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S    c $DB/ "`` F T* 2     c $D/ "`  F V*(2     c $J/ "`  / V*(2     6";U,$@  0T  <v޽h @ ? f̙f[S___PPT103+MD' = @B D' = @BA?%,( < +O%,( < +D ' =%(%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*+ Whirlpool  0 c[ (     6",$@ 0  B? ??"@  X Click to edit Master title style!!    c $d " P   [#Click to edit Master subtitle style$$    c $ "``  X* 2     c $ "`   Z*(2     c $ "`   Z*(2     6"; ,$@  0T  <v޽h @ ? f̙f___PPT10+SD' = @B De' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* +0 JBD(   D D NR^|O|O -  ^ x*   e44ee D Nc^|O|O 2 - ^ z*   e44eed D c $ ?$  ^ D Ne^|O|O H ] ^ 8________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________   D Txr^|O|O s  ^ x*   e44ee D Ttw^|O|O s2  ^ z*   e44eeH D 0ηo~ ? ̙3380___PPT10.UV  $( t; 4  C x|O|O Ԕ?-   t*    e44ee6  C x|O|O Ԕ?2 -  v*    e44ee4  C x|O|O Ԕ?s   t*    e44ee6  C xX(|O|O Ԕ?s2   v*    e44eeH  0ηo~ ? ̙3380___PPT10.UV790 `X ( #e#eCi#e#e#eCi#e    0/P n4CAUSAL INFERENCE AS A MACHINE LEARNING EXERCISE5 50  0d`/ 2 HJudea Pearl Computer Science and Statistics UCLA www.cs.ucla.edu/~judea/I I$H  0޽h ? f̙fy___PPT10Y+D=' = @B + 0    9 (    s *^ u OUTLINE < 00      0M^Q,$D0 w+ Learning: Statistical vs. Causal concepts,,  ,  B  N D1?"hhL  N^?"#',$D 0 " Causal models and identifiability #!# &      ]  N^?"; ,$D 0 / Learnability of three types of causal queries:$0! 20 &     #    N0*^?"b 8 p$D 0<4___PPT9 #Effects of potential interventions,"$! 2$  $   @`  N^?" 8f t$D 0@8___PPT9 *Queries about attribution (responsibility)"+! 2+  +   @`   N^?" 8=t$D 0@8___PPT9 )Queries about direct and indirect effects"*! 2*  *   @`H  0޽h ? f̙f 0 _ W @ .X(  X" ,X@  fZ G H$I@1Ԕ?"Dy X s *X^ ( {%TRADITIONAL MACHINE LEARNING PARADIGM&&$B X N D1?" &X Hd^?" D <Data(2 'X BX^?"   @Learning (2  (X H,^?"w Q(P) (Aspects of P)\ )X Z|^Ԕ?"fU ] XP Joint Distribution(B *X T DԔ?"X B +X@ T DԔ?" f .X T@2?"e Om )Ye.g., Learn whether customers who bought product A would also buy product B. Q = P(B|A)Z1H X 0޽h ? &X(X,X f̙f 0   P \ (  \" \@  fZ G HN$I`Ԕ?"Dy  \ s *2 ( rTHE CAUSAL ANALYSIS PARADIGM$B \ N D1?" \ H2?" D <Data(2 \ BP2?"   @Learning (2  \ H2?"w+ Q(M) (Aspects of M)\ \ Z2Ԕ?"tK ] [M Data-generating Model(B  \@ T DԔ?"y  \ T-?"& Om iSome Q(M) cannot be inferred from P. e.g., Learn whether customers who bought product A would still buy A if we double the price. Data-mining vs. knowledge mining$2=B  \ T DԔ?"P H \ 0޽h ? \\\ f̙f"0 `'`P>(  ` ` s *- ( rTHE SECRETS OF CAUSAL MODELS$B ` N D1?"  ` Z-?"D$_ 0Causal Model = Data-generating model satisfying:61 $ %` T-?"=$D<4___PPT9 \Modularity (Symbol-mechanism correspondence) Uniqueness (Variable-mechanism correspondence)]] @`H ` 0޽h ? f̙f0 Q(    s *- ( rTHE SECRETS OF CAUSAL MODELS$B  N D1?"  Z$-?"D$_ 0Causal Model = Data-generating model satisfying:61 $ z e  e,$D 0  H/?"~  .Causal Model Joint Distribution PR = f (CP,PS, e1) P(PR, PS, CP, ME, QS) QS = g(ME, PR, e2); P(e1, e2) Q1: P(QS|PR=2) computable from P (and M) Q2: P(QS|do(PR=2) computable from M (not P)>        N e   e   NxԔ?"^  U f 2     Hm?"  ]Quantity Sold (QS).B   T DԔ?"mG H    N0Ԕ?"   U g 2  B   T DԔ?"u v B   T DԔ?"uC vC B  T DԔ?"ucvc  T?"b ^ Cost Proj.    TH:?"xb  V Prev. Sale    Tĕ:?"e b  ] Others (e1)6    TL?"   ^ Others (e2)6   B  T DԔ?"[ [   H:?"% VMarketing (MS), B  T DԔ?" v v B  T DԔ?"2  N Ԕ?"    B:?"9   U PRICE (PR) .  2  T`:?"=$D<4___PPT9 \Modularity (Symbol-mechanism correspondence) Uniqueness (Variable-mechanism correspondence)]] @`H  0޽h ? f̙f___PPT10+ǜD' = @B DW' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+m 0   pY (    s *<:p 8FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES,98   9  B  N D1?"(z [5   /[ ,$D0  Z:Ԕ?"a 8Data  Z:Ԕ?"Hf  Hjoint distribution  Tx:?"n 5  X$inferences from passive observations%%  Tt:?"[f m5Probability and statistics deal with static relations66B   T DԔ?"3B   T DԔ?" @r   3 r GDHTI#Ԕ?"8Z`r   3 r G+HUIAԔ?" b`    T:?"^ T9 A Probability    TX:?"R9 @ Statistics    Z:?"h   ; H  0޽h ?/@   f̙f 0 N F   (      s *:p 8FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES,98   9  B   N D1?"(z [5    /[ ,$D0   Zt:Ԕ?"a 8Data   Z:Ԕ?"Hf  Hjoint distribution   T\:?"n 5  X$inferences from passive observations%%   T :?"