ࡱ> yvwx`!rC0&dTvf: [X+ss:与~(sy"+8|-_*T TX@NJM׀ӼYøcd&Cq*ZG 8{HЅۣ |,oHfnj_jBP~nbu`5cy9 *,J= 2! 7b@207₦F.p[M `pgdbR ,.Iehԡ_ MB`!ayi=#λJI[ )(xڝSNA3cwAHcň>j5 ά0M!l#1OL,-,҂J ج& BD'̹3gιw. @G !D"{>/cb8$gd0hܤΫ 5HNV [?"ļ?p㓧̍c:Xѷqؽ~YtﵱX1g uIϩX2!RP^2LxSSR3NUΨTf; s'cEsR,y&c%%tdQUpT%UU$T|bG57kd3~3exb2 yTUЃ O&p\}+Lp1G'F b.<@-" ƟQyAax7lM?2_,~#XD$2?9KcBܤ\ ^g[q GFrAs8wv0o8?221)W2ePdk9υØb$`!ylb3{v+@C0= xcdd`` @c112BYL%bpux Ӵ@Fh)P5Kʨ@|8,_ߖD_& ^&<4ss؛  0y{yI)$5aȟ @ ]` B0z3U`!nm/V,Z\(`.GQxcdd``v`d``baV d,FYzP1n: zB@?b 1@0&dT20L @201d++&1Dl\ _o ٳUsi#L\ f%'M"9LMB?_#֑ӡ'c 0f$Ռ -5@Kp4X>7b٘%P 27)?a8CF;1 8>)v0o8021)W2؃ePdk/ 3Xvw`!IǔrWx!'"J z`Bf@2xcdd`` @c112BYL%bpuLETE4\HEM>?0@penR~WOqD@[0~e_,P>;I K;xo2!n(L#a|f5|8"_(GA< ~UGPp,L'( 1(xڥKK@g7mM҂⫔DGԻh}EXCLT4Z(!NDeY".Wfm<B,L? q0ͼNfDq<~8BW+fVsiHX?`RHaL> QuՎCXc~CX>nLbLfs>x>N*m׿}9F -xXr]{YMQ휟=uj'Rf1V0fsdDNsjRUI=PbH҃'"X_LU:PaM,J.)1@=Qu/K|իВX_LP""} #xtqo-%<@}1o" `!1'dx;<(Y5 H)RFxcdd``c 2 ĜL0##0KQ* W`*d3H1)fY; PT obIFHeA*CPD.#l,Z 0ro *'] ZZsy"+8|-M 1d6]# |3 ~E9W@טJ0~k3VQ q_cܝX3t@&$WL ~6)%?я]L wW&cAKSaq]] sDFc,ʿʏ/E"2jPb07F_6c@b:C /Nn pޑĀ/dVNd-ܞf^_BX 8X(Y \l8Fv0o`dbR ,.Ie 2C  0Ou@lza$E2fY_,I!)( >|`!u's2/` @+ xڕ?K@{[UA!up],*NVThB tp;8Vp<oб_@(HQϑ:xp%=ܟ0D^B GBƼmNʦwR@&D883`TϢDׅ5$x를yfNY1a N*wW >E-QNB"֕X7IyL9Zf?- oJo|S"MX9ڇIŨV+?d$Y,\qSB|K2g؏cG 9C~/|}>u>rؗ geѠcX+mUU9 q>gFZĈ9rVj{F`!hCsD 4C:` 0xcdd`` @c112BYL%bpu*x{IB3DtY¶KO7W'e} %h//'E0؟v39:?w //ݗZ%z~ y|m [d"Y5/4mzGσFٿ?e*WCͲ\'p^U~dy =JfE? W{[ 7:Bl)}IUids\2(Z#L~]`!ܑxI 9%h`  6xS&xڥ?hQ5^ ibcKccjmjšBl)&ȭR+xvqvrtr*(E ~ b;x)]Hxu#H6VDQܕBZ*(qZآU. bsf|Ώr5=0YY8^vE)p^;67 p-ޝ^Wipy%bºVw؋y&]v:nVouS@xh)""=٠eČl3W|Zu^ZdvP9?09p=hhr= c`qYoi{d,͙ga?~3Fv})p܌(ao yg=~~| YoiOzK{9&[ڝ#O؟xKtlgЛYoG:Waw1 ~Fe?#+Iy~F~}59OȀ8*qTO{[@mj"R}~ZWd>h_7`! XdY9Sb,s٫т@@ 0/dPVxڝ/ALt Q4^%GDhEh=IBҋ]9C#*添|vv \ap-Θb<6Ve}W$<:jsB5Th bcոa> AFq|7nT *߈ǢD@gCLqB (Td,WZ seFJ.hAKεLLJ% A.`Pdk9υC-`!mTX%d~@:@XMZ8 xcdd``fb``baV d,FYzP1n:&6+! KA?H1Y10|*冪aM,,Ht 01d++&1Dl\ _'}b*F\ 'UUAk F[\ #jI9 >\Y_> \F/X@FsAC `CPF&&\F= @ ]` B0zelE( 4  'x'Equation Equation.30,Microsoft Equation 3.0{'Equation Equation.30,Microsoft Equation 3.0|' Equation Equation.30,Microsoft Equation 3.0}' Equation Equation.30,Microsoft Equation 3.0~' Equation Equation.30,Microsoft Equation 3.0' Equation Equation.30,Microsoft Equation 3.0' Equation Equation.30,Microsoft Equation 3.0'Equation Equation.30,Microsoft Equation 3.0'Equation Equation.30,Microsoft Equation 3.0'Equation Equation.30,Microsoft Equation 3.0'Equation Equation.30,Microsoft Equation 3.0'Equation Equation.30,Microsoft Equation 3.0'Equation Equation.30,Microsoft Equation 3.0'Equation Equation.30,Microsoft Equation 3.0'Equation Equation.30,Microsoft Equation 3.0'Equation Equation.30,Microsoft Equation 3.0'Equation Equation.30,Microsoft Equation 3.0' Equation Equation.30,Microsoft Equation 3.0'!Equation Equation.30,Microsoft Equation 3.0'#Equation Equation.30,Microsoft Equation 3.0'$Equation Equation.30,Microsoft Equation 3.0'%Equation Equation.30,Microsoft Equation 3.0'&Equation Equation.30,Microsoft Equation 3.0'(Equation Equation.30,Microsoft Equation 3.0')Equation Equation.30,Microsoft Equation 3.0/ 0DTimes New Romanppϒ  0DArialNew Romanppϒ  0" DWingdingsRomanppϒ  00DTimesngsRomanppϒ  0 @DArial Unicode MSpϒ  0"PDMonotype SortsSpϒ  0`DSymbole SortsSpϒ  0 C0.  @n?" dd@  @@`` ,,p 0   )Z n9t |P20.B4  \a$} rtB! $K305>&  >E4#Ej;7z'&$G4*;1)  ,,#35555))) |H()*'&(& @ q #2   11 $ %!>> /8s/8> FT%.|" &xzazze2-T)    '7  L !!!!! -,7A  9;87  !'"""""""##"""""" (1a  0( s  )% *#AD 0 -$ M 7E0#!L4? 5 ."+#C%&#Tv  34  r  *  7?&))/7 p .   !; p>  X!   9 q, q MC i*G &  7-    &    *   )L E @@^ B//,,//   //,(  ;U > 5  9 RL9C 9t 2$?" dd@,?dd@   @ ` p?" dd@   @@``PR    @ ` ` p>> f^ d(  d d 67 " 7 T Click to edit Master title style! !$ d 07 "  7 RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S d 67 #" `^` 7 F*  d 67 #" `^  7 H*  d 6Nz #" ` 7 H* V d s *h޽h ?#" @@ MMM̙80___PPT10.ݿ 8,  Clouds  0 1)0h(  h h B! ?#" ` ! T Click to edit Master title style! ! h 0! " `   ! W#Click to edit Master subtitle style$ $ h 6! #" `^` ! F*  h 6"! #" `^  ! H*  h 6&! #" `^  ! H* V h s *h޽h ?#" @@ MMM̙80___PPT10.ݿ 8, 0 TLD(   D D NP|@|@ -    x*   e44ee D NP|@|@ 2 -   *   e44eed D c $ ?$    D NXP|@|@ H ]   8________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________   D TЗP|@|@ s    x*   e44ee D TP|@|@ s2  P z*   e44eeH D 0ηo~ ? ̙3380___PPT10.bUp,Q  ,( t; 4  C x$!|@|@ Ԕ?-  # t*    e44ee>  C x2!|@|@ Ԕ?2 - ! ~*   e44ee4  C x=!|@|@ Ԕ?s  ! t*    e44ee6  C xI!|@|@ Ԕ?s2  ! v*    e44eeH  0ηo~ ? ̙3380___PPT10.bUQ  0 ((    0pz(h f, ON MEASUREMENT BIAS IN CAUSAL INFERENCE- -,"  0z u JJudea Pearl University of California Los Angeles (www.cs.ucla.edu/~judea/).K8nKZK$&2      H  0޽h ? f̙f___PPT10i.:U+D=' = @B +e!  0 vn@ (   y  H9 ?"z,$@ 0 iGiven causal diagram Z-unobserved (latent) Find P(y | do (x)) We know that But Minimize the biasj&1    6   @`  s * PH {THE MEASUREMENT BIAS PROBLEM, $$   B  N D1?"   c $A x'??"` `P  x'$D 0dA   c $A* {'??"`P `  {'$@ 0dA  HP?"j ?Z"0 2F *    x b 2   N ?"2   N ?"62  B N ?"6N *    *    T ?" * ;W02   N ?"FB   T DԔ?"@B  B T DԔ?"0B   T DԔ?"hh   T` ?" *  ;X0   T4T ?" *  ;Y0B   T DԔ?"0H  0޽h ? f̙f___PPT10o+ŃDS' = @B D' = @BA?%,( < +O%,( < +DE' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* AO%(D' =-o6Bdissolve*<3<* AOD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* OR%(D' =-o6Bdissolve*<3<* ORD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* RV%(D' =-o6Bdissolve*<3<* RVD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* VX%(D' =-o6Bdissolve*<3<* VXD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* Xj%(D' =-o6Bdissolve*<3<* XjD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* +  0 !((  ( ( s *Q  cOUTLINE,((   B ( N D1?"LL  ( TDK ?"Rl D<4___PPT9 BEffect Restoration using Matrix Inversion Example: Restoration in binary models Extension to multivariate confounders Effect Restoration in linear models The 3-proxy principle and its variations Model testing with measurement errors $1} @`H ( 0޽h ? MMM̙___PPT10i. o+D=' = @B +E+   0 RJ  (    s *T n x*MEASUREMENT BIAS AND EFFECT RESTORATION++$ +  B  N D1?"h   To0 2%0H- R 0  `A- |'? ?d"`  |'$D 0  d  ZT?" < ,$ 0 N(local independence)0  Tĝ?" ,$ 0 AAssume:0l  ,$D 0. 2  `A& }'? ?d"`  }' d  Tl?" C Solution: 0 H  0޽h ? MMM̙___PPT10s.+@D' = @B DZ' = @BA?%,( < +O%,( < +D+' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*Ds' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*++0+0 ++0+0 ++0+0 +2   0 .. 28-(   + T|?" ;Y0B  T D>?"  B  T D>?"   