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S    c $A "`` A T* 2     c $L A "`  A V*(2     c $`A "`  A V*(2     6";U,$@  0b  <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 qi (     6",$@ 0  BP ? ??"@   X Click to edit Master title style!!    c $  " P    [#Click to edit Master subtitle style$$    c $  "``   X* 2     c $8  "`    Z*(2     c $L  "`    Z*(2     6"; ,$@  0b  <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 NrA|L|L -  A x*   e44ee D NpA|L|L 2 - A z*   e44eed D c $ ?$  A D NPA|L|L H ] A 8________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________   D TA|L|L s  A x*   e44ee D TܗA|L|L s2  A z*   e44eeH D 0ηo~ ? ̙3380___PPT10.bUp,Q $( t; 4  C x |L|L Ԕ?-    t*    e44ee6  C x| |L|L Ԕ?2 -   v*    e44ee4  C x%G|L|L Ԕ?s    t*    e44ee6  C xlA|L|L Ԕ?s2  A v*    e44eeH  0ηo~ ? ̙3380___PPT10.bUQ<0^N0 KC0$ (  $  $ 0RAT r(THE MATHEMATICS OF CAUSAL RELATIONS .) &00 $ 0G   i/Judea Pearl Department of Computer Science UCLA0 0$H $ 0޽h ? f̙fy___PPT10Y+D=' = @B +d0 @ (     Tt-A?"Kf Statistical vs. Causal Modeling: distinction and mental barriers Formal semantics for counterfactuals: definition, axioms, graphical representations Graphs and Algebra: Symbiosis translation and accomplishments! &.  ! &.         @`  s *UA u OUTLINE < 00   B  N D1?"hhH  0޽h ? f̙fy___PPT10Y+D=' = @B +" 0^N0 1 ) P /X (  X X s *A PP *TRADITIONAL STATISTICAL INFERENCE PARADIGM++$B X N D1?"8  t  /Xt  " ,XB  fZ G H$I@1Ԕ?"y[ &X HA?"  @Data (2 'X BD?"> $  E Inference (2 7 (X H8?"[ Q(P) (Aspects of P)z  )X Z Ԕ?"ftU  ^P Joint Distribution.B *X T DԔ?"X 00B +XB T DԔ?" 0f0 .X T?" Oc IYe.g., Infer whether customers who bought product A would also buy product B. Q = P(B | A)Z1H X 0޽h ? &X(X,X f̙fy___PPT10Y+D=' = @B +0^N0 `K@(  B G T Do?"$ V B H T Do?"$ V  & N?" bX 9What happens when P changes? e.g., Infer whether customers who bought product A would still buy A if we were to double the price.; ( s *@+" 8FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES,98   9  B ) N D1?"?(? = T 0?"?[fs m5Probability and statistics deal with static relations66" >@ # lZ GXH$I+Ԕ?"Bl^ ? Nd5?"B @Data (2 @ H\9?" T c  E Inference (2 n A N7?"m; *Q(P) (Aspects of P) ( B  `EԔ?" ` .P Joint Distribution@B C@ Z DԔ?"R B D Z DԔ?"g E TLԔ?"& Q nP Joint Distribution>B F  ` D)?"D  I HR?"L$  Bchange (2B J Z DԔ?" H  0޽h ? ?A> f̙fy___PPT10Y+D=' = @B +0^N0 p@ @(   B ;   ` D)?"D B <  T Do?"$ V B =  T Do?"$ V B ?  Z DԔ?"    s *0Y" 8FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES,98   9  B   N D1?"?(?<   ZL`?"G / 6Note: P (v) P (v | price = 2) P does not tell us how it ought to change e.g. Curing symptoms vs. curing diseases e.g. Analogy: mechanical deformationR(N 1  Nt?"rx <What remains invariant when P changes say, to satisfy P (price=2)=1F" 3 @ # lZ GXH$I+Ԕ?"8l^ 4  N<?"8 @Data (2 5  HX?" T Y  E Inference (2 n 6  N?"c; *Q(P) (Aspects of P) ( 7   `쏻Ԕ?" V  .P Joint Distribution@B 8 @ Z DԔ?"R B 9  Z DԔ?"g :  ThԔ?"& G  nP Joint Distribution> >  H?"B $  Bchange (2H   0޽h ? 4 6 3  f̙fy___PPT10Y+D=' = @B + 0^N0 !    (    s *H9 ?FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT),@?   @  B  N D1?"p8 ? ?R  TT?"  pCAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables,qj  T譻?" = 2STATISTICAL Regression Association / Independence  Controlling for / Conditioning Odd and risk ratios Collapsibility,v je  Tа?"?09D<4___PPT9 +Causal and statistical concepts do not mix.",1, @`=  Z?" H@8___PPT9 U "5 @`W   Hǻ?"  t$D 0@8___PPT9 U "1 @`]  Nϻ?"c ]t$D 0@8___PPT9 U "5 @`H  0޽h ? f̙fy___PPT10Y+D=' = @B +0^N0 6<(  F ? + ?R , Tݻ?"  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 . TL?"?09D<4___PPT9 +Causal and statistical concepts do not mix.",1, @`m 0 N?"f `t$D 0@8___PPT9 e 25 @`   H ?" _ t$D 0@8___PPT9 [Causal assumptions cannot be expressed in the mathematical language of standard statistics."\1\ @`" ) s *9,$D 0 8FROM STATISTICAL TO CAUSAL ANALYSIS: 2. MENTAL BARRIERS,98   9  B * N D1?"p,$D 0z e  1 e ,$D 0 2 T ?" H@8___PPT9 ^No causes in  no causes out (Cartwright, 1989)205 @` 3 T ?"Q ^  i1statistical assumptions + data causal assumptions22 4 T ?" / e)  Jcausal conclusions 5 T ?"j C  <  6 T$ ?" a  N }077H  0޽h ? f̙fy___PPT10Y+D=' = @B +0^N0 ;3 P(   Tl f -  f -,$D 0   T+ ?"f -ZRJ___PPT9,$  Non-standard mathematics: Structural equation models (Wright, 1920; Simon, 1960) Counterfactuals (Neyman-Rubin (Yx), Lewis (x Y))5o1U,b @`T   # ~X   ZC  Co?S"?D 2B   T Do?"JF ?   ?R   TG ?"  pCAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables,qj   T(E ?" = 2STATISTICAL Regression Association / Independence  Controlling for / Conditioning Odd and risk ratios Collapsibility,v je   THZ ?"?09D<4___PPT9 +Causal and statistical concepts do not mix.",1, @`  H] ?" _ H@8___PPT9 [Causal assumptions cannot be expressed in the mathematical language of standard statistics."\1\ @`  s *\g 9 8FROM STATISTICAL TO CAUSAL ANALYSIS: 2. MENTAL BARRIERS99  9  B   N D1?"pF e     e     Tu ?" H@8___PPT9 ^No causes in  no causes out (Cartwright, 1989)205 @`    Tx ?"Q ^  i1statistical assumptions + data causal assumptions22    Tx| ?" / e)  Jcausal conclusions   Th ?"j C  <    T ?" a  N }077Z   S  9??pp  9H  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    S (   "  @  fZ GH I+Ԕ?"dl^B  N D1?"KK  H\ ?"d @Data (2  B0 ?"> T  E Inference (2 /  H ?"; Q(M) (Aspects of M)z   Zh Ԕ?"   PData Generating ModelB  @ T DԔ?"R   Tt ?"v O! &M  Oracle for computing answers to Q s. e.g., Infer whether customer u who bought product A would still buy A if we were to double the price.# B  T DԔ?"g  N Ԕ?"& r o Joint Distribution >B  T DԔ?"   s *4  x@@ sTHE STRUCTURAL MODEL PARADIGM$  ZȭE?"  CM&H  0޽h ?    f̙f___PPT10i.iK+D=' = @B +0^N0 " #(  x # B A  ?"  Z  Ԕ?%   OZ(     Z  Ԕ??  OY(     Z  Ԕ?0W OX(     Z  Ԕ??   SINPUT(     Z|  Ԕ?? '  TOUTPUT(     Z  Ԕ?+: |.FAMILIAR CAUSAL MODEL ORACLE FOR MANIPILATION //( /  B ! N D1?"hhB  s *3޽h ? f̙fy___PPT10Y+D=' = @B +s 0^N0  z (, (  , , s *d   nSTRUCTURAL CAUSAL MODELS(B , N D1?""