[f m5Probability and statistics deal with static relations66B   T DԔ?"3B   T DԔ?" @r   3 r GDHTI#Ԕ?"8Z`r   3 r G+HUIAԔ?" b`    T|:?"^ T9 A Probability     T:?"R9 @ Statistics     Z0?"h Fx lCausal analysis deals with changes (dynamics) i.e. What remains invariant when P changes. P does not tell us how it ought to change e.g. Curing symptoms vs. curing diseases e.g. Analogy: mechanical deformationz[+QO zH   0޽h ?/@   f̙f,0 <(  < < s *0p 8FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES,98   9  B < N D1?"(z [5  < /[ ,$D0 < Z0Ԕ?"a 8Data < ZP0Ԕ?"Hf  Hjoint distribution < TI0?"n 5  Y%predictions from passive observations&& < T0?"[f m5Probability and statistics deal with static relations66B  < T DԔ?"3B  < T DԔ?" @r  < 3 r GDHTI#Ԕ?"8Z`r  < 3 r G+HUIAԔ?" b`   < T0?"^ T9 A Probability   < TH0?"R9 @ Statistics   z   < h r,$D 0 < Z0Ԕ?"lw  B Causal Model  N " ' <   < T0?"" N  8Data < T82?"" ' FCausal assumptionsN    < z u < T2?"  J PH@___PPT9" Effects of interventions8 0  @`v < T*?" O *}TLD___PPT9& Causes of effects< 10  @`F < T$*?" H@8___PPT9 \ Explanations 1  @` < Tp*?" F  e-Causal analysis deals with changes (dynamics)..B < T DԔ?" Y B < T DԔ?"mYmB < T DԔ?" ~ " <  f GHI4Ԕ?" ` < TX*?" ? Experiments  H < 0޽h ??` < <<< f̙f___PPT10+N+B#style.visibility<*<%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<+q 0 !    (    s *@*9 ?FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT),@?   @  B  N D1?"p8 ? ?R  T0*?"  pCAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables,qj  T*?" = 2STATISTICAL Regression Association / Independence  Controlling for / Conditioning Odd and risk ratios Collapsibility,v je  TY?"?09D<4___PPT9 +Causal and statistical concepts do not mix.",1, @`=  Z?" H@8___PPT9 U "5 @`W   H?"  t$D 0@8___PPT9 U "1 @`]  N?"c ]t$D 0@8___PPT9 U "5 @`H  0޽h ? f̙f0 `X6(  F ? + ?R , T?"  pCAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables,qj - TH?" = 2STATISTICAL Regression Association / Independence  Controlling for / Conditioning Odd and risk ratios Collapsibility,v je . TD?"?09D<4___PPT9 +Causal and statistical concepts do not mix.",1, @`, 0 NԾ?"f $D 0RJ___PPT9,$  xNon-standard mathematics: Structural equation models (SEM) Counterfactuals (Neyman-Rubin) Causal Diagrams (Wright, 1920)F5_1_L' @`   HT?" _ t$D 0@8___PPT9 [Causal assumptions cannot be expressed in the mathematical language of standard statistics."\1\ @`) ) s *9,$D 0 ?FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT),@?   @  B * N D1?"p,$D 0z e  1 e ,$D 0 2 T ?" 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F     Z  H҅?" ,$D0 Definition: A causal model is a 3-tuple M = V,U,F with a mutilation operator do(x): M Mx where:ti "_ @`  NŅ?",$D 0 .(i) V = {V1& ,Vn} endogenous variables, (ii) U = {U1,& ,Um} background variablesTO @^  N?" ,$D 0 (iii) F = set of n functions, fi : V \ Vi U Vi vi = fi(pai,ui) PAi V \ Vi Ui Uh^  P  B + Z DԔ?" zB ,@ Z DԔ?" 2   Z Ԕ?" 6     Z4?"    BU1$ 2  Z Ԕ?"    Z?" ~  BU2$ L h 0  #  c  2  T Ԕ?"h 0   T?"   AI$ L 0 0  #  r 2  T Ԕ?"0 0   T$?"k *  AW$ L  h8 # c 2  T Ԕ?" h8  T*?"  43 AQ$ L (   # r2  T Ԕ?"(    Tt/?"z (  AP$ B  Z DԔ?" B @ Z DԔ?" B   Z DԔ?"& 66B ! Z DԔ?" B  Z DԔ?" zB "@ Z DԔ?" 2 '  ` Ԕ?"w ^B ( Z DԔ?"& ) Z6?"$$k QPAQ2 . c $A" ?? S[4x d` / c $A ?? |  H  0޽h ? f̙f0 OG (   F     Z  HZ?" ,$D0 Definition: A causal model is a 3-tuple M = V,U,F with a mutilation operator do(x): M Mx where:ti "_ @`  NH?",$D 0 .(i) V = {V1& ,Vn} endogenous variables, (ii) U = {U1,& ,Um} background variablesTO @^  Nq?" ,$D 0 (iii) F = set of n functions, fi : V \ Vi U Vi vi = fi(pai,ui) PAi V \ Vi Ui Uh^  P    N䞆?" @ ,$D 0 X*(iv) Mx= U,V,Fx, X V, x X where Fx = {fi: Vi X } {X = x} (Replace all functions fi corresponding to X with the constant functions X=x) b-6    N༆?" X,$D 0 K .   s *h ( rCAUSAL MODELS AND MUTILATION$B   N D1?"H  0޽h ? f̙f%___PPT10+AsD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+0 + 0 H @ ""(    s *І ( rCAUSAL MODELS AND MUTILATION$B  N D1?" F     Z  H?" ,$D0 Definition: A causal model is a 3-tuple M = V,U,F with a mutilation operator do(x): M Mx where:ti "_ @`  N?",$D 0 .