Z A!?"8 (   s *$ v(EFFECT RESTORATION IN BINARY MODELS))  )  B  N D1?"h& 0  `A' ~'? ?d"`  \ ~' dB  Z DԔ?"(8 B   N DԔ?" B   N D>?" 0 B   T DԔ?"0   Z A!?"@  0   s *A( '? ?"`  x 'd  T?"J>j 910  T?" nB  910  c $A% '??"`T \ x 'dAg  c $A) '??"`\ S x 'dAgN   0e0e    B9C_DE@F  Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||9-!adiA /> W _ @     "<  O( N @  0e0e    B9C_DE@F  Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||9-!adiA /> W _ @     "<  W (   Tb?"O QI  E undefined 0   T0?"O 9I  E undefined 0 F       . 2  `A" '? ?d"`   ' d2  N ?"V  Lz w    ,$D 0. 2  `A+ '? ?d"`w  ' d2  N ?">  4   H2?"b $Weight distribution from cell (x,y)L%0  ! T?"" ;W02 " N ?"d2 # N ?">dB $ T DԔ?"82 % N ?".BxB &@ T DԔ?"h(2 '@ N ?".xB ( T DԔ?"`p` ) T8?"X7 ;Z0 * T8G?"N7 ;X0B , T DԔ?"(B . T DԔ?"8   1p 81  Hd?S"?  To cell (x,y,z0)^0  0 T 1?" !F   2 @  1 3 H_?S"?  To cell (x,y,z1)^0  4 T 1?" B 5 T DԔ?"0 0 H 7  0e0e    BCtDE@F A Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||"EXk>l_ptppp,p_pppp @     "< 8 N 8  0e0e    BCtDE@F  Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||"EXk>l_ptppp,p_pppp @     "<  H  0޽h ? MMM̙E=___PPT10..+HKD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*+!  0 80 (4 (  B ) T DԔ?"0    s *J WHAT IF Z IS MULTI-VARIATE?L  $($$   B  N D1?" P  0ض ` zIf Z is high-dimensional, most cells will be empty of samples and, even if P(w | z) is known, P(x,w,y) cannot be estimated{0  G           ,K @h`B @  ` DԔ?"`` B @  ` DԔ?"(pX B   ` DԔ?"` B  @  ` DԔ?"p B  @  ` DԔ?"  B    ` DԔ?"  0 B    ` DԔ?"8 x( 8 B    ` DԔ?"@ 8 @ B @  ` DԔ?"p@B @  ` DԔ?"80   H4 ?"0 P  HZ6*    H  ?"(=\H HZ3*    HD ?"P<p HZ2*    H ?"Ed(  HZ5*    H ?" HZ1*    H ?" q  7X   H8 ?"0 P  7Y   H  ?"  @  HZ4*    Z ?"S E W G 82  N ?"( :j   Hs?"l  LW1.0 2  % Hw?"  LW3.0 2 B & T DԔ?" h B ' T DԔ?"8p ( H|?" L  LW4.0 2 B * T DԔ?"0  + H?"& LW2.0 2 B , T DԔ?"0 B . T DԔ?"H`  / Hl?"$ &D  LW5.0 2 2 0 N ?"  J 2 1 N ?"r "2 2 N ?"2 3 N ?"$fX2 4 N ?"P  H  0޽h ? 33___PPT10i.`[+D=' = @B +!  0 7/ %Q(   . s *(/A/ 6PROPENSITY SCORE ESTIMATOR (Rosenbaum & Rubin, 1983)77$ 7  B / N D1?"B 0@ Z Do?";a iB 1@ Z Do?"xM B 2 Z Do?";a  /B 3@ Z Do?";= )B 4@ Z Do?"?_( B 5 Z Do?"?(  B 6 Z Do?" D B 7 Z Do?"  B 8@ Z Do?";QB 9@ Z Do?" E ~ : H  ?" 3  ; H ?"  %  HZ6*   < H( ?"Xx HZ3*   = H ?"?^; HZ2*   > HL ?"w?^ HZ5*   ? H ?"$C  HZ1*   @ H0 ?" >(  7X  A HP ?"   7Y  B H ?" %  HZ4*  l @  Q@ ,$D  0B PB T DԔ?" J( N @  C @ B D T DԔ?"` R B E T DԔ?" @B FB T DԔ?". @B GB T DԔ?"F :L H Tp?"v _  7Lz $  I $ ,$D 0 J c $A/ '??"`Y$ t x 'dPN K y K K y( L Th?"YP6 *Adjustment for L replaces Adjustment for ZH+ M c $A1 '??"`YL yx 'd N T?"K Rk >Theorem:   O T?"E P(y | do(x)) = ?$H  0޽h ? f̙f0(___PPT10.LL+hD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =-o6Bwipe(up)*<3<*QD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =-o6Bdissolve*<3<*I+M  0 d\ )K(    s *: zPROPENSITY SCORE RESTORATION,     B  N D1?"    0<@ $  GFrom observed samples (x,w,y) to synthetic samples (x,z,y), to L(z), toH0     > @h` ' c $A2 '??"` 0x 'dB ( T DԔ?"t B )@ Z DԔ?"<< B *@ Z DԔ?"L4 B + Z DԔ?"< t B ,@ Z DԔ?"( tB -@ Z DԔ?"dt B . Z DԔ?"t B / Z DԔ?" T B 0 Z DԔ?"  B 1@ Z DԔ?"(l`B 2@ Z DԔ?"h   3 HU ?" i   HZ6*   4 Ht ?"8 HZ3*   5 HV ?" ( HZ2*   6 H<7 ?"!@ HZ5*   7 H ?" HZ1*   8 H) ?" dM  7X  9 H0G ?"   7Y  : HU ?"  HZ4*   ; TZ?"