angeles (www.cs.ucla.edu/~judea)
CAUSAL INFERENCE IN THE EMPIRICAL SCIENCES Judea Pearl University of California Los Angeles (www.cs.ucla.edu/~judea) OUTLINE Inference: Statistical vs. Causal distinctions and mental barriers Formal semantics
More informationANALYTIC COMPARISON. Pearl and Rubin CAUSAL FRAMEWORKS
ANALYTIC COMPARISON of Pearl and Rubin CAUSAL FRAMEWORKS Content Page Part I. General Considerations Chapter 1. What is the question? 16 Introduction 16 1. Randomization 17 1.1 An Example of Randomization
More informationEconometric Causality
Econometric (2008) International Statistical Review, 76(1):1-27 James J. Heckman Spencer/INET Conference University of Chicago Econometric The econometric approach to causality develops explicit models
More informationSingle World Intervention Graphs (SWIGs):
Single World Intervention Graphs (SWIGs): Unifying the Counterfactual and Graphical Approaches to Causality Thomas Richardson Department of Statistics University of Washington Joint work with James Robins
More informationIntroducing Proof 1. hsn.uk.net. Contents
Contents 1 1 Introduction 1 What is proof? 1 Statements, Definitions and Euler Diagrams 1 Statements 1 Definitions Our first proof Euler diagrams 4 3 Logical Connectives 5 Negation 6 Conjunction 7 Disjunction
More informationLearning Semi-Markovian Causal Models using Experiments
Learning Semi-Markovian Causal Models using Experiments Stijn Meganck 1, Sam Maes 2, Philippe Leray 2 and Bernard Manderick 1 1 CoMo Vrije Universiteit Brussel Brussels, Belgium 2 LITIS INSA Rouen St.
More informationThe Two Sides of Interventionist Causation
The Two Sides of Interventionist Causation Peter W. Evans and Sally Shrapnel School of Historical and Philosophical Inquiry University of Queensland Australia Abstract Pearl and Woodward are both well-known
More informationCS 5522: Artificial Intelligence II
CS 5522: Artificial Intelligence II Bayes Nets Instructor: Alan Ritter Ohio State University [These slides were adapted from CS188 Intro to AI at UC Berkeley. All materials available at http://ai.berkeley.edu.]
More informationExchangeability and Invariance: A Causal Theory. Jiji Zhang. (Very Preliminary Draft) 1. Motivation: Lindley-Novick s Puzzle
Exchangeability and Invariance: A Causal Theory Jiji Zhang (Very Preliminary Draft) 1. Motivation: Lindley-Novick s Puzzle In their seminal paper on the role of exchangeability in statistical inference,
More informationBayesian Networks. Vibhav Gogate The University of Texas at Dallas
Bayesian Networks Vibhav Gogate The University of Texas at Dallas Intro to AI (CS 6364) Many slides over the course adapted from either Dan Klein, Luke Zettlemoyer, Stuart Russell or Andrew Moore 1 Outline
More informationThe Doubly-Modifiable Structural Model (DMSM) The Modifiable Structural Model (MSM)
unified theory of counterfactual reasoning hristopher G. Lucas cglucas@cmu.edu Department of Psychology arnegie Mellon University harles Kemp ckemp@cmu.edu Department of Psychology arnegie Mellon University
More informationOUTLINE CAUSAL INFERENCE: LOGICAL FOUNDATION AND NEW RESULTS. Judea Pearl University of California Los Angeles (www.cs.ucla.