(i) V = {V1& ,Vn} endogenous variables, (ii) U = {U1,& ,Um} background variablesTO @^  N?" ,$D 0 (iii) F = set of n functions, fi : V \ Vi U Vi vi = fi(pai,ui) PAi V \ Vi Ui Uh^  P     N5?" @ ,$D 0 `(iv) >B   T DԔ?" zB  @ T DԔ?" 2   T Ԕ?" 6     T;?"    BU1$ 2   T Ԕ?"    T@?" ~  BU2$ L h 0  #  c  2  T Ԕ?"h 0   TF?"   AI$ L 0 0  #  r 2  T Ԕ?"0 0   TpK?"k *  AW$ L  h8 # c 2  T Ԕ?" h8  TP?"  43 AQ$ 2  T Ԕ?"r  T V?" AP$ B  T DԔ?" B  T DԔ?" zB @ T DԔ?" B  T DԔ?"B  T[?"G!, RP = p00  c $A# ?? S[x d   ZaԔ?" }E  LMp. B ! N DԔ?"w . RB "@ N DԔ?"w . RH  0޽h ? f̙fL0 p!%(    s *xh ( rCAUSAL MODELS AND MUTILATION$B  N D1?" F     Z  H(?" ,$D0 Definition: A causal model is a 3-tuple M = V,U,F with a mutilation operator do(x): M Mx where:ti "_ @`  N|?",$D 0 .(i) V = {V1& ,Vn} endogenous variables, (ii) U = {U1,& ,Um} background variablesTO @^  N?" ,$D 0 (iii) F = set of n functions, fi : V \ Vi U Vi vi = fi(pai,ui) PAi V \ Vi Ui Uh^  P  B  T DԔ?" zB  @ T DԔ?" 2   T Ԕ?" 6     TT?"    BU1$ 2   T Ԕ?"     Tx?" ~  BU2$ L h 0  #  c  2  T Ԕ?"h 0   T?"   AI$ L 0 0  #  r 2  T Ԕ?"0 0   T?"k *  AW$ L  h8 # c 2  T Ԕ?" h8  TX?"  43 AQ$ 2  Z Ԕ?"r  Z?" AP$ B  T DԔ?" B @ T DԔ?" B  T DԔ?"& 66B  T DԔ?" B  T DԔ?" zB @ T DԔ?"  # c $A  ?? S[4x  d` $ c $A  ?? |     % N?" @ ,$D 0 `(iv) >H  0޽h ? f̙f0 RJ  (   F     Z  H?" ,$D0 Definition: A causal model is a 3-tuple M = V,U,F with a mutilation operator do(x): M Mx where:ti "_ @`  N?",$D 0 .(i) V = {V1& ,Vn} endogenous variables, (ii) U = {U1,& ,Um} background variablesTO @^  N%?" ,$D 0 (iii) F = set of n functions, fi : V \ Vi U Vi vi = fi(pai,ui) PAi V \ Vi Ui Uh^  P    NxS?" @ ,$D 0 X*(iv) Mx= U,V,Fx, X V, x X where Fx = {fi: Vi X } {X = x} (Replace all functions fi corresponding to X with the constant functions X=x) b-6   N,r?" X,$D 0 NDefinition (Probabilistic Causal Model): M, P(u) P(u) is a probability assignment to the variables in U. %2   s *  rPROBABILISTIC CAUSAL MODELS$B   N D1?"H  0޽h ? f̙f%___PPT10+92D' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* +8+0+ 0 +- 0 0  ](    s *R o!CAUSAL MODELS AND COUNTERFACTUALS""( "    Hd?"z  |Definition: Potential Response The sentence:  Y would be y (in unit u), had X been x, denoted Yx(u) = y, is the solution for Y in a mutilated model Mx, with the equations for X replaced by X = x. ( unit-based potential outcome ) 00       $ ,`3[ @`B  N D1?"{h{   `D?"8 X  =  1    `P?"  =  1H  0޽h ? f̙f0 g _ @ $  (  $ $ s * : o!CAUSAL MODELS AND COUNTERFACTUALS""( "   $ H ?"b  |Definition: Potential Response The sentence:  Y would be y (in unit u), had X been x, denoted Yx(u) = y, is the solution for Y in a mutilated model Mx, with the equations for X replaced by X = x. ( unit-based potential outcome ) 00       $ ,`3[ @`B $ N D1?"chc $  `8?" @  =  1el h   $h ,$D 0  $ s *A 0??,X  x 0d  $  `p?"h  a'Joint probabilities of counterfactuals:(0(H $ 0޽h ? f̙f___PPT10+l!D' = @B DW' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* $D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* $+j0 2 * P (0  (  ( ( s *: o!CAUSAL MODELS AND COUNTERFACTUALS""( "   ( H?"b  |Definition: Potential Response The sentence:  Y would be y (in unit u), had X been x, denoted Yx(u) = y, is the solution for Y in a mutilated model Mx, with the equations for X replaced by X = x. ( unit-based potential outcome ) 00       $ ,`3[ @`B ( N D1?"chc  ( s *A 2??X , x 2d  (  `t?"h  a'Joint probabilities of counterfactuals:(0(.l   ( ,$D 0 ( s *A ,??x x ,d (  `H?" @  HIn particular:0  ( s *A (??Dl Ld x (dOH ( 0޽h ? f̙f___PPT10+D' = @B DW' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(+/0 `B(    s *P t&CAUSAL INFERENCE MADE EASY (1985-2000)''( '     T$?" k |$D0H@___PPT9" ]Inference with Nonparametric Structural Equations made possible through Graphical Analysis. d31+0 + @`B  N D1?"shs  NY?" D $D 0LD___PPT9& XMathematical underpinning of counterfactuals through nonparametric structural equations T-1,0  , @`  Z+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+0 +%20 11p==e1(    TPn?",jf Z NON-PARAMETRIC STRUCTURAL MODELS!!(V  To?"A (Given P(x,y,z), should we ban smoking? )  T`|?"  (Px = u1, z = ax + u2, y = bz + g u1 + u3.)  T?"n PLFind: a b Find: P(y|do(x))'B  N D1?"hh  T؜?" r k-x = f1(u1), z = f2(x, u2), y = f3(z, u1, u3).".  