~ (, N ?"n Definition: A structural causal model is a 4-tuple V,U, F, P(u), where V = {V1,...,Vn} are observable variables U = {U1,...,Um} are background variables F = {f1,..., fn} are functions determining V, vi = fi(v, u) P(u) is a distribution over U P(u) and F induce a distribution P(v) over observable variablesI54!5@ t<}H , 0޽h ? f̙fy___PPT10Y+D=' = @B +-0^N0 <4 0 (    s **  H {%STRUCTURAL MODELS AND CAUSAL DIAGRAMS&&$B  N D1?"  T86 ?"`  TTThe arguments of the functions vi = fi(v,u) define a graph vi = fi(pai,ui) PAi V \ Vi Ui U Example: Price  Quantity equations in economics 3P$&  <B + Z DԔ?"  B ,@ Z DԔ?"B t (B 2   Z Ԕ?"G O    ZT ?"Y Z  BU1$ 2  Z Ԕ?"G 7   ZY ?"Y   BU2$ L h 0  # G  2  T Ԕ?"h 0   T^ ?"   AI$ L 0 0  # G  2  T Ԕ?"0 0   TLd ?"k *  AW$ L  h8 # /  2  T Ԕ?" h8  Ti ?"  43 AQ$ L (   # /  2  T Ԕ?"(    Tn ?"z (  AP$ B  Z DԔ?" m8 i B @ Z DԔ?" ](i B   Z DԔ?" ( B ! Z DԔ?" m ! B  Z DԔ?"  B "@ Z DԔ?"B t (B 2 '  ` Ԕ?"  B ( Z DԔ?"| l|  ) Zv ?"   MPAQ. . c $A" ??"` S[4x d\H  0޽h ? f̙fy___PPT10Y+D=' = @B +0^N0 (T(  B  N D1?"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   T ?"   AI$ L 0 0  #  r 2  T Ԕ?"0 0   TT ?"k *  AW$ L  h8 # c 2  T Ԕ?" h8  T ?"  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  ?? |    & H ?"@v\  vLet X be a set of variables in V. The action do(x) sets X to constants x regardless of the factors which previously determined X. do(x) replaces all functions fi determining X with the constant functions X=x, to create a mutilated model Mx 7 P-QM @` ( s *  ( x"STRUCTURAL MODELS AND INTERVENTION##$H  0޽h ? f̙fy___PPT10Y+D=' = @B +0^N0   $(  B  N D1?"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   T ?"   AI$ L 0 0  #  r 2  T Ԕ?"0 0   TP ?"k *  AW$ L  h8 # c 2  T Ԕ?" h8  TD ?"  43 AQ$ 2  T Ԕ?"r  T ?" AP$ B  T DԔ?" B  T DԔ?" zB @ T DԔ?" B  T DԔ?"B  T ?"G- NP = p0,  c $A# ??"` S[x dA   Z8 Ԕ?" }F  HMp* B ! N DԔ?"w . RB "@ N DԔ?"w . R # H3 ?"@v\  vLet X be a set of variables in V. The action do(x) sets X to constants x regardless of the factors which previously determined X. do(x) replaces all functions fi determining X with the constant functions X=x, to create a mutilated model Mx 7 P-QM @` $ s *L  ( x"STRUCTURAL MODELS AND INTERVENTION##$H  0޽h ? f̙fy___PPT10Y+D=' = @B +0     @6 (     s *( 0 o!CAUSAL MODELS AND COUNTERFACTUALS""$ "    HR ?""  Definition: The sentence:  Y would be y (in situation u), had X been x, denoted Yx(u) = y, means: The solution for Y in a mutilated model Mx, (i.e., the equations for X replaced by X = x) with input U=u, is equal to y. 0      ,U9T @`B  N D1?"chc~2  H 33?"![6  Z4U ?"{ "  =  1   s *A ??"` RZx d   `xY ?"| "  c'Joint probabilities of counterfactuals: (0(B  T Do?"CCB  T Do?"  Z_ ?"N UThe super-distribution P* is derived from M. 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TT`TTT`TD`TT`TD`T_w`L   S4 # O}cx T4  c BdC)DEF1?c(c @`B U4  01? ^2 V4 61?{UwL 3 W4 # [P X4  c B.CaDEF1?`-` @3`B Y4 01?%.`B Z4 01?%.L  [4 # ;kH \4  c B7CaDEF1?`6` @`B ]4 01?`B ^4 01?2 _4 # C_TENGH I`TJ_TQ1? _TT_TT`T_T_TT`T_TG_`^2 `4 68c?NL * g E a4 # ^%  b4  c BCODEF1?,N,, @ g E`B c4  08c?* / /L  @E d4 # ^ s e4  c BCODEF1?,N,, @@E`B f4  08c? //L :c g4 # lc h4  c BYCDEF1?> 2X> @:c`B i4 08c?zc`B j4 08c?zcXB k4 08c? J^2 l4 68c? u 1^2 m4 68c?