OUTLINE CAUSAL INFERENCE: LOGICAL FOUNDATION AND NEW RESULTS Judea Pearl University of California Los Angeles (www.cs.ucla.edu/~judea/) Statistical vs. Causal vs. Counterfactual inference: syntax and semantics
More informationA Decision Theoretic Approach to Causality
A Decision Theoretic Approach to Causality Vanessa Didelez School of Mathematics University of Bristol (based on joint work with Philip Dawid) Bordeaux, June 2011 Based on: Dawid & Didelez (2010). Identifying
More informationIntroduction to Probabilistic Graphical Models
Introduction to Probabilistic Graphical Models Kyu-Baek Hwang and Byoung-Tak Zhang Biointelligence Lab School of Computer Science and Engineering Seoul National University Seoul 151-742 Korea E-mail: kbhwang@bi.snu.ac.kr
More informationTechnical Track Session I: Causal Inference
Impact Evaluation Technical Track Session I: Causal Inference Human Development Human Network Development Network Middle East and North Africa Region World Bank Institute Spanish Impact Evaluation Fund
More informationBayesian Networks. Vibhav Gogate The University of Texas at Dallas
Bayesian Networks Vibhav Gogate The University of Texas at Dallas Intro to AI (CS 4365) Many slides over the course adapted from either Dan Klein, Luke Zettlemoyer, Stuart Russell or Andrew Moore 1 Outline
More informationCarnegie Mellon Pittsburgh, Pennsylvania 15213
A Tutorial On Causal Inference Peter Spirtes August 4, 2009 Technical Report No. CMU-PHIL-183 Philosophy Methodology Logic Carnegie Mellon Pittsburgh, Pennsylvania 15213 1. Introduction A Tutorial On Causal
More informationCausal Discovery. Richard Scheines. Peter Spirtes, Clark Glymour, and many others. Dept. of Philosophy & CALD Carnegie Mellon
Causal Discovery Richard Scheines Peter Spirtes, Clark Glymour, and many others Dept. of Philosophy & CALD Carnegie Mellon Graphical Models --11/30/05 1 Outline 1. Motivation 2. Representation 3. Connecting
More informationProbabilistic Models. Models describe how (a portion of) the world works
Probabilistic Models Models describe how (a portion of) the world works Models are always simplifications May not account for every variable May not account for all interactions between variables All models
More informationStructure learning in human causal induction
Structure learning in human causal induction Joshua B. Tenenbaum & Thomas L. Griffiths Department of Psychology Stanford University, Stanford, CA 94305 jbt,gruffydd @psych.stanford.edu Abstract We use
More informationComments on The Role of Large Scale Assessments in Research on Educational Effectiveness and School Development by Eckhard Klieme, Ph.D.
Comments on The Role of Large Scale Assessments in Research on Educational Effectiveness and School Development by Eckhard Klieme, Ph.D. David Kaplan Department of Educational Psychology The General Theme
More informationCS 188: Artificial Intelligence Fall 2009
CS 188: Artificial Intelligence Fall 2009 Lecture 14: Bayes Nets 10/13/2009 Dan Klein UC Berkeley Announcements Assignments P3 due yesterday W2 due Thursday W1 returned in front (after lecture) Midterm
More informationCausal Inference in the Presence of Latent Variables and Selection Bias
499 Causal Inference in the Presence of Latent Variables and Selection Bias Peter Spirtes, Christopher Meek, and Thomas Richardson Department of Philosophy Carnegie Mellon University Pittsburgh, P 15213
More informationCS 343: Artificial Intelligence
CS 343: Artificial Intelligence Bayes Nets Prof. Scott Niekum The University of Texas at Austin [These slides based on those of Dan Klein and Pieter Abbeel for CS188 Intro to AI at UC Berkeley. All CS188
More informationCSE 473: Artificial Intelligence Autumn 2011
CSE 473: Artificial Intelligence Autumn 2011 Bayesian Networks Luke Zettlemoyer Many slides over the course adapted from either Dan Klein, Stuart Russell or Andrew Moore 1 Outline Probabilistic models
More informationCausal Inference. Prediction and causation are very different. Typical questions are:
Causal Inference Prediction and causation are very different. Typical questions are: Prediction: Predict Y after observing X = x Causation: Predict Y after setting X = x. Causation involves predicting
More informationLocal Characterizations of Causal Bayesian Networks
In M. Croitoru, S. Rudolph, N. Wilson, J. Howse, and O. Corby (Eds.), GKR 2011, LNAI 7205, Berlin Heidelberg: Springer-Verlag, pp. 1-17, 2012. TECHNICAL REPORT R-384 May 2011 Local Characterizations of
More informationOutline. CSE 473: Artificial Intelligence Spring Bayes Nets: Big Picture. Bayes Net Semantics. Hidden Markov Models. Example Bayes Net: Car
CSE 473: rtificial Intelligence Spring 2012 ayesian Networks Dan Weld Outline Probabilistic models (and inference) ayesian Networks (Ns) Independence in Ns Efficient Inference in Ns Learning Many slides
More informationSTA441: Spring Multiple Regression. This slide show is a free open source document. See the last slide for copyright information.