TĞ?" `V  `(Linear Analysis Nonparametric Analysis))   T?" ;U2   N jJ ?"2   N jJ ?"c2   N jJ ?"c%]2   N jJ ?"c 5 B @ Z DjJ?" WB  Z DjJ?":]2  N jJ ?"OB  Z DjJ?"WB  T DjJ?"6 B  T DjJ?".  T̿?" c |  ;X  TD?"<  ;Z  T$?"  ;Y  T?"  91  TL?"r<$l ;U  T?" 92  Tt?"C p   ?Smoking  T?"C   G Tar in Lungs  T?"C   >Cancer2  N jJ ?"E}!B @ Z DjJ?"LTi  T$?" ;U   T?"e  93 ! T?"  ;U2 " N jJ ?"2 # N jJ ?"2 $ N jJ ?"E}2 % N jJ ?"UB &@ Z DjJ?" >B ' Z DjJ?"Z2 ( N jJ ?"B ) Z DjJ?"B * T DjJ?"VB + T DjJ?"N , T?"  ;X - T<?"\2  ;Z . T?"  ;Y / T?"@:: 91 0 Tl?"\D ;U 1 T?" 92 2 T ?"s G  ?Smoking 3 T?"s   G Tar in Lungs 4 T,?"s  G  >Cancer 5 N?"$ Pa2 6 T?"|H Pb22 7 N jJ ?"eQB 8@ Z DjJ?"Ct 9 T?":"  ;U : T"?"B < 93 ; <& ` I @ Hf1*   < <*Pp Hf2*   = <0 Hf3*  H  0޽h ? f̙f20 11<=P1(    T5?"jd 92 : <<:8 Hf2*  V  T>?"u  (Given P(x,y,z), should we ban smoking? )  TH?"y  (Px = u1, z = ax + u2, y = bz + g u1 + u3.)  T^?"v{H tFind: a b Find: P(y|do(x)) = P(Y=y) in new modell;B  N D1?"00  To?"y r $,x = const. z = f2(x, u2), y = f3(z, u1, u3).-  T\?"x `V  `(Linear Analysis Nonparametric Analysis))   T?" ;U2   N jJ ?"2   N jJ ?"+c2   N jJ ?"+%]c2   N jJ ?"+ 5 cB  Z DjJ?":%2  N jJ ?"OB  Z DjJ?"GB  T DjJ?"I6 IB  T DjJ?"O.O  T?"J hD  ?X = x  T ?"V<P  ;Z  T?"PJ  ;Y  T?" 91  T?":<$4 ;U  T?" p  ?Smoking  T<?" y  G Tar in Lungs  T?"   >Cancer2  N jJ ?"E}B @ Z DjJ?"LT1  Tį?" ;U  TX?"e 93   T?" ;U2 ! N jJ ?"2 " N jJ ?"[2 # N jJ ?"[E}2 $ N jJ ?"[UB %@ Z DjJ?">OB & Z DjJ?"ZU2 ' N jJ ?"GB ( Z DjJ?"wOB ) T DjJ?"yVyB * T DjJ?"N + Tؼ?"zt  ;X , Tx?"\2  ;Z - T?"z  ;Y . T|?": 91 / NT?"j\Dd ;U 0 N,?" 92 1 T?";   ?Smoking 2 T?";   G Tar in Lungs 3 T?";    >Cancer 4 T ?"$ Pa2 5 N?"|H Pb22 6 N jJ ?"eB 7@ Z DjJ?" th 8 T?":"  ;U 9 T?"   93 ; < Hf3*   < T?"B Z LEARNING THE EFFECTS OF ACTIONS!!( = T?"&  HD*H  0޽h ? f̙f 0    6 (      s * e eIDENTIFIABILITY,B   N D1?"    < 7Definition: Let Q(M) be any quantity defined on a causal model M, and let A be a set of assumption. Q is identifiable relative to A iffp  -     <`   ;  o   N#?"   for all M1, M2, that satisfy A. 2   N,?"  =   2    <3X   !P(M1) = P(M2) Q(M1) = Q(M2)`" ffffffffffffffffffffH   0޽h ? f̙f0 (    (      s *2 e eIDENTIFIABILITY,B   N D1?"    <$T 7Definition: Let Q(M) be any quantity defined on a causal model M, and let A be a set of assumption. Q is identifiable relative to A iffp  -  )   <e` ,$D 0 In other words, Q can be determined uniquely from the probability distribution P(v) of the endogenous variables, V, and assumptions A. "     <t+   !P(M1) = P(M2) Q(M1) = Q(M2)`" ffffffffffffffffffffo   Nt?"   for all M1, M2, that satisfy A. 2   N(?"  =   2H   0޽h ? f̙f%___PPT10+ǜD' 5= @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* +8+0+ 0 + 0   9(    s *p e eIDENTIFIABILITY,B  N D1?"   <p 7Definition: Let Q(M) be any quantity defined on a causal model M, and let A be a set of assumption. Q is identifiable relative to A iffp  -  o  N?"   for all M1, M2, that satisfy A. 2    <h+   !P(M1) = P(M2) Q(M1) = Q(M2)`" ffffffffffffffffffff  B% 8^ JPA: Assumptions encoded in the diagram Q1: P(y|do(x)) Causal Effect (= P(Yx=y)) Q2: P(Yx =y | x, y) Probability of necessity Q3: Direct Effect( A. S ),M J @`   s *A ??Px d   ZP)?" I  E In this talk:H  0޽h ? f̙fr0 "(    s *` y+THE FUNDAMENTAL THEOREM OF CAUSAL INFERENCE,,( ,  <  HlC?"B!  Causal Markov Theorem: Any distribution generated by Markovian structural model M (recursive, with independent disturbances) can be factorized as Where pai are the (values of) the parents of Vi in the causal diagram associated with M. \0 ($ $ ',5 _U @`B  N D1?"#h#  c $A ??L x dA  TE?" T  ] >10H  0޽h ? f̙f0   E (    s * P y+THE FUNDAMENTAL THEOREM OF CAUSAL INFERENCE,,( ,  <  HO?"2  Causal Markov Theorem: Any distribution generated by Markovian structural model M (recursive, with independent disturbances) can be factorized as Where pai are the (values of) the parents of Vi in the causal diagram associated with M. \0 ($ $ ',5 _U @`B  N D1?"h  c $A ??L x dAz 4 T@   T,$D 0  c $A ??p@x d(  NtT?"4 T Corollary: (Truncated factorization, Manipulation Theorem) The distribution generated by an intervention do(X=x) (in a Markovian model M) is given by the truncated factorization 0 _+w 3H  0޽h ? f̙f___PPT10+y[D' = @B DW' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+0 >6%)(    s *o@z w)RAMIFICATIONS OF THE FUNDAMENTAL THEOREM **( *  B  N D1?",$D08l 1  (Y ,$D0  Tt?"0* ZU (unobserved). 