^ ^2 n4 68c?Nd2 o4 <1?Uw6 p4 TX ?"8} {E NGenotype (Unobserved)B q4 N D1?"8p8H 4 0޽h ?4  4  4 4 *4 64 B4 N4  R4  _4  f̙f___PPT10i.w`+D=' = @B +-0^N0 ((@67e((    T8) ?"0Q QTHE BACK-DOOR CRITERION%  Tt2 ?"q Graphical 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   HJ  ?"H %Dh  HZ6*    HO  ?"H Uth  HZ3*    HTT  ?"h  HZ2*    HY  ?"8  X  HZ5*    H]  ?""aB HZ1*    Hb  ?"  8  7X   HXf  ?"H uT h  7Y   H0j  ?"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  ?"^   # Hp  ?"^ e~  HZ6*   $ Hu  ?"^ ~  HZ3*   % Hz  ?"~ HZ2*   & H  ?"N n  HZ5*   ' H  ?"8X HZ1*   ( H8  ?". ` I N  7X  ) B ?"H h  7Y  * HL  ?"   HZ4*  2 + N o?" @X  , H  ?"pO GZ*  B -@ N Do?" l O  7 O,$D0 . HH  ?"O  H~Moreover, P(y | do(x)) = P(y | x,z) P(z) ( adjusting for Z) @    / BX ?" y  7z 8 h 5 1 H ?"? DGx&  2 N ?"h8 \ 4 0 T ?"}3 S G 4 3 N ?"\T  <3v޽h @ ? f̙f5-___PPT10 +D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-o6Bdissolve*<3<*7+K 0 bZ  (   r  S 8      s *p  Nn Z"RECENT RESULTS ON IDENTIFICATION##F$B  N D1?"LwLH  0޽h ? f̙f___PPT10i.p+D=' = @B +0   |(     s *l r v(STRUCTURAL ANALYSIS: SOME USEFUL RESULTS))$ )  B  N D1?"h  T ?"___PPT9 XVComplete formal semantics of counterfactuals Transparent language for expressing assumptions Complete solution to causal-effect identification Legal responsibility (bounds) Non-compliance (universal bounds) Integration of data from diverse sources Direct and Indirect effects, Complete criterion for counterfactual testability-02")2-02  "  )$$((,,00244 @`H  0޽h ? f̙f___PPT10i.(+D=' = @B +(0^N0 7/(    s *(  ; ;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   N ?"{ h ,$D 0 U 6} H  0޽h ? f̙fy___PPT10Y+D=' = @B +0^N0 y(    s *  ; ;DETERMINING THE CAUSES OF EFFECTS (The Attribution Problem)<< ("    B  N D1?"u  T: ?"h  EYour Honor! My client (Mr. A) died BECAUSE he used that drug. l,} | } , @`H  C Ajp2xB   N\B ?" ,$ 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) > 0.50P.} X|    TO ?",$0 HPN =(H  0޽h ? f̙fU M ___PPT10- +eD' S = @B D\' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+p+0+0 ++0+0 +u0^N0 |,(  , , 0[  E THE PROBLEM   (B , N D1?"hhz , 0Pc x PH@___PPT9" TSemantical Problem: What is the meaning of PN(x,y):  Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur. "   v 2   @` , Nx ?"  ,$D 0 U "!  @`T , <޽h @ ? ̙33y___PPT10Y+D=' = @B +0^N0  (    0h  E THE PROBLEM   (B   N D1?"hh.   0(k PH@___PPT9" vSemantical Problem: 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: Computable from M     2    @`   c $A ??"` | x dH  0޽h ? f̙fy___PPT10Y+D=' = @B + 0^N0  f(    0  E THE PROBLEM   (B  N D1?"hhz  0 x PH@___PPT9" TSemantical Problem: What is the meaning of PN(x,y):  Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur. "   v 2   @`sz  7    7,$D0   Np ?" 7H@8___PPT9 rUnder what condition can PN(x,y) be learned from statistical data, i.e., observational, experimental and combined.ns! S @`   Z ?" _  KAnalytical Problem: H  0޽h ? f̙fUM___PPT10-+fED' = @B D' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +&0^N0 5- 0 (    s *(  'TYPICAL THEOREMS (Tian and Pearl, 2000).