/  W G 82 < N ?" "  = H$a?"H  LW1.0 2  > Hde?"  LW3.0 2 B ? T DԔ?"D l B @ T DԔ?"Lf A H4j?" (  LW4.0 2 B B T DԔ?"H  C Hn?"b LW2.0 2 B D T DԔ?"H B E T DԔ?"$  F Hs?"  LW5.0 2 2 G N ?"  2 H N ?"N 2 I N ?"T2 J N ?"B2 K N ?" J H  0޽h ? MMM̙___PPT10i.` [+D=' = @B +   0 }!8 (    s *XR t&EFFECT RESTORATION IN LINEAR MODELS''$ '  B  N D1?" h   T?"Z<Pz ;W02  N ?" UP2  N ?" UB  T DԔ?"(//2  N ?"3B  @ T DԔ?" 62  @ N ?"+vB   T DԔ?"6   T?"*$J ;Z0   T?"Zz ;X0  T ?"ZdCz ;Y0  T?"<Q Lc1.0B  T DԔ?" &N  T?" Lc2.0  T?"2)R Lc3.0  T?"r Lc0.0  T\#?"x  =  0 / c $A '??"`d  '$D 0d 4 c $A '??"`^" *x 'dDl   8 ,$D 0 5 T(?" =  Z The pivotal parameter needed is !0! 6 c $A '??"`( <x 'd 7 c $A '??"`Px 'dH  0޽h ? MMM̙0(___PPT10.6+>D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-o6Bdissolve*<3<*/D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-o6Bdissolve*<3<*8+@-  0 }#u#`!08 #(      s *6z x*EFFECT RESTORATION FROM A SECOND PROXY++$ +  B   N D1?" h  8 *  8 *    Tz?"Zz ;W02   N ?"P2   N ?"B   T DԔ?"(2   N ?"B  B T DԔ?" 2  B N ?"8 B   T DԔ?"   Tz?"*J ;Z0   Tz?"Zz ;X0   Tz?"& Z z ;Y0   Tz?" Lc1.0B   T DԔ?"     Tz?" Lc2.0   Tz?"2R Lc3.0   T`z?"r Lc0.0   T8z?"f  ?(a) 0:  *   # *  ,$@ 0   Tz?"^G ;V02   N ?"/P2   N ?"DB   T DԔ?" (i2   N ?"  B  B T DԔ?" 2  B N ?"PB   T DԔ?"    Tz?"*J ;Z0 !  Tz?" Z z ;X0 "  Thz?">Zz ;Y0 #  Taz?" + Lc1.0B $  T DԔ?" ( %  Tez?" Lc2.0 &  Hjz?"&:;Z Lc4.0 '  Tpnz?"r Lc0.02 (  N ?"\ )  Tsz?"^ r ;W0B * B T DԔ?"(  +  Thwz?" :Z Lc3.0 ,  T@|z?"  ?(b) 02 /  T ?" L 0  c $A '??"` 6 x 'd2 1  N ?"*  t  5  c $A '??"` B  '$D 0dA 7  c $A '??"`&<^ '$D 0dH   0޽h ? MMM̙k c ___PPT10C .& +EeiD ' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*5 %(D' =-o6Bdissolve*<3<*5 D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*7 %(D' =-o6Bdissolve*<3<*7 +$  0N0 !-$ h(  $  $ s *1 l sTHE THREE-PROXIES PRINCIPLE$B $ N D1?" $ Z:?" w  ;V02 $ T ?".$ o x2 $ T ?" B $ Z DԔ?"PI 2 $ T ?"  M B $ @ Z DԔ?"H( P  $ Zl@?"R>  r ;Z0 $ ZD?"   ;X0 $ ZH?"V k  Lc1.0 $ NM?"bf {  Lc4.0il 0 H]  ($ H0 ] ,$D  02 $ B T ?"E S@ 0 H]  '$ 0 H] B $  Z DԔ?"0 P  $  ZX ?"~ ]  ;Y0B $  Z DԔ?"@ Hh  $  Z?" +  Lc2.0 $  ZР?"  Lc0.02 $ T ?"  $ Z?"^ r  ;W0B  $ @ Z DԔ?"P I  !$ Z(?"b   Lc3.0 #$  c $A6 '??"` x 'dB $$ @ Z DԔ?"@ ,$@  0l O ` *$  O`,$D 0`  $  Z,?"O K  *Cai and Kuroki (2008) c0 is identifiable`+  ( &$  s *A4 '??"`` `x 'dl  p  -$  p ,$D  02 +$  T Ԕ?"p p 2 ,$  T Ԕ?"  H $ 0޽h ? f̙fJ B ___PPT10" +--mD ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*($ %(D' =-s6Bwipe(left)*<3<*($ D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-$ %(D' =-o6Bwipe(up)*<3<*-$ D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$$ %(D' =-o6Bwipe(up)*<3<*$$ D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<**$ %(D' =-o6Bdissolve*<3<**$ +B   0 **!.B I*(   2   HLȝ?"2~ ,$@ 0 xSolution: Test if c0 = 0 Theorem 1: If a latent variable Z d-separates two measured variables, X and Y, and Z has a proxy W, W = cZ + e, then cov(XY) must satisfy: cov(XY)=cov(XW) cov(WY) / c2 var(Z) Corollary: is testable if k=c2 var(Z) is estimable Example: Given W and V, k is estimable, andLB0   #  #  6>   T8?"jv ;W02   N ?"D`2   N ?"DB   T DԔ?"8ii2   N ?"" m B  @ T DԔ?"0H p2  @ N ?"e   T`U?"~ o Z unobservedF0    T[?"j.   ;X0   T_?"j} ;Y0   Tc?"v  Lc1.0B   T DԔ?"0`   Ti?"6K Lc2.0   Tm?"BNcb Lc3.0l P p > P p,$D  0B   T DԔ?"P (p(   Tf?" Lc0.0   s * l P ~(MODEL TESTING WITH MEASUREMENT ERRORS)) B   N D1?"   Tu?"~ ZProblem: Test if ,0    0 ZA3 '? ?d"`}5  ' dR  0  `A7 '? ?d"`A 01  '$D 0 dl J b ?  ,$@  0 $  Zl?"FR/r ;V02 %  T ?"  2 &  T ?",VwB '  Z DԔ?" Q`2 (  T ?"VB ) B Z DԔ?" P2 * B T ?"V8B +  Z DԔ?" ,  Zx?"J j  ;Z0 -  Z?"*J ;X0 .  Zm!?"&*J ;Y0 /  Zlo!?"   Lc1.0B 0  Z DԔ?" P 1  Zt!?"   Lc2.0 2  NDx!?" #* Lc4.0 3  Z{!?"Bb Lc0.02 4  T ?"D^ 5  Zz!?"FRZr ;W0B 6 B Z DԔ?" ` 7  Z!?" * Lc3.0  <  s *A '? ?"` k '$D  0dH   0޽h ? MMM̙y___PPT10Y.r+DMD-' = @B D' = @BA?%,( < +O%,( < +D"' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*> %(D' =-s6Bwipe(left)*<3<*> D ' =%(D< ' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bdissolve*<3<* DY' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* B%(D' =-o6Bdissolve*<3<* BD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*? %(D' =-o6Bdissolve*<3<*? D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*< %(D' =-o6Bdissolve*<3<*< +  0 !,(  , , s *x! g CONCLUSIONS,  ((   B , N D1?"LL , T@!?"Y\TL___PPT9.& qEffect restoration is feasible Rests on two principles 2.1 Matrix inversion in discrete models Requires synthetic population and propensity score estimation 2.2 3-proxies per latent in linear models proxies can be decoupled by clever conditioning 3. Conditional independence tests can be replaced by tetrad-like tests (using 2SLS) for model testing f710i07i @`H , 0޽h ? MMM̙80___PPT10. t$ 0 4(  d  c $D$     s * DH ]    H  0ηo~ ? ̙3380___PPT10. Hv9xX_hSWιmkӚv+tO"JA/B4m#:|66d݃ CI "o2|Rs7Roj*& ;|ko=83h+(6ǜjd_&Z* СXC2_~s_?m휛nlʼNZ:F RԚNR?ecpxa8CPvg {77c9% uc0aϓ]܃4]6=Kedqb'-PJR?|)%<R/ӭ]i־ !x&;/T2]!ddI/T 7bY fu 6UՓ".o=&GS/J~)'߸یAGpQ1}F#1J[-3 QqYy^[^Gpg)˘E)XRzLCs-v09Xsq}qhH#CmjGq ;Ƣ><>eO\T/#IS^z<&[.(g}$b"z020bx}r{v$.S?[M8dDŽl^,:`#[@#eI>$&71hT J;u۽CYIwM[@=;g.vPlߏQ5/ozVn'/ w&#tBk~}2ow9-,3xXMhSY>/m80jT0 mqLY')7 722 "Tԅ 7"R܈0{}}tvrͻsw9}ՖG[Al xm WL@6fIÒЂ0/tAG!o ߤ`*B&eαN~k|X[l ^ҚNҁ3Z[f2,Po'c?.yFD>(yJ]::cRw  owC͜2>7sp bx󃸏$!bqׯY<3yy!9}A5*_Lⷦ25`*vTh~R9+p)9SF)ۤ#Ê?#l>!͟岟uz|iu&db7WpD@8o͌ x;^0!&T;|S Qd[d#,˩}mJ8ņ4g{4i1p_cTR~FR?rm6ywĭ+_F`sOs)w_C QXNa=8RX[/{\-(ʙ)#,/zǡ1':ԗ9el9-wxN/ ~s`N_|{Oǰӯ tB;}hHycEo**.O"nXr'mZr#O=f",O)Yrzoir<|vrQwf\{#9W뻋cKR0oi_E͏ >c󗃹ꥫ<`*~MoŨY\?DuօDbg4o{+EcoS73?j.S]&xX]lTE>nۥ--cB¶DjbZic-hh@4ヱbGmJjE FV$Q,ۮgtvK@;a;gf;s^x\oiR'4$3|&JRDCJe=vC @`{ ry9Ϗs?ךI3yF|󳗉 78n܇^܇("BL}t#>2QWZMBb\II勀bۊ`&Η,/lmva{zٖm)r\LH քDjb 5[ha+./>h0<1Q>DJ"C`CH(B33qfmwKV4ws338}#'*Z.=Qy9 0WnOLL Dfm ?@? TC.9?;+3$*?6o|.ZSsz CwF>uY蛩KGg-oe<.x)c%|G^ejm|aV`?|9?o| "< {t/<?:_čˡH`O?'^)3"{6.j : NkWWۭt$ӣL4f;x$gnl/dzal୒*|UVB.C_P<,>.[Pݍk`ϐeǮΡ`MJ*x;5cгd_FnFFNk9ʩGyZtB]_pRؘwllo8a-'^8 /BF\S>)98$qD=(L*pPv=ļq(%MbOF 'ѝʞ8!"ǣD%w\>cO>'<煈x5d[NpnO My=в~~w3%niO*`NOcb܄5YWHrE]o7,,'fQ6wC[o<.#< Kb ᾮv%vpr)l=G7w8^m尴7Ҟgy1MSqFɮFv;z!`_g8]KٽWZBVbٽ3^i ϩE1a7t{'=lS3rtk庽[;ϔcʇ u1L/~qT&;Ws{$h"^T ako-Cx*L)_1<]&]2]~ԏ CJ]>9._ۏ{>n2e-6՚،'1 W-܇W~6; w-&`G}S'Uz1Z`ʛb\3?NuL_нgߴWQ.ȞA_腨qP]aG"s ʓzk֮W_^ngnT\O}d*2p%=;}"UQ`baQz+x۱_.\v#0*6ܳ=Cy|H-S-N]Av8;_F7NTrp3΍?cA hc+bGo|">j;8?:,ުYD8OxV ߘ1Ib| a5a a-G[$kQ#};xl$MXTioiCfv$JK;|?<=qF :8%$Yf್6 (UMد?ExK?lTr-C#V $H@QP ic m"?0xx ?Z.p[Wkfgξy;~}ηhH92Bu }\.]+(@.zT36}Pa9b.hhf=ypL9tL]Ae3_d$+s< :}V԰m*7$u;;4#{YQ@n316Ua:GX[cspyv?R__w5]|r>؍0 !QqBi)*kF R};e'1T3 x|֊v4қׁ~ޖd@?