2  N jJ ?"?2  N jJ ?"2  N jJ ?"u2  N jJ ?"M  B   Z DjJ?"1B   T DjJ?" B   T DjJ?" ~   T{?"   ?X = x   T?"b  ;Z  T?"8  ;Y  T|?" %g  ?Smoking  Tt?"   G Tar in Lungs  TЋ?"2 Bg  >Cancer  Th?" `& Z ZU (unobserved). 2  N jJ ?"79o2  N jJ ?"9 2  N jJ ?" 2  N jJ ?"m B B Z DjJ?"[ B  Z DjJ?".aB  T DjJ?"  B  T DjJ?"@    T?"   ;X  T?"   ;Z  T?"X 7  ;Y  T?" E  ?Smoking  T<?"/ 1  G Tar in Lungs   Th?"R b  >CancerV ! T?" &Given P(x,y,z), should we ban smoking?z' # Z?" \H1  9  $ ZP?" 1  9  ' T?"  ),$D 0 9 H  0޽h ? f̙f$0 ((1(    s *@z w)RAMIFICATIONS OF THE FUNDAMENTAL THEOREM **( *  B  N D1?",$D0Fz 1   Y ,$D0  T?"0* ZU (unobserved). 2  N jJ ?"?2  N jJ ?"2  N jJ ?"u2   N jJ ?"M  B   Z DjJ?"1B   T DjJ?" B   T DjJ?" ~   T?"   ?X = x  T<?"b  ;Z  Th?"8  ;Y  T?" %g  ?Smoking  T?"   G Tar in Lungs  T?"2 Bg  >Cancer  T?" `& Z ZU (unobserved). 2  N jJ ?"79o2  N jJ ?"9 2  N jJ ?" 2  N jJ ?"m B B Z DjJ?"[ B  Z DjJ?".aB  T DjJ?"  B  T DjJ?"@    T,?"   ;X  T?"   ;Z  T?"X 7  ;Y  T?" E  ?Smoking   T?"/ 1  G Tar in Lungs ! TL?"R b  >CancerV " T?" &Given P(x,y,z), should we ban smoking?z' z \  #  \,$D 0 $ T?"\   FPre-intervention % T?"   GPost-intervention & c $A ??\ x d ' c $A ?? x d ( T?"  ),$D 0 9 H  0޽h ? f̙f___PPT10+o݄D' = @B DW' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*#D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*#+8&0 ((K(    s *d @z w)RAMIFICATIONS OF THE FUNDAMENTAL THEOREM **( *  B  N D1?",$D0Fz 1   Y ,$D0  T%?"0* ZU (unobserved). 2  N jJ ?"?2  N jJ ?"2  N jJ ?"u2   N jJ ?"M  B   Z DjJ?"1B   T DjJ?" B   T DjJ?" ~   T4-?"   ?X = x  T\1?"b  ;Z  T4?"8  ;Y  T8?" %g  ?Smoking  T<?"   G Tar in Lungs  T??"2 Bg  >Cancer  TB?" `& Z ZU (unobserved). 2  N jJ ?"79o2  N jJ ?"9 2  N jJ ?" 2  N jJ ?"m B B Z DjJ?"[ B  Z DjJ?".aB  T DjJ?"  B  T DjJ?"@    TH?"   ;X  TL?"   ;Z  TP?"X 7  ;Y  T|S?" E  ?Smoking   TtW?"/ 1  G Tar in Lungs ! TX?"R b  >CancerV " T^?" &Given P(x,y,z), should we ban smoking?z' z \  #  \,$D 0 $ Tg?"\   FPre-intervention % Tl?"   GPost-intervention & c $A ??\ x d ' c $A ?? x d ( TXq?"(,$D 0 SCTo compute P(y,z|do(x)), we must eliminate u. (graphical problem).D H  0޽h ? f̙f%___PPT10+ D' t= @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(+8+0+(0 +r&0 "&&46%(    T?"0Q,$D 0 QTHE BACK-DOOR CRITERION%  TP?"^q cGraphical test of identification P(y | do(x)) is identifiable in G if there is a set Z of variables such that Z d-separates X from Y in Gx.! B  N D1?"hL @0 0  # 0 x B B Z Do?"@ B B Z Do?"@ ` B  Z Do?"@0 B  B Z Do?"`@0 B  B Z Do?"@ 0 B   Z Do?"0  B   Z Do?"  B   Z Do?"  B B Z Do?"@0 P B B Z Do?"  z  <  ?" 0   Hl ?"H %Dh  HZ6*    Hx ?"H Uth  HZ3*    HX ?"h  HZ2*    H ?"8  X  HZ5*    HD ?""aB HZ1*    H ?"  8  7X   H, ?"H uT h  7Y   H ?"8 UtX  HZ4*  B @ Z Do?"N B @ Z Do?"~ . B  Z Do?"p B @ Z Do?"p B @ Z Do?"N p. B  Z Do?"N p^ B  Z Do?"^ ^ B  @ Z Do?"p B !@ Z Do?" ^ x " B  ?"^   # H, ?"^ e~  HZ6*   $ H, ?"^ ~  HZ3*   % H ?"~ HZ2*   & H ?"N n  HZ5*   ' H ?"8X HZ1*   ( H\ ?". ` I N  7X  ) B?"H h  7Y  * H ?"   HZ4*  2 + N o?" @X  , Hd ?"pO GZ*  B -@ N Do?" 2 . B ?" R4,$D 0 ~ ^  F h 0  1 H?"? bGx*  2 N ?"hF \ 3  4 T?"3 9 G  5 N ?"\H  0޽h ? f̙f_/0 ')) 66((    TD ?"0Q,$D 0 QTHE BACK-DOOR CRITERION%  T?"^q cGraphical test of identification P(y | do(x)) is identifiable in G if there is a set Z of variables such that Z d-separates X from Y in Gx.! B  N D1?"hL @0 0  # 0 x B B Z Do?"@ B B Z Do?"@ ` B  Z Do?"@0 B  B Z Do?"`@0 B  B Z Do?"@ 0 B   Z Do?"0  B   Z Do?"  B   Z Do?"  B B Z Do?"@0 P B B Z Do?"  z  <  ?" 0   H' ?"H %Dh  HZ6*    H|, ?"H Uth  HZ3*    H(1 ?"h  HZ2*    H5 ?"8  X  HZ5*    H: ?""aB HZ1*    H|? ?"  8  7X   HTC ?"H uT h  7Y   HG ?"8 UtX  HZ4*  B @ Z Do?"N B @ Z Do?"~ . B  Z Do?"p B @ Z Do?"p B @ Z Do?"N p. B  Z Do?"N p^ B  Z Do?"^ ^ B  @ Z Do?"p B !@ Z Do?" ^ x " B  ?"^   # H4 ?"^ e~  HZ6*   $ HR ?"^ ~  HZ3*   % HQ ?"~ HZ2*   & H4> ?"N n  HZ5*   ' HY ?"8X HZ1*   ( HL^ ?". ` I N  7X  ) Bb?"H h  7Y  * Hk ?"   HZ4*  2 + N o?" @X  , HDp ?"pO GZ*  B -@ N Do?" Gl   6 ,$D 0  . BTx ?" ,$D 0 N~Moreover, P(y | do(x)) = P(y | x,z) P(z) ( adjusting for Z) @    / <h?" I  7z 8 h 5 1 H?"