(( &        HD ?":@ ;Bounds given combined nonexperimental and experimental data <1< @`B  N D1?"  c $A ??"`(E5 x d  s *A! ??"`  x d  T ?" a  u7 Identifiability under monotonicity (Combined data) "81 28    ` ?"E  Z corrected Excess-Risk-Ratio"0 2H  0޽h ? f̙fy___PPT10Y+D=' = @B +<0^N0 KC ((  ( ( s *D  ^ /CAN FREQUENCY DATA DECIDE LEGAL RESPONSIBILITY?00(B ( N D1?"FF ( N ?" ,$D 0 oNonexperimental data: drug usage predicts longer life Experimental data: drug has negligible effect on survivalHp%!( @`| ( T  ?",$D0 | Experimental Nonexperimental do(x) do(x) x x Deaths (y) 16 14 2 28 Survivals (y) 984 986 998 972 1,000 1,000 1,000 1,000 ffffffffff ffffffffff#f fffffff ( N7 ?"  p$D 0<4___PPT9 ~He actually died<%  @` ( N> ?"b  t$D 0@8___PPT9 zHe used the drug by choice.% 2 @`  ( s *A ??"`"Bx dA  ( TA ?"q ] %m Court to decide (given both data): Is it more probable than not that A would be alive but for the drug?(%F$( 4  ( ZtL ?" n ,$D 0 h Plaintiff: Mr. A is special.,!%H ( 0޽h ? f̙fy___PPT10Y+D=' = @B +0^N0 0 D (  D  D s * e 'TYPICAL THEOREMS (Tian and Pearl, 2000).(( &       D He?":@ ;Bounds given combined nonexperimental and experimental data <1< @`B D N D1?" D  c $A ??"`(E5 x d D  c $A! ??"`  x d D Nȶe?" a  u7 Identifiability under monotonicity (Combined data) "81 28 D Z,e?"E  Z corrected Excess-Risk-Ratio"0 2H D 0޽h ? f̙fy___PPT10Y+D=' = @B + 0^N0 ~(    s *S  Bg $SOLUTION TO THE ATTRIBUTION PROBLEM*%$(($  B  N D1?"hhpz   ,$D0  Tpt ?" n WITH PROBABILITY ONE 1 P(yx | x,y) 1 *| 5}  $ @`X  C (A jp1-red`# >  Z@ ?"m&,$D 0 r0 Combined data tell more that each study alone&1} 21H  0޽h ? f̙f%___PPT10+AsD'  = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+0 +0^N0 @ H z(  H  H s *4g  a CONCLUSIONS  ,B H N D1?" H HT g?"Q  mStructural-model semantics, enriched with logic and graphs, provides: Complete formal basis for causal reasoning Unifies the graphical, potential-outcome and structural equation approaches Powerful and friendly causal calculusVI0}1}G @`H H 0޽h ? f̙fy___PPT10Y+D=' = @B +0^N0    (     Z  Ԕ?R iDIRECT AND INDIRECT EFFECTS(   B  N D1?"  HA?"t \p What is the semantics of direct and indirect effects? Can we estimate them from data? Experimental data? 4654354   *  '  @`H  0޽h ? f̙fy___PPT10Y+D=' = @B +0 0  P(     H ?"$ VNF___PPT9(  0Direct (or indirect) effect may be more transportable. Indirect effects may be prevented or controlled. Direct (or indirect) effect may be forbiddenbm0-1m- @`  Hȫ ?" PWHY DECOMPOSE EFFECTS?,B  N D1?"p(p  T ?"l :Pill  T ?" s 5  @ Thrombosis    T ?"\ | ? Pregnancy  2  T o?"   2   T o?"s 2   T o?" 6B   Z Do?"/  B   Z Do?"I  B   @ Zm Do?"; F #  T ?"9  Y  9+  T ?" k  L-.  T ?"9 > " Y  9+2  T o?"Z  2  T o?" s M 2  T o?"w  B  Z Do?"  B  Z Do?" A wB  @ Zm Do?". ; F   T ?"- M  >Gender  Td ?"C  <Hiring  TD ?" /  C Qualificationb  <v޽h @ ?"` f̙f___PPT10i.cU!S +D=' = @B +0^N0 @  *(      c $A ??"`T x dAC  s *`  4< KTOTAL, DIRECT, AND INDIRECT EFFECTS HAVE SIMPLE SEMANTICS IN LINEAR MODELSfL$$$ $0$B  N D1?"tt  T ?"wt 9X  T( ?"  9Z  T ?"5  9Y  Tp ?"F  9c  T ?"6W;} 9a  T ?"n :b B  T DԔ?"Q B  T DԔ?"! B  @ T DԔ?"