=>c-VY\ vuߘw7 Hq!Q:Lp|zKcAl[^TqZG*= 8 k`% i{J]SX:嶿 g(8(xpR JO"CJ:Ċҹ<%ZLAͪ'e[ |~ޤj3%'bcQGfH~i%9>A51C&>nAUo];skCj2:0K|^VYӁ wBX&d[ٷ]FD$t RD\hml+Ukɔ"[~ov7+Mc7j,rm!7 f=?O݅?,v2K[s@t~R9IxWkA3fkV łQ<46H>&5s< [=zETЃ/Eh7_vI*"-7of~o7=W?&@ȾeYJou@ )qTS:YP) Ez֓͗TY0 N:Pbߏ9~l}C!;Ϡ臄n@|.l$O<`0LCaguG n0%ln_S6ᦿYUnut"*o+߄= bs;[?I_fg{׿j:b KQy0ޚc-e⧊"dPv]aF`XY1 u68QrLR<^cAk߄*PVss,rD4~dOzo^FU(:DJ#>.=JOVKe3co'Q]Ag"j%&wiii2 Blċr7ƣc ;}YJ׳kxGxD_Zqw8:imM=S&wxmS1,l>8׋ܧrLD:scb8]Vq䠟ɦt%,NY2fŘ)RxWMLQ *"HY 5@@QuM<OHR&$ɳ&ī<&"WEM cDmy^Y %RC ;73vf>U~. m[c8((kX,FK.,*  C|"%; V[;fg+oŔULRi' 8?=Ao1 7S_)T;ٓcT\>F{ei)O9UWcKbfdO9ݗTN5dU~:HՃBփH;ccWwefp*ұI)P9-KU@f3t11>Ï(IES8S@:T0)2!y=D]aJܥH+P8QX\+y%sA<yQv?Z-2oG&5 ۿY9"lR1š+S-F[Qr6WcHDV4T[6EZ8L}GQxT-e;rr3nLGy4'^~;8 ;ް?8774}5qXꢡ~{ZgҬ5SΑ ux]csI褔T\#ޤqܦQ #1X?jjlhp_%'6S)m[hMW@%a􂒩|y>NOEa/w;2gqC`"p9DZ ^1DSp"u'rɉCJ~i7ۧS]Sدdm?h_2?grW oýW5'/hg"搿 }/ך[zwYCfz|zoJ  ?I ߃&K̫$S윭sasYhi[o_:E{ sxX_HSQιN.әЊ}H 0̰Y[gfD% zbSec== rsngٝR7;s[|]\HZ5P8.)lij@?h+lH!״9ڟ2%S&(ʙ 6vy ٛ{g%\L~{+ٸ {V'DĥU,t!g`vv.e2ї6w0mʙ Bͳ"?|99Lk(f [w Q>NU?JB@IEi3Q܀1y㿄R0z2#cAoџR13PXzR׊}6 k+j^"-Y c+0<ؠ9`xot4J9PwMss_Flbgp1|5ى^X1#W$9Ȣ1*=X5:TuuC`IL=;*Embc[qg"?.9]Lѫ,O:N=be= 4WGk_5*[Uݣus]ϥO G!A=1-; FJtGH,xXkLTG>^]YʊDXW* !iFmBjS&?l64إ&JKYs ]ikÐo|3sΜ{Ι^&ڱa@ \ `S.TY ~7A>@`+<$˗؃]s_K:P0|jǚ/*-l q /=N~?#۪?e}P SK2b0 "_8:':Q>lOG ]^3m8oǿa7S0ڪv| F"l0|_J솽P_qPyDg/G'-VG<wy7)Н3?9LcAOdD "HGd F#2|` " 11111 &!r^dD.b 1 g>| U$Ee>4wX\kh޲ ;/w Vm{$7d7eu 7zПJ#'dw$ [S"IFIl*El]C=> zi0[,!z~daƁ`hm~g|JQ0$šAezy{H#1ʼnxT|$i;6a44ЦvOVF:^K2|nzN#p!5G" *v+CHCkk* UL,Wt2Ul-H"! +%mvs\\0bXFbfX&kqĦb.hL l_؅(ʉ2풭K.%`͞ezCs6iֻyF޸c9FۙFۄF[rڋpYs}!nUr'2T_X{Nm]]ۖu#[r4ڛBu~HTʻkMF<`d^S @UPmmΘ{Io>O&ůoKը㴳 s$YZ)4r %7lnY5X)YXl>{m>`hHZ-7f3.ƙ l&g/H@t0xxZZ &6խFv Gξ{?,723:.'"bGfw{Lys1e3C'<0Oox9:X;!"Oet?aXvQC؏a?X^į8.s<ە CuD6^79_jwIy{*}6%0g{)rt'3"5yt&닉Y7hL],ۮ|Y$/ۖ/ x{\Q_`oX=ok&72g 󃳒}eYL*z ysfy1aiߚ缞į#hwbÎkro$jgs>wX ~d;!oJT ~$T>NT~NybScWv|'J pj_g Wgk@?8Ek/av%G]W(ʟ m?M \gRl1{nR xsH}9}(>v|Ώ;6v'x#wXt?S*% xXMlE~3k;:ΏiEUih(MqIU)5)HH$ 0vӸdJUB=Tc{=VH*R% H\ʁ ET0ofg8m?hgͼy{3wuw._mhR .[aJP 0ۖWVVIceeSJB`ϓ`dºNk7m! !{_D-R S$D i(Ay>Sp c f>kV~@яz:W8`v8"^6}?}o|nalG!*{K)\r;?B7BIOd^W?]_ ",JxMDri-sjYԃ[ͧ=XJ3+\~)O,Gav GJ6bRY\'x΂VCM~dN:I,:t92Q3W`ozφεg(k> m*ڛ/gkfLaGgwl'+m.{n$@??p9l?uz!PrMw1=goFr$M!x}{0bfxC >c6t1F pGb'ɉpX*L{v`pjC zkSv-n>0Ɩ*NW ;]W;PӀ~hkh"]b)yڹ@hta ܪ b?M+O[FRǐ_~ǔ/;#:7vRdk H6WZ_!F(!u&- E›a v>Upd,1DۿgNxq}O;/ċ/"U|[do#>ck6;~W;~8<^|EpsKc:M~mɥz{'F\:: ߫i ("_ |p1:=#{G]S8~C^@g7Zs{,γv~iT yUv/ImӇ{TxKőgTG=Γ?۪_UqJVxQSqGUv2bď''Y6ke58 XW;8E݋zKl\\ xX]HT[^{g̣3BKM{˨Iԇm&'sʎHBEO[CEx/Av/DA9shQXaYkퟵK+4L%cX9o*LR%9/sJ!X X{z(|U9'#dBouuG'7Bŝb08>{QdZ^}fgFC~. SN7ջ6moKִT/Ns/t󟝥+gjzCn7&×iQz)qݒO0FV xGX{PJ3%1t}S‰A|[\rOG9óPcW;;J`ɑ0wFX3[KcoYa2OqўdbëjBbwK97 [Ȉ3Ke+KQ[b}3mҮ[ t&]~H^«ՄmڻR^'o ߷z3~8usTU5qpwxo(//+?*,xX[l[^{5տE"~=RƤ탧czT:2$!^G/;I-/$G> ".%wr:`qx1< K=B/CmKoz/9pWXκiN- .Dy$:L/I~!|YcUЏLAA92gXbQȣǀ ~r94tȹ{{Zp1㹠hF)RUH"hBx+v |ckUlCƧ#p|]s˱%GYIp_ M |1VxF?tV$i39f~,Jh{ٕ uno]`lǵKh5鲂nюcSl/_8o?]G;+5$or $'\yeO )ڎ<@rPEaʽ1'@&Hm\ʩI$ovgUѴ@)V)}W)Qn z*R4%($inu9@%6(pMg }Nn[ V;fͭJ&O=VG ACW Gm˫ 5J ! eB Nm ͈kmۄм}pKhR-be\P,$D3Qr5f{_b]`Șbx #7\exM[0ex_0>D xwq"x[lTE[vO-Tش"E@H%(6mݾ`wR5-ibC hH~ $&FQi=gf6L93̜=gnG.7d2ؕ1&&''iH6w:۠vi!6'󯤇$iu @HS/ALtL=5$:o685xFC [ڕ`v}[Pn3ABf\Z'abYLOE{`.`:RB8ceDγ"7בNf/ȼN8 N3wPPRnٙlL|/P4n1Ek2g6>s-#ܱP=3٬?h ^N\PSٓUGW̲1Gx^ ֐`_RF?rDa77-x@lƒ~ @PkW9 ;I,3 n_;uȞ`6!y!?