STA441: Spring 2018 Multiple Regression This slide show is a free open source document. See the last slide for copyright information. 1 Least Squares Plane 2 Statistical MODEL There are p-1 explanatory
More informationAdvances in Cyclic Structural Causal Models
Advances in Cyclic Structural Causal Models Joris Mooij j.m.mooij@uva.nl June 1st, 2018 Joris Mooij (UvA) Rotterdam 2018 2018-06-01 1 / 41 Part I Introduction to Causality Joris Mooij (UvA) Rotterdam 2018
More informationThe Role of Assumptions in Causal Discovery
The Role of Assumptions in Causal Discovery Marek J. Druzdzel Decision Systems Laboratory, School of Information Sciences and Intelligent Systems Program, University of Pittsburgh, Pittsburgh, PA 15260,
More informationConditional probabilities and graphical models
Conditional probabilities and graphical models Thomas Mailund Bioinformatics Research Centre (BiRC), Aarhus University Probability theory allows us to describe uncertainty in the processes we model within
More informationComputational Tasks and Models
1 Computational Tasks and Models Overview: We assume that the reader is familiar with computing devices but may associate the notion of computation with specific incarnations of it. Our first goal is to
More informationCS 188: Artificial Intelligence Spring Announcements
CS 188: Artificial Intelligence Spring 2011 Lecture 16: Bayes Nets IV Inference 3/28/2011 Pieter Abbeel UC Berkeley Many slides over this course adapted from Dan Klein, Stuart Russell, Andrew Moore Announcements
More informationCausal discovery from big data : mission (im)possible? Tom Heskes Radboud University Nijmegen The Netherlands
Causal discovery from big data : mission (im)possible? Tom Heskes Radboud University Nijmegen The Netherlands Outline Statistical causal discovery The logic of causal inference A Bayesian approach... Applications
More informationIntervention and Causality. Volker Tresp 2015
Intervention and Causality Volker Tresp 2015 1 Interventions In most of the lecture we talk about prediction When and how a can use predictive models for interventions, i.e., to estimate the results of
More informationChapter Three. Hypothesis Testing
3.1 Introduction The final phase of analyzing data is to make a decision concerning a set of choices or options. Should I invest in stocks or bonds? Should a new product be marketed? Are my products being
More informationCausal Mechanisms Short Course Part II:
Causal Mechanisms Short Course Part II: Analyzing Mechanisms with Experimental and Observational Data Teppei Yamamoto Massachusetts Institute of Technology March 24, 2012 Frontiers in the Analysis of Causal
More informationProbability Rules. MATH 130, Elements of Statistics I. J. Robert Buchanan. Fall Department of Mathematics
Probability Rules MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Introduction Probability is a measure of the likelihood of the occurrence of a certain behavior
More informationThe roots of computability theory. September 5, 2016
The roots of computability theory September 5, 2016 Algorithms An algorithm for a task or problem is a procedure that, if followed step by step and without any ingenuity, leads to the desired result/solution.
More informationExperimental Designs for Identifying Causal Mechanisms
Experimental Designs for Identifying Causal Mechanisms Kosuke Imai Princeton University October 18, 2011 PSMG Meeting Kosuke Imai (Princeton) Causal Mechanisms October 18, 2011 1 / 25 Project Reference
More informationDirected Graphical Models
CS 2750: Machine Learning Directed Graphical Models Prof. Adriana Kovashka University of Pittsburgh March 28, 2017 Graphical Models If no assumption of independence is made, must estimate an exponential
More informationComputer Systems on the CERN LHC: How do we measure how well they re doing?
Computer Systems on the CERN LHC: How do we measure how well they re doing? Joshua Dawes PhD Student affiliated with CERN & Manchester My research pages: http://cern.ch/go/ggh8 Purpose of this talk Principally
More informationCausal Models. Macartan Humphreys. September 4, Abstract Notes for G8412. Some background on DAGs and questions on an argument.
Causal Models Macartan Humphreys September 4, 2018 Abstract Notes for G8412. Some background on DAGs and questions on an argument. 1 A note on DAGs DAGs directed acyclic graphs are diagrams used to represent
More informationChallenges (& Some Solutions) and Making Connections
Challenges (& Some Solutions) and Making Connections Real-life Search All search algorithm theorems have form: If the world behaves like, then (probability 1) the algorithm will recover the true structure
More informationInferring Causal Structure: a Quantum Advantage
Inferring Causal Structure: a Quantum dvantage KR, M gnew, L Vermeyden, RW Spekkens, KJ Resch and D Janzing Nature Physics 11, 414 (2015) arxiv:1406.5036 Katja Ried Perimeter Institute for Theoretical
More informationCounterfactual Undoing in Deterministic Causal Reasoning
Counterfactual Undoing in Deterministic Causal Reasoning Steven A. Sloman (Steven_Sloman@brown.edu) Department of Cognitive & Linguistic Sciences, ox 1978 rown University, Providence, RI 02912 USA David
More informationCS 188: Artificial Intelligence Fall 2008
CS 188: Artificial Intelligence Fall 2008 Lecture 14: Bayes Nets 10/14/2008 Dan Klein UC Berkeley 1 1 Announcements Midterm 10/21! One page note sheet Review sessions Friday and Sunday (similar) OHs on
More informationCAUSALITY. Models, Reasoning, and Inference 1 CAMBRIDGE UNIVERSITY PRESS. Judea Pearl. University of California, Los Angeles
CAUSALITY Models, Reasoning, and Inference Judea Pearl University of California, Los Angeles 1 CAMBRIDGE UNIVERSITY PRESS Preface page xiii 1 Introduction to Probabilities, Graphs, and Causal Models 1
More informationLecture Notes 22 Causal Inference
Lecture Notes 22 Causal Inference Prediction and causation are very different. Typical questions are: Prediction: Predict after observing = x Causation: Predict after setting = x. Causation involves predicting
More informationMediation for the 21st Century
Mediation for the 21st Century Ross Boylan ross@biostat.ucsf.edu Center for Aids Prevention Studies and Division of Biostatistics University of California, San Francisco Mediation for the 21st Century
More informationWhen Should We Use Linear Fixed Effects Regression Models for Causal Inference with Panel Data?