!    T ?"6 , .8z = bx + e1 y = ax + cz + e2  T ?"   va + bcT  Z ?"-0 t G bc(  Z ?" 6  L 9aH  0޽h ? f̙fy___PPT10Y+D=' = @B + 0^N0   P  z (     T ?" J <<z = f (x, e1) y = g (x, z, e2)   c $A ??"`p $ x d  T) ?"wt 9X  T ?"  9Z  TP+ ?"  9YB  T DԔ?"XQ XB  T DԔ?"!B   @ T DԔ?"!     s *>   ^SEMANTICS BECOMES NONTRIVIAL IN NONLINEAR MODELS (even when the model is completely specified)0_1$. B   N D1?"44   T4B ?". g  ,   TD ?"  1a cDependent on z?8   TPJ ?"p G RVoid of operational meaning?H  0޽h ? f̙fy___PPT10Y+D=' = @B + 0^N0   `   (     THP ?" J <<z = f (x, e1) y = g (x, z, e2)  Tl] ?"wt 9X  T@a ?"  9Z  Te ?"  9YB  T DԔ?"XQ XB  T DԔ?"!B  @ T DԔ?"!    s *i   -THE OPERATIONAL MEANING OF DIRECT EFFECTS..$B   N D1?"444   Hls ?"\   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    S  ??pp  Z    S  ??pp     c $A ??"`9Jx dH  0޽h ? f̙fy___PPT10Y+D=' = @B + 0^N0   p    (     T ?" J <<z = f (x, e1) y = g (x, z, e2)  T( ?"wt 9X  T ?"  9Z  Tȝ ?"  9YB  T DԔ?"XQ XB  T DԔ?"!B  @ T DԔ?"!    s *x   /THE OPERATIONAL MEANING OF INDIRECT EFFECTS00$B   N D1?"44   H ?"\   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  ? `    c $A ?? |      c $A ??"`yFx dH  0޽h ? f̙fy___PPT10Y+D=' = @B +# 0^N0 0(  (     H ?"P k!POLICY IMPLICATIONS (Who cares?)."( (  Z o?"; o m  M f ( Ez B! c   ! Bc,$D0   T ?"B! A  NGENDER,   T ?"Z ! A  S QUALIFICATION*    T ?" C c <HIRINGB  T Do?" } B   T Do?" _ % B   T Do?"w . . V B   @ T Do?" % B   N D1?"p   H ?"G,$D 0 $What is the direct effect of X on Y?`% @`J  Z ?"},$D 0 ~FThe effect of Gender on Hiring if sex discrimination is eliminated.GGB  N DԔ?"0,$D 0  H ?"*N,$D 0 Bindirect  2   T ?"! A  ;X  T ?"! j S A  IZ,  T ?"C c GY*B  @ N DԔ?"8    T ?" _  <IGNOREH  0޽h ? f̙f{___PPT10[+ D '  = @B D ' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* 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<* D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +p+0+ 0 ++0+ 0 + 0^N0    G (     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?""  T,$ ?"V t   HNO DIRECT EFFECT   0A ?Ԕ?"` 0 @(x dH  0޽h ? f̙fy___PPT10Y+D=' = @B + 0^N0   $ b (  $  $ 0 (b p6SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS7 7(B $ N D1?"hh  $ 0D   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 @` $  c $A ??"`T 4x d $  c $A ??"`@ Vx d $  c $A ??"` 9iI x dH $ 0޽h ? f̙fy___PPT10Y+D=' = @B +0  ( (  ( T ( C ,Ar273-fig2S :p ( TS ?"   @Example:  B ( N D1?" ( HW ?"W 4Theorem: If there exists a set W such that 5  @`7 ( HZ ?" WGRAPHICAL CONDITION FOR EXPERIMENTAL IDENTIFICATION OF AVERAGE NATURAL DIRECT EFFECTS>X$ (  c $A ??"` x d (  c $A ??"`x0x dH ( 0޽h ? f̙fy___PPT10Y+D=' = @B +0^N0  z(     s *Џ   a CONCLUSIONS  ,B  N D1?"  Ht ?"Q  mStructural-model semantics, enriched with logic and graphs, provides: Complete formal basis for causal reasoning Unifies the graphical, potential-outcome and structural equation approaches Powerful and friendly causal calculusVI0}1}G @`H  0޽h ? f̙fy___PPT10Y+D=' = @B +b 0 "t(  t^ t S Ds    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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