? bGx*  2 N ?"h8 \ 4 0 Tp?"3 9 G  3 N ?"\T  <3v޽h @ ? f̙f___PPT10+bD' = @B DW' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*6D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*6+0 4,0(    Twawa1?qH iRULES OF CAUSAL CALCULUS(   B  N D1?"hh  T?" 0  :Rule 1: Ignoring observations P(y | do{x}, z, w) = P(y | do{x}, w) Rule 2: Action/observation exchange P(y | do{x}, do{z}, w) = P(y | do{x},z,w) Rule 3: Ignoring actions P(y | do{x}, do{z}, w) = P(y | do{x}, w) &&& *     -      PPPPPPPP    PPPPPPPPPPPP     PPPPPPPPPPPPQ  `  c $A D??4 T < D  s *A E?? x EdA4B  # g \ ] 4B  # g  A    6X >  IZ,    6X   IX,    6   IG,    6X   IW,    6  v  IX,   6 +  IZ,   6@ e  IY,   6    G)*   6   G,*   6 e  G|*   6H$  G(*   6" %  H *   6&   Hif*   6o h|  G^*   65o 3|  G^*   s *A F?? x FdAH  0޽h ? f̙fZT0  TT@ooR(    T<wawa1?ek n DERIVATION IN CAUSAL CALCULUS!!$ !    Z@wawa1?i  QSmoking     ZEwawa1? ;  MTar     ZtIwawa1? PCancer   2   BuTENG HgIQ8c? vTuT>TvTuT>T`TuT>T`T6 z2  3 B_TC_TENG(H I_TJ_TQ8c? =TIT=TIT_T_T=TIT_T_T\ z2  3 BNCSENG HfINJSQ8c? NX'6FNX'6FNS'6FNS    s BtCDEF1?&9s & @BIXB   08c?Jb   Z@Pwawa1? dP (c | do{s}) = St P (c | do{s}, t) P (t | do{s})D3```` ````````` 3     Zqwawa1?   d= Ss St P (c | do{t}, s) P (s | do{t}) P(t |s)l3   ````  ````` 3     Z wawa1?H f  VP= St P (c | do{s}, do{t}) P (t | do{s})) ``````````````` )    Zwawa1? 4  F= St P (c | do{s}, do{t}) P (t | s)$ `````````` $  6  Zwawa1?"  @  8= St P (c | do{t}) P (t | s)D `````     Zwawa1?_ } <N= Ss St P (c | t, s) P (s) P(t |s)(       (  >  Zwawa1?! ^= Ss St P (c | t, s) P (s | do{t}) P(t |s)&0      ```` 0    Z wawa1?Q ZProbability Axioms     Z|wawa1? Q  ZProbability Axioms     Zwawa1?=P[  NRule 2     Zwawa1? P  NRule 2     ZLwawa1? P;  NRule 3     Zwawa1?MPk NRule 3     Z #wawa1? P NRule 2   d2  <1? 1R6 d2  <1? q6 d2  <1? Qr6 2   BTENG IDQ1? TT`TTT`TD`TT`TD`TW`x# L  #  fL;   c BdC)DEF1?c(c @p`B  01?l^2   61?>Vx^L J3 !# \  " c B.CaDEF1?`-` @J3`B # 01?%J.`B $ 01?%J.d2 % <1? 1R d2 & <1? q ^2 ' 61? Vx 2 ( # C_TENGH I`TJ_TQ1? _TT_TT`T_T_TT`T_T Xp L   A  )#  ~l  * c BmC)DEF1?l(l @ A `B + 01? 4 4 d2 , <1? Qr L    -#  Dly  . c B.CaDEF1?`-` @  `B / 01?  `B 0 01?  ^2 1 61? 6X/ d2 2 <1? q/ d2 3 <1? Qr/ 2 4  BTENG IDQ1? TT`TTT`TD`TT`TD`TP `x L     5#  ~l8  6 c BmC)DEF1?l(l @  `B 7 01?   ^2 8 61?7 VxW L C <  9#  \  : c B7CaDEF1?'`6'` @C < `B ; 01?%C . `B < 01?%C . d2 = <1?>`d2 > <1?v^2 ? 61?Vx2 @  BTENG IDQ1? TT`TTT`TD`TT`TD`T/hL  A# pU B c BmC)DEF1?l(l @r`B C 01?nL "< D# \ E c B7CaDEF1?'`6'` @"<`B F 01?."7b`B G 01?."7bL " H# Eu I c B7CaDEF1?`6` @"`B J 01?.n`B K 01?.n2 L # C_TENGH I`TJ_TQ1? _TT_TT`T_T_TT`T_T+Phd2 M <1?S6Xsd2 N <1?Svs^2 O 61?XVxx2 P  BTENG IDQ1? TT`TTT`TD`TT`TD`T`x`L   Q# O~dx R c BdC)DEF1?c(c @`B S 01? ^2 T 61?{VxL 3 U# \P V c B.CaDEF1?`-` @3`B W 01?%.`B X 01?%.L  Y# <lH Z c B7CaDEF1?`6` @`B [ 01?`B \ 01?2 ] # C_TENGH I`TJ_TQ1? _TT_TT`T_T_TT`T_TH``^2 ^ 68c?NL * g E _# ^&  ` c BCODEF1?,N,, @ g E`B a 08c?* / /L  @E b# ^ t c c BCODEF1?,N,, @@E`B d 08c? //L :c e# mc f c BYCDEF1?> 2X> @:c`B g 08c?zc`B h 08c?zcXB i 08c?J^2 j 68c? v 1^2 k 68c?^ ^2 l 68c?Nd2 m <1?Vx6 n T>?"8~ |E NGenotype (Unobserved)B o N D1?"8q8H  0޽h ?(4@L P ] f̙fw+0 '++P5S($ )(  ( ( HE?"8r XA RECENT IDENTIFICATION RESULT(B ( N D1?" ( HJ?"{  &Theorem: [Tian and Pearl, 2001] The causal effect P(y|do(x)) is identifiable whenever the ancestral graph of Y contains no confounding path ( ) between X and any of its children.+4O  @` +( ZtO? "B b 3Y,2 #(@ s  B+ZCENGH\IQԔ? k+Z3k+Z3`T3`T"  P $( N  ? "0 p %( N  ? "p  &( N  ? "P B '( Z DԔ? " ` B (( Z DԔ? " @ )( Z0U? "z  3X *( Z|i? "   >Z1  ,( Zm? "Yd y 5(b) /( N  ? " F ~  0( Zq? "  >Z2 2 2(@  P. 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(#( 2( 5(7(B(D(M(N(O( f̙f 0   ` | (  | | s *t u OUTLINE < 00     | 0|Q,$D0 w+ Learning: Statistical vs. Causal concepts,,  ,  B | N D1?"hhL | NL?"#',$D 0 " Causal models and identifiability #!# &      ] | N?"; ,$D 0 / Learnability of three types of causal queries:$0! 20 &     #   | Nt?"b 8 p$D 0<4___PPT9 +Distinguishing direct from indirect effects",! 2,  ,   @` | N?" 8f t$D 0@8___PPT9 *Queries about attribution (responsibility)"+! 2+  +   @`q  | N?" 8=t$D 0@8___PPT9 i "! 