When Should We Use Linear Fixed Effects Regression Models for Causal Inference with Panel Data? Kosuke Imai Department of Politics Center for Statistics and Machine Learning Princeton University Joint
More informationBehavioral Data Mining. Lecture 19 Regression and Causal Effects
Behavioral Data Mining Lecture 19 Regression and Causal Effects Outline Counterfactuals and Potential Outcomes Regression Models Causal Effects from Matching and Regression Weighted regression Counterfactuals
More informationCMPT Machine Learning. Bayesian Learning Lecture Scribe for Week 4 Jan 30th & Feb 4th
CMPT 882 - Machine Learning Bayesian Learning Lecture Scribe for Week 4 Jan 30th & Feb 4th Stephen Fagan sfagan@sfu.ca Overview: Introduction - Who was Bayes? - Bayesian Statistics Versus Classical Statistics
More informationModels of Causality. Roy Dong. University of California, Berkeley
Models of Causality Roy Dong University of California, Berkeley Correlation is not the same as causation. 2 Conditioning is not the same as imputing. 3 Stylized example Suppose, amongst the population,
More information6.3 How the Associational Criterion Fails
6.3. HOW THE ASSOCIATIONAL CRITERION FAILS 271 is randomized. We recall that this probability can be calculated from a causal model M either directly, by simulating the intervention do( = x), or (if P
More informationBayes Nets. CS 188: Artificial Intelligence Fall Example: Alarm Network. Bayes Net Semantics. Building the (Entire) Joint. Size of a Bayes Net
CS 188: Artificial Intelligence Fall 2010 Lecture 15: ayes Nets II Independence 10/14/2010 an Klein UC erkeley A ayes net is an efficient encoding of a probabilistic model of a domain ayes Nets Questions
More informationDAGS. f V f V 3 V 1, V 2 f V 2 V 0 f V 1 V 0 f V 0 (V 0, f V. f V m pa m pa m are the parents of V m. Statistical Dag: Example.
DAGS (V 0, 0,, V 1,,,V M ) V 0 V 1 V 2 V 3 Statistical Dag: f V Example M m 1 f V m pa m pa m are the parents of V m f V f V 3 V 1, V 2 f V 2 V 0 f V 1 V 0 f V 0 15 DAGS (V 0, 0,, V 1,,,V M ) V 0 V 1 V
More informationTreatment Effects. Christopher Taber. September 6, Department of Economics University of Wisconsin-Madison
Treatment Effects Christopher Taber Department of Economics University of Wisconsin-Madison September 6, 2017 Notation First a word on notation I like to use i subscripts on random variables to be clear
More informationCS 5522: Artificial Intelligence II
CS 5522: Artificial Intelligence II Bayes Nets: Independence Instructor: Alan Ritter Ohio State University [These slides were adapted from CS188 Intro to AI at UC Berkeley. All materials available at http://ai.berkeley.edu.]