+_ 160W`UwDِ[Y7b(TEy‹&&yblnoAPǗ \ [ۊGNT'BZ%%:KY!upS^˃L*57DTҳìFu{ގv_ӳPMgSVt8O7iI9\;g~,<#c3Ww-h x)c -%ڳ%l? |kro*JX%*Xu )~G[L{|ķ#H3o2"oS#qģC/gTMvXrx%^w 9rɎlbQ>%CWzE|_?ͣ1F[i4sm#zʺM77 .Ǟ_Iߊ;3?Z8mGMz p^ҁj8*9vď퐡Gb>|/ hBLxtl#[4CȷɷkȷsđOZwyt|57m?oYMKN>TdˍF|b0xKhSYFSNV.Xj3 #(`[lt!6mWZ7ąnEg%.a3Qxλ}q;&w9sej{ e w[u.N~偮fKϰ1:SLb LaTPBWg HMd=&!YIN!va|9;#M}7z =ڞT"譿Fc8?x=m#Y./f!{;$ d%}6H^~QKtE0d'#;{RCVݿ5λonbygΖ'?a$K3Fq\s?{A"6Xߴ +^Nϳ_mӈC_:_ݱ8B@pη{1'3Frf9SQ ^~}HN"xWoL[U?2^ LL mN`&Cdc| :h톐}`,&fq1onll~D&'>lF_D{J99{s|M*2`OI48dx^S BPXPf9Ńڈ{G'y]oŗ ۭv3a/jDQAH4{ Q8> CCJÌHL_:m #pa$d15mu61cuکMc"r'(WA>R :`znT;XN"0/K)&SqGxPә86&-#rT/eۅhC4^R1m${r7ʘrIP>L¿b{( #Z\¯>Tq+psF5+ѹh46Iy,G3UcRՠp$j}`857ꥬWi7o?O=ռg`Cg?m9[o,IJ ]stj| [=0KZ:Ґ#A}~j}FGYKMJ0)',gz?~] nIoF^#7a*^L=ӓ@_:2b@ݤi{>|8+ !W~*0+-S[[< 3(fuR'E8t' W7]lo%Mtzc#0dKs-[gٮR{-&{e+dIպ[3<[ٲT<[u-vWD:d7<ˏ]+ϭ GdsN{TzϥrӝOүk>~U0ݼ4 ZC2RKN ζΧ91^ge8c&O"1 FQa1 W12F'jFXXʸxcc%cZƻak1xc-0>z: 2,l7&Wo[o5O;==_H2oh\b&'!HnxXMLA~3[@@5qě Q0P `-bH<O&PQ&j'.Bx2}.U`To3̛77ݘrC Rb(41Q^q8-NI%mEO hgŠ +BZw2~6aQV2$gX+o~yPCXjg,J_*qrC'ΡN[ʅ|//%@|(?2l"ebۦI?[8os1y%9ϊ<֩:'R7_dOf_q އrQP+:D&JZv3E19- 8.9='OJse]3 H ("ܑ[*_T{O\o<][ELIBjW{0*^ 18Ei:4WôWNV'"wBV򘪗YӼ\$&|D Sg wT3?ZԿWh2{[%>_>|xXAhA3kLŦ5S/R(%"`I&5. -xēzoB={^Az7ORggbVw%R3ͻGkO @z -:ĕ8$uFA* GO'RۃkBٞ ZS.4z'?ƪUֶN׌jJUt  8s:c_gaDZAJK_ÕOA&e~9d!WFU- kqBw~A@DJwlJ՞ݸn N;/tL0( v&W2.A޸54]gO:r,ncn} nycRVߘi g0Nn8 /R&b~jV!wjdDGPE3kVS&K2ÊW~X#|(8xlhDRe#&s+'oxzwY{^Q73H\S+V#aGX;:X]7ffK/ò˚v?č]R*bzEx) <3T6e@t(I'"#Ww ˸k;m6ss%_̧KF,e9c*4)pcM%]R~Ew^xX]LW>΀2в U,!v@H4M hiD4`H4d7neVppX|k}>M4ߌ}0i>4D}ÔAUts{νMVm 2(2WBD2lTr@ :'  CSi)wNϧD\ßy6טS*DeKpF －(p ۾=ϠEeG`'8-S|tV.<os g(  ]!_N_D_fEx/Cm̏a bH#)k?ߴw*TC,ӧ?ƧzM_⛠\m%P+1%1ní!{؎&p2fZ9ᡄpDUǤ~co؇W4H0Bg"@A1K8LJyeVUP-1"2mDӜ2Nۑ:co>Xxx!# [Qӎghj~4zÌ0SiЫ)%o\ߥć4֕9%RTO+#uj4V3} f'ԕnFj9˖MDK+I^tw.etH?Oh\漣g,cn*8ͻN_cL3 f6=6v*7 R> J#1de3ݏl#O<<<]GczNj xlpЫKy=O+uU'g^2qwfMyyrkδ[*v~l{.O4p\4`wtK{0s:^W,#y+^4{=9rj1)G'+Xէ4HF\XQ6rD5$ 㼶+5,&T\ǘ:K-ljutSK߲uجufu^W"x[lTE[vO-Tش"E@H%(6mݾ`wR5-ibC hH~ $&FQi=gf6L93̜=gnG.7d2ؕ1&&''iH6w:۠vi!6'󯤇$iu @HS/ALtL=5$:o685xFC [ڕ`v}[Pn3ABf\Z'abYLOE{`.`:RB8ceDγ"7בNf/ȼN8 N3wPPRnٙlL|/P4n1Ek2g6>s-#ܱP=3٬?h ^N\PSٓUGW̲1Gx^ ֐`_RF?rDa77-x@lƒ~ @PkW9 ;I,3 n_;uȞ`6!y!?+_ 160W`UwDِ[Y7b(TEy‹&&yblnoAPǗ \ [ۊGNT'BZ%%:KY!upS^˃L*57DTҳìFu{ގv_ӳPMgSVt8O7iI9\;g~,<#c3Ww-h x)c -%ڳ%l? |kro*JX%*Xu )~G[L{|ķ#H3o2"oS#qģC/gTMvXrx%^w 9rɎlbQ>%CWzE|_?ͣ1F[i4sm#zʺM77 .Ǟ_Iߊ;3?Z8mGMz p^ҁj8*9vď퐡Gb>|/ hBLxtl#[4CȷɷkȷsđOZwyt|57m?oYMKN>TdˍF|bxX_hSWιiLn㚦RQؗ(s]k,?MbhhWcc0| =tQ! U66Msν6&Lk˹||wrJ:ovRo9 PߊR/5蹢=' ܪP:>sw.7cgo}X31~KvUa:I[SeN Uq8 ICW)O2y/X~Tm(py ˨j<EGY R:4|P=l?DL"/sl^5uFqwkM e+@u7aˉ,!%=&=3{d^ h:(۫T݁u4"z~'ﴺhWzo]<ޛLY#n=T" ^'==zp F1/QZߔ5mEPgFP8Lu7Oq.[Za ҈|\܌?ǿ׉?hžl'<\HΌ|ܮf}LCyv%mާ#~Iai@=,/bW ٜsCeHL؝6p\+r^ܹ'ŠD&\26{D({c&›m?;H(X#cc?Z43a?@+O!oqy`}M0ϜJc1U9%+GG\kr4Ճ VŦ:Dl?t?CuEYX~w,oo%P;}ԇ@[v^AnhLjH^Z}kDžd0"Jc:Ϡh2̮Ƅ`0;p$p窔9QOw%rhKb_oan䐵mP [ވcH&]ga{~ -a9f7vٯ[Ii<׸yr;̉ߩV{{{;\~_ ,W/w78w\}ƿK=KxMKTQ?X(Q1K)N.e$q%-A6-Ė/E]}l*{o|4csx{9|9g@gA`jC2)bn5 0vXq+D` 9?}xȼ|Awqth[>e7>u %Z>|'F` L~f!fnDu+k sc\7 3;h> eq?E73W5Hv\?a5'oں-{s`ǹ>}鐬{! B?& J-Jϗ-\1Nֈ*o.B  cȿO-bOHdDx_,d#,lQW1SdPuT!Q5)sоLjO,njq$5 EL2FVLJ=]fxE+7Ri qwgH9.~;Z??l_QrZl7\.I1Jܫ) ^? _+^.芤"U!9Dzo o`?@};.[=nPMuyhG_Su rЄ'T{@@>hkQ+,,Q_5qBiEǸr `,D/(: B@XՕd޻jx=}>kȘڪkx)?26bD+OKuD$25VXzAʢ`&tƆed:F Y+~@TV,k+tțjs˪}6}_Z*CZNYj!5\{L'mQN ZGL!C\ct,D)>{(?Z Kcy3=Dc'c}G/>qQ_O q-IxKlUp]$N(.