2     @`H | 0޽h ? f̙f0 7/p(    s *x ; ;DETERMINING THE CAUSES OF EFFECTS (The Attribution Problem)<< ("    B  N D1?"u  T?"h  EYour Honor! My client (Mr. A) died BECAUSE he used that drug. l,} | } , @`H  C Ajp2xB   N0?"{ h ,$D 0 U 6} H  0޽h ? f̙fD 0 W(    s * ; ;DETERMINING THE CAUSES OF EFFECTS (The Attribution Problem)<< ("    B  N D1?"u  TT?"h  EYour Honor! My client (Mr. A) died BECAUSE he used that drug. l,} | } , @`H  C Ajp2xB   N$?" ,$D 0  Court to decide if it is MORE PROBABLE THAN NOT that A would be alive BUT FOR the drug! P(? | A is dead, took the drug) > 0.50D.} X|  H  0޽h ? f̙f%___PPT10+y[D' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+0 +0 ,8(  , , 0|+ E THE PROBLEM   (B , N D1?"hhb , 0Dx PH@___PPT9" XTheoretical Problems: What is the meaning of PN(x,y):  Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.     v 2  @`] , NTF?"  t$D 0@8___PPT9 U "!  @`T , <޽h @ ? ̙33{0 +#(    0\ E THE PROBLEM   (B  N D1?"hhv  0,< PH@___PPT9" lTheoretical Problems: What is the meaning of PN(x,y):  Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur. Answer:      2  @`  c $A ?? d&x dH  0޽h ? f̙f 0 ]U(    0o E THE PROBLEM   (B  N D1?"hhb  0dx PH@___PPT9" XTheoretical Problems: What is the meaning of PN(x,y):  Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.     v 2  @`  N?" t$D 0@8___PPT9 rUnder what condition can PN(x,y) be learned from statistical data, i.e., observational, experimental and combined.fs! T @`H  0޽h ? f̙f%___PPT10+ǜD' m= @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+0 +0 %(    0p C ]#WHAT IS LEARNABLE FROM EXPERIMENTS?$ $(B  N D1?"hh  N?"*(w,$D 0  Simple Experiment: Q = P(Yx= y | z) Z nondescendants of X. Compound Experiment: Q = P(YX(z) = y | z) Multi-Stage Experiment: etc& j; L , S @`H  0޽h ? f̙f0  ((  ( ( s * ^ 0 CAN FREQUENCY DATA DECIDE LEGAL RESPONSIBILITY?11(  0B ( N D1?"FF ( N?" ,$D 0 oNonexperimental data: drug usage predicts longer life Experimental data: drug has negligible effect on survivalHp%!( @`H ( T?",$D0 | Experimental Nonexperimental do(x) do(x) x x Deaths (y) 16 14 2 28 Survivals (y) 984 986 998 972 1,000 1,000 1,000 1,000 fffffffff ffffffff$f fff ff ( N ?"  p$D 0<4___PPT9 ~He actually died<%  @` ( N?"b  t$D 0@8___PPT9 zHe used the drug by choice.% 2 @`z y  ( q ,$D 0  ( c $A ??"Bx d  ( N|?"y e m Court to decide (given both data): Is it more probable than not that A would be alive but for the drug?~(%F$( 4  ( Z?" n ,$D 0 h Plaintiff: Mr. A is special.,!%H ( 0޽h ? f̙f0  3(    s *p 'TYPICAL THEOREMS (Tian and Pearl, 2000).(( &      5  H<%?":@ ;Bounds given combined nonexperimental and experimental data <1< @`B  N D1?"  c $A% ??==x d]l     ,$D 0       ,$D 0  c $A! ??p  x d=  NH)?" A  7 Identifiability under monotonicity (Combined data) "81 28, .   Z1?"E ,$D 0 Z corrected Excess-Risk-Ratio"0 2H  0޽h ? f̙f___PPT10+pD' = @B DW' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* +P0 c(    s *L@ Bg + SOLUTION TO THE ATTRIBUTION PROBLEM (Cont)@,*((+  B  N D1?"hhz   ,$D0  TT?" b WITH PROBABILITY ONE P(yx | x,y) =1 | /}   @`X  C (A jp1-red`# ,  NH?"~,$D 0 l* From population data to individual case&+} 2+>  ZK?"m,$D 0 r0 Combined data tell more that each study alone&1} 21H  0޽h ? f̙f%___PPT10+AsD' 5= @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+0 + 0    9 (    s *b u OUTLINE < 00      0hQ,$D0 w+ Learning: Statistical vs. Causal concepts,,  ,  B  N D1?"hhL  Ntm?"#',$D 0 " Causal models and identifiability #!# &      ]  Nq?"; ,$D 0 / Learnability of three types of causal queries:$0! 20 &     #    N?"b 8 p$D 0<4___PPT9 #Effects of potential interventions,"$! 2$  $   @`  N?" 8f t$D 0@8___PPT9 *Queries about attribution (responsibility)"+! 2+  +   @`   N,?" 8=t$D 0@8___PPT9 )Queries about direct and indirect effects"*! 2*  *   @`H  0޽h ? f̙fY0  \(  \ \ Z< Ԕ?; aQUESTIONS ADDRESSED,   B \ N D1?" \ H?"t \p What is the semantics of direct and indirect effects? Can we estimate them from data? Experimental data? 4654354   *  '  @`H \ 0޽h ? f̙f: 0   Pz (  P P Tx?" J <<z = f (x, e1) y = g (x, z, e2) P T?"wt 9X P T@?"  9Z P Tȡ?"  9YB P T DԔ?"XQ XB P T DԔ?"!B P@ T DԔ?"!   P s *$  -THE OPERATIONAL MEANING OF DIRECT EFFECTS..$B  P N D1?"444  P H?"