More informationCS 188: Artificial Intelligence Spring Announcements
CS 188: Artificial Intelligence Spring 2011 Lecture 14: Bayes Nets II Independence 3/9/2011 Pieter Abbeel UC Berkeley Many slides over the course adapted from Dan Klein, Stuart Russell, Andrew Moore Announcements
More informationSingle World Intervention Graphs (SWIGs): A Unification of the Counterfactual and Graphical Approaches to Causality
Single World Intervention Graphs (SWIGs): A Unification of the Counterfactual and Graphical Approaches to Causality Thomas S. Richardson University of Washington James M. Robins Harvard University Working
More informationCAUSAL MODELS: THE MEANINGFUL INFORMATION OF PROBABILITY DISTRIBUTIONS
CAUSAL MODELS: THE MEANINGFUL INFORMATION OF PROBABILITY DISTRIBUTIONS Jan Lemeire, Erik Dirkx ETRO Dept., Vrije Universiteit Brussel Pleinlaan 2, 1050 Brussels, Belgium jan.lemeire@vub.ac.be ABSTRACT
More informationPotential Outcomes and Causal Inference I
Potential Outcomes and Causal Inference I Jonathan Wand Polisci 350C Stanford University May 3, 2006 Example A: Get-out-the-Vote (GOTV) Question: Is it possible to increase the likelihood of an individuals
More informationPackage CausalFX. May 20, 2015
Type Package Package CausalFX May 20, 2015 Title Methods for Estimating Causal Effects from Observational Data Version 1.0.1 Date 2015-05-20 Imports igraph, rcdd, rje Maintainer Ricardo Silva
More informationLearning Causal Bayesian Networks from Observations and Experiments: A Decision Theoretic Approach
Learning Causal Bayesian Networks from Observations and Experiments: A Decision Theoretic Approach Stijn Meganck 1, Philippe Leray 2, and Bernard Manderick 1 1 Vrije Universiteit Brussel, Pleinlaan 2,
More informationRespecting Markov Equivalence in Computing Posterior Probabilities of Causal Graphical Features
Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence (AAAI-10) Respecting Markov Equivalence in Computing Posterior Probabilities of Causal Graphical Features Eun Yong Kang Department
More informationInferring Causal Phenotype Networks from Segregating Populat
Inferring Causal Phenotype Networks from Segregating Populations Elias Chaibub Neto chaibub@stat.wisc.edu Statistics Department, University of Wisconsin - Madison July 15, 2008 Overview Introduction Description
More informationThe Causal Inference Problem and the Rubin Causal Model
The Causal Inference Problem and the Rubin Causal Model Lecture 2 Rebecca B. Morton NYU Exp Class Lectures R B Morton (NYU) EPS Lecture 2 Exp Class Lectures 1 / 23 Variables in Modeling the E ects of a
More informationFormalizing Probability. Choosing the Sample Space. Probability Measures
Formalizing Probability Choosing the Sample Space What do we assign probability to? Intuitively, we assign them to possible events (things that might happen, outcomes of an experiment) Formally, we take
More informationTaming the challenge of extrapolation: From multiple experiments and observations to valid causal conclusions. Judea Pearl and Elias Bareinboim, UCLA
Taming the challenge of extrapolation: From multiple experiments and observations to valid causal conclusions Judea Pearl and Elias Bareinboim, UCLA One distinct feature of big data applications, setting
More informationGraphical models and causality: Directed acyclic graphs (DAGs) and conditional (in)dependence
Graphical models and causality: Directed acyclic graphs (DAGs) and conditional (in)dependence General overview Introduction Directed acyclic graphs (DAGs) and conditional independence DAGs and causal effects
More informationcausal inference at hulu
causal inference at hulu Allen Tran July 17, 2016 Hulu Introduction Most interesting business questions at Hulu are causal Business: what would happen if we did x instead of y? dropped prices for risky
More informationCRITICAL NOTICE: CAUSALITY BY JUDEA PEARL
Economics and Philosophy, 19 (2003) 321 340 DOI: 10.1017/S0266267103001184 Copyright C Cambridge University Press CRITICAL NOTICE: CAUSALITY BY JUDEA PEARL JAMES WOODWARD California Institute of Technology
More informationProbability. We will now begin to explore issues of uncertainty and randomness and how they affect our view of nature.
Probability We will now begin to explore issues of uncertainty and randomness and how they affect our view of nature. We will explore in lab the differences between accuracy and precision, and the role
More informationFrom Causality, Second edition, Contents
From Causality, Second edition, 2009. Preface to the First Edition Preface to the Second Edition page xv xix 1 Introduction to Probabilities, Graphs, and Causal Models 1 1.1 Introduction to Probability
More informationCausal Effect Identification in Alternative Acyclic Directed Mixed Graphs
Proceedings of Machine Learning Research vol 73:21-32, 2017 AMBN 2017 Causal Effect Identification in Alternative Acyclic Directed Mixed Graphs Jose M. Peña Linköping University Linköping (Sweden) jose.m.pena@liu.se
More informationInference and Explanation in Counterfactual Reasoning
Cognitive Science (2013) 1 29 Copyright 2013 Cognitive Science Society, Inc. All rights reserved. ISSN: 0364-0213 print / 1551-6709 online DOI: 10.1111/cogs.12024 Inference and Explanation in Counterfactual
More information