߅j%*AЀV₄Ţxɮl E'$J/!G"PQyﭵ^l&X7̼GW[cnӀLd!Rsf4X0.+PBW[wNϿ!?]YaSáO=K`3 &pO2U,C JssWk?xU N*s8 &y=&0.Ѕ o#8U2w#8i?`yx1m5'%;n8=NA/MuTLq dߗW$Ș:AU}3 >pKփ[@?ӄ c)>ƻtYg eS)Yb jTsֺ_36TɌAQ,HH}x!3R9,#-|kv)-!}`fݞX-!ZO=Pg25ۣ\^u5wxYobghca ~#}~RqʎuⵑvM;x&iokRF>eyLu&sQ2)wS}k^NyՒ\\>iF#~_ɍۻ^Z\Nrճޒ{F]TB|LTl bA #~`WNrc[,_:hbfw1o[ðZkX^?'{[s#xf5'5dgϧ>Y`T 8|gG޻ZƘfCed8|ȃ}uC O5ĊBZ[Vt֤ -.Ur[c"R>W#DyԌ4`57V5]-O>|.a*f~ N~^pJfHף1) ;4cxbZ?>p3򿧈{%;1+. !sY븢.("*X}E.Wj{{5M!{$H\VI28-&+zrZ{  CϣU~K X!?c獓f!;YD<݃摭״{ojWNDΜ+ !a*NO{Z Ie+1!UkoXI WosJ\%NDey.oRN%XzEdR.S%8>R9n?bj]wFٍFh(H|OF.6k/QGyT2xY_l[W8^ӺiV,+^ pn7IסA .l.mUe jOH& ! N T $/g/BwN7 k]|ο|w=㷯 Ew}}͗0wh0V?ЇRO"dQ ilu{a^y*=\:fH'mO#n <9>C[kZi!s]n {yӰACoZsi<_hw!K{z}@x Pg}?l4OE)؆1~c]=e@~/lc)bv$o.g@g~O#CtA8oG'#I`x 8 |O#@ p 8 Y_琞b(I@؞-'*~[S~CkzA?wi{}ܿ? 3=rՐd".j1Y֍"vL +n 1H#Ly~eI{:_r#36U)_t 5Twr\fJX.|8RzDNɒ$ `d#h˲k Szz$a$,=aHFJ1  MC5UUR)>7^"Sw*B\ظ*'eT!f)!&KZM0R(qmԌnZo%=ϖ+_. ιZĴ0 !q"b!>G@wh;܃iZ8R#'2EIGF 1FvC_]sϑpG˃=]{6ajkZ7e?^Ci>W`w@?~CԫZE?%zi}|Nœ 66tfWF*R~*6m=9?xJ+ò8 E$FNbce!9W2=s24.7sS3U(4' NByNV6\Te<.7W-_BVzHad2~wq1@L7IA6ajPE[Ճ9H=T04d,j՝?hÿ0YԷ]-Y,9QugWHe4g Kn4 ,ƙ(/IԘse1"oˆf13qiW!fXFôg!Z}2oؼ[\86dup4#Bf3L][p[re/Lcgeq0|!ٖq?f}5R[|qOEۧqS ǧwpܿ42$`yHV>fqC;bU~}(.wVvz#W^_I/7DkO/ }7P`|ɩLړ SW'9L݁bITzRXq&_ɏ`Davra\ͽ3XGby|%L >w0L}tb~q%_s}x,wACWMFCblx1 EMFk6F(GDICx!b $$=='h% % V0xx x % % $$AA" FGDICF(GDICwoyFGDICRp@"Arial Arial"(`@"Arial 0 m[ÒO0?0U0%0(خÒlɒɒ\p[7ğ#m0#Ò0Pdv%    TTtwA0AwLP1    TTsvA0AvLP1 % ( F(GDIC 1FGDICRp"Arial Unicode MS Arial (z(`"Arial 0 Tm[ÒO0?0U0%0(خÒlɒɒ\p[7ğ#m0#Ò0Pdv%    T#A0A!LxON MEASUREMENT BIAS % ( Rp"Arial Unicode MS Arial (z(`"Arial 0 Tm[ÒO0?0U0%0(خÒlɒɒ\p[7ğ#m0#Ò0Pdv%    T"/A0A-LxIN CAUSAL INFERENCE!| % ( F(GDICCoFGDICRp@"Arial. z,z z( 8m0-g0-zruU0xh0-?0uU00uU0T\kK0`~ 0`- 0?0U0%0(خ(h0Arial,z|(`dv%    T;CdLA0A;K LdJudea Pearl % ( Rp@"ArialS zL ,z z(%8m0-g0-zruU0xh0-?0uU00uU0T\kK0`~ 0`- 0?0U0%0(خ(h0Arial,z|(`dv%    T(NvWA0A(VL|University of California % ( Rp@"Arial/ z\,z z(18m0-g0-zruU0xh0-?0uU00uU0T\kK0`~ 0`- 0?0U0%0(خ(h0Arial,z|(`dv%    T:YdbA0A:a LdLos AngelesA % ( Rp@"Ariala zT,z z(K8m0-g0-zruU0h0-?0uU00uU0T\kK0`~ 0`- 0?0U0%0(خ(h0Arial,z|(`dv%    TT!c$lA0A!kLP( % ( Rp@"Ariala h0-?0uU00uU0T\kK04|0< A|H]| 8< 8< خ(h0Aria?0U0%0(x| `dvp7^Zw%p7 D[!dv%    T$cxlA0A$kLxwww.cs.ucla.edu/~judea % ( Rp@"Ariala A|H]| 8< 8< خ(h04|0< A|H]| 8< 8< x| `dvp7^Zw?0U0%0( D[!dvp7^Zw(%p7 D[!dv%    TXyc}lA0AykLP/) % (   xh--$xx--'@"Arial??-.  2 w1H..  2 v1H."System-"Arial Unicode MS??-. '2 !ON MEASUREMENT BIAS.-"Arial Unicode MS??-. '2 -IN CAUSAL INFERENCE.-@"Arial.??-. 2 K; Judea Pearl.-@"ArialS??-. +2 V(University of California.-@"Arial/??-. 2 a: Los Angeles.-@"Ariala??-.  2 k!(H.-@"Ariala??-. (2 k$www.cs.ucla.edu/~judea.-@"Ariala??-.  2 ky/).-՜.+,0    GOn-screen ShowUCLAo ' Times New RomanArial WingdingsTimesArial Unicode MSMonotype SortsSymbolCloudsMicrosoft Equation 3.0Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9 Slide 10 Slide 11 Slide 12 Slide 13  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles _K{KaoruKaoru  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTVWXYZ[\]^_`abcdeghijklmopqrstuzRoot EntrydO)Pictures.Current UsernSummaryInformation(U!PowerPoint Document(o{DocumentSummaryInformation8f