\   Natural Direct Effect of X on Y: The expected change in Y per unit change of X, when we keep Z constant at whatever value it attains before the change. In linear models, NDE = Controlled Direct Effect;Z  P S  ??pp  Z  P S  ??pp   P c $A ??9Fx dH P 0޽h ? f̙f4 0   0 Lt (  L L T?" J <<z = f (x, e1) y = g (x, z, e2) L Td?"wt 9X L T?"  9Z L TD?"  9YB L T DԔ?"XQ XB  L T DԔ?"!B  L@ T DԔ?"!   L s *|  /THE OPERATIONAL MEANING OF INDIRECT EFFECTS00$B  L N D1?"44 L H0?"\   Natural Indirect Effect of X on Y: The expected change in Y when we keep X constant, say at x0, and let Z change to whatever value it would have under a unit change in X. In linear models, NIE = TE - DE F  ? ` L c $A ?? |   L c $A ??yFx dH L 0޽h ? f̙f 0   @|% (  | | N(?"p  ``The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of different race (age, sex, religion, national origin etc.) and everything else had been the same  [Carson versus Bethlehem Steel Corp. (70 FEP Cases 921, 7th Cir. (1996))] x = male, x = female y = hire, y = not hire z = applicant s qualificationsj 0!8      @`  | s */ M KLEGAL DEFINITIONS TAKE THE NATURAL CONCEPTION (FORMALIZING DISCRIMINATION),L.& B | N D1?"" | Z>o?" ^ XR ,2YxZx = Yx, YxZx = Yx H | Tf?"V t   HNO DIRECT EFFECTH | 0޽h ? f̙f> 0   PT~ (  T T 0j(b p6SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS7 7(B T N D1?"hhf T 0  Consider the quantity Given M, P(u), Q is well defined Given u, Zx*(u) is the solution for Z in Mx*, call it z is the solution for Y in Mxz Can Q be estimated from data? $$ 4$$9>FH@ @` T c $A ??T 4x d T c $A$ ??@ Vx d T c $A& ?? 9iI x dA3H T 0޽h ? f̙f 0 VN`T(  T T Z~ Ԕ?f bANSWERS TO QUESTIONS,   B T N D1?" T NT?"tD,$D 0 V Graphical conditions for estimability from experimental / nonexperimental data.f25%4  %,  T NP?"( Do ,$D 0 B "5P T Nԑ?"4D{ ,$D 0 0 Graphical conditions hold in Markovian models"151 H T 0޽h ? f̙f%___PPT10+{(AD' }= @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*TD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*T+8+0+T0 ++ 0   p4_ (  4 4 H?"0 ^$IDENTIFICATION IN MARKOVIAN MODELS%%,2 4 T o ?" 2 4 T o ?"H M l2 4 T o ?"( KB 4 Z Do?" ( PB  4 Z Do?" 7  4 Z?"$ k ;X  4 Z?"b   ;Y  4 Z?"v  ;ZB  4@ Z Do?"AA LB 4 N D1?"  4 c $A ??G x d 4 c $A ?? #x dH 4 0޽h ? f̙f 0 ' (    Z Ԕ?f bANSWERS TO QUESTIONS,   B  N D1?"  Nl?"tD,$D 0 V Graphical conditions for estimability from experimental / nonexperimental data.f25%4  %,   N?"( D ,$D 0 m Useful in answering new type of policy questions involving mechanism blocking instead of variable fixing.5594# P   NP?"4D{ ,$D 0 0 Graphical conditions hold in Markovian models"151 H  0޽h ? f̙f%___PPT10+ԾahD' q= @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+0 +y 0 ) !    (    s *(/,$D 0 y CONCLUSIONS < 00     0(/ $D 0TL___PPT9.& General theme: Define Q(M) as a counterfactual expression Determine conditions for the reduction If reduction is feasible, Q is learnable. Demonstrated on three types of queries:  )! h)    @`B  N D1?"hh,$D 0  c $A ??x dA,   B1^{ \ Q1: P(y|do(x)) Causal Effect (= P(Yx=y)) Q2: P(Yx = y | x, y) Probability of necessity Q3: Direct EffectT. AS ,% J @`   s *A ??=x dAzH  0޽h ? f̙f0 >( } X  C D$   ^  c $m^DH ]  ^ : H  0ηo~ ? ̙33 0 D<t(  t^ t S Ds   6 t c $D L   xHere is a causal model we all remember from high-school -- a circuit diagram. There are 4 interesting points to notice in this example: (1) It qualifies as a causal model -- because it contains the information to confirm or refute all action, counterfactual and explanatory sentences concerned with the operation of the circuit. For example, anyone can figure out what the output would be like if we set Y to zero, or if we change this OR gate to a NOR gate or if we perform any of the billions combinations of such actions. (2) Logical functions (Boolean input-output relation) is insufficient for answering such queries (3)These actions were not specified in advance, they do not have special names and they do not show up in the diagram. In fact, the great majority of the action queries that this circuit can answer have never been considered by the designer of this circuit. (4) So how does the circuit encode this extra information? 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