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Experimental non-Identifiability  can there be two causal models which are experimentally indistinguishable while producing different path-specific effects?@Hu 1, n~+ Methodology@Path-specific effects we are interested in are expressions in counterfactual logic (with associated probabilities). We convert these expressions into DAGs with sets of  cut edges. Our subsequent reasoning and results are stated in terms of these graphs rather than the underlying logic.1 &Effect-invariant Graph Transformations - Main ResultApplying the two rules results in one of two cases: Case 1: we obtain a  kite pattern. Then the path-specific effect is not identifiable. D4?.Main Result (cont.)YCase 2: all blocked edges emanate from the root node. Then the effect is identifiable. 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B  p T?D33Ԕ?"0?@NNN?NZ   p 0T H ;AZT0 2  p 0> n . A Pneumonia 0 2   p 0$d wG $  C Antibiotics 0 2  p 0n`{. C Headaches 0(2  p 0 `   E Painkillers 0(2  p 0 d  @Survival 0 2 2 p N??"6?@`NNN?NtI2 p N??"6?@`NNN?N  2 p N??"6?@`NNN?Nf  `2 p s *"`a2 p N??"6?@`NNN?NI2 p N??"6?@`NNN?N1 RB p@ s *D> OZ  p 0 L  v>Painkiller contribution to the total effect of AZT on survival?0 2? p 0|]`  ,$ 0 C Antibiotics 0 2 B p@ T?D33Ԕ?"0?@NNN?NIV ,$@  0B p@ T?D33Ԕ?"0?@NNN?N1O ,$@  0B p T?D33Ԕ?"0?@NNN?N O  ,$@  0B p@ T?DԔ?"0?@NNN?N R ,$@  0B p T?D33Ԕ?"0?@NNN?NP;,$@  0B p T?D33Ԕ?"0?@NNN?N;.,$@ 0B  p@ T?D33Ԕ?"0?@NNN?N;@ ,$@ 0B !p T?D33Ԕ?"0?@NNN?NV ,$@ 0 "p 0<D,$ 0 ;AZT0 2 #p 0(j%*,$ 0 A Pneumonia 0 2  $p 04j *,$ 0 C Headaches 0(2  %p 0[ $  ,$ 0 E Painkillers 0(2  &p 0,#  ,$ 0 @Survival 0 2 2 'p N??"6?@`NNN?N8E,$@ 02 (p N??"6?@`NNN?N G ,$@ 02 )p N??"6?@`NNN?N* ,$@ 02 *p s *"` G],$@ 02 +p N??"6?@`NNN?NUE,$@ 02 ,p N??"6?@`NNN?N c - ,$@ 0B -p s *D> g` ,$@ 0B .p c $D2 2 H,$@ 0  /p 0~ u  ,$ 0 w?Antibiotics contribution to the total effect of AZT on survival@0 2@B 0p s *D>HB,$@ 0B 1p s *D>W,$@ 0B 2p@ s *D>wT5,$@! 0 3p jACheckmark J\$D  0U 0   X 0  4p# " '`,$D#  0T 5p c $X99?X 0  6p BoCDELFVoxiYJ ; -$.9ENZdo(,@`Hl  7p . Bl C| DEhFr7"  y dN;%$##""" * 7 B "'-E9FVj5m C v C o  2 Z 2 g    D ~ 0 = 4I tO T V :T uQ J B "7 \*    2 f   l + I Z % Y"9Fj.Wo^QF/<\2+qC _~4#_XmE9crT&zS,: S  p    m ?    ~- K8 B H M uQ =R Q L yE ;: -   L    qz <V -   { Pq (>  f-wcT{F<<50}/:2:zI^? u_L~6]!Ha4&) l?1!3Hac|8 {  < k5 p   C ~? k  A    ^4 I [ Ej t z 7| z t )j x[ I 4 Y   - n k ? *  d p  5 0 \ { 8   c$ 9 K Z 1d j l ?j d Z KK 9 $ c    D \ 0  E  d * m = n - ~YaH3x!)768@`"`X 0  8p BCDEHFR.#8 Tq!Bb &(@` UH p 0޽h ? 3ff~BB___PPT10B.W{PS+#tDA' = @B D@' = @BA?%,( < +O%,( < +D#' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0p%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*p%(D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*pD' =1:B true*a3>Bstroke.on<*pD#' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1p%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*0p%(D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*pD' =1:B true*a3>Bstroke.on<*pDn' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*3p%(D' =+4 8?`CB4*#ppt_wBCB#ppt_wB*Y3>B ppt_w<*3pD' =+4 8?`CB4*#ppt_hBCB#ppt_hB*Y3>B ppt_h<*3pD$' =%(D?$' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/p%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D#' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*-p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2p%(D]' =4@BBBB%(D' =,54*3>#Bstroke.color=@BPB<*pD' =1:B true*a3>Bstroke.on<*pDn' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*4p%(D' =+4 8?`CB4*#ppt_wBCB#ppt_wB*Y3>B ppt_w<*4pD' =+4 8?`CB4*#ppt_hBCB#ppt_hB*Y3>B ppt_h<*4p++0+p\ ++0+"p\ ++0+#p\ ++0+$p\ ++0+%p\ ++0+&p\ ++0+/p\ +2 - u$m$'.$(  ^F ;P  ' G@ B ( T?D33Ԕ?"0?@NNN?N;PB ) T?DԔ?"0?@NNN?N.;B *B T?D33Ԕ?"0?@NNN?N@; 2 + N??"6?@`NNN?N8Er  S  M }     0@'\`   C Antibiotics 0 2 B @ T?D33Ԕ?"0?@NNN?NIV B @ T?D33Ԕ?"0?@NNN?N1O B  T?D33Ԕ?"0?@NNN?N O  B @ T?DԔ?"0?@NNN?N R B   T?D33Ԕ?"0?@NNN?NV    0T-\D ;AZT0 2  0N\j%* A Pneumonia 0 2   0R\j * C Headaches 0(2   0.\[ $   E Painkillers 0(2   00\   @Survival 0 2 P8 ;P  &G@ B   T?D33Ԕ?"0?@NNN?N;PB   T?DԔ?"0?@NNN?N.;B  B T?D33Ԕ?"0?@NNN?N@; 2  N??"6?@`NNN?N8E2  N??"6?@`NNN?N G 2  N??"6?@`NNN?N* `2  s *"` G]2  N??"6?@`NNN?NUE2  N??"6?@`NNN?N c -   0\ u   w?Antibiotics contribution to the total effect of AZT on survival@0 2@RB @ s *D>wT5n Z X 0  # " '`T  c $X99?X 0   BoCDELFVoxiYJ ; -$.9ENZdo(,@`Hl   . Bl C| DEhFr7"  y dN;%$##""" * 7 B "'-E9FVj5m C v C o  2 Z 2 g    D ~ 0 = 4I tO T V :T uQ J B "7 \*    2 f   l + I Z % Y"9Fj.Wo^QF/<\2+qC _~4#_XmE9crT&zS,: S  p    m ?    ~- K8 B H M uQ =R Q L yE ;: -   L    qz <V -   { Pq (>  f-wcT{F<<50}/:2:zI^? u_L~6]!Ha4&) l?1!3Hac|8 {  < k5 p   C ~? k  A    ^4 I [ Ej t z 7| z t )j x[ I 4 Y   - n k ? *  d p  5 0 \ { 8   c$ 9 K Z 1d j l ?j d Z KK 9 $ c    D \ 0  E  d * m = n - ~YaH3x!)768@`"`X 0   BCDEHFR.#8 Tq!Bb &(@` U % <\ ,$ 0 JR-Recanting Witness0 , 0t^\P  7 ,$ 0 C R behave as I0 2 - 0\ p ,$ 0 DR behave as II0 2: . 0\p,$ 0 *P(RX,RX*) is not experimental identifiable`+0 2NN#H  0޽h ? 3ff~  ___PPT10 .|`=R+vD ' = @B DL ' = @BA?%,( < +O%,( < +D' =%(D' =%(D/' =4@BB#BB%()?)?D' =.K7 BBBBB]M -4.72222E-6 -4.95829E-6 L -0.54878 0.06048 *3>*B ppt_xB ppt_y=@0BBAApBBB;<<*&D0' =4@BBBB%(D' =0l9 CCBB*<3<*&D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(Ds' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(++0+%\ ++0+,\ ++0+-\ ++0+.\ + 0 {0x(  xX x C @   { x S  @ 0   Here is the motivating example for the remainder of the slides. It would be appropriate to discuss here, in English, the medical story this graph represents. Also mentioning natural direct and indirect effects on this graph would be good as a preliminary.H x 0޽h ? 3380___PPT10.f{^ 0 @|n(  |X | C @    | S @ 0   pHere we need to discuss the question which necessitates path-specific effects. In this case, the doctor might be interested in the interrations of AZT and painkillers (resp. antibiotics) on survival. Thus he might want to compare the direct effect of AZT to the path specific effect in the situation where painkillers (resp. antibiotics) are administered as if AZT was _not_ given. It would be good to point out that while these two examples look very similar, one of them will turn out to be problematic, while the other will not.> H | 0޽h ? 3380___PPT10.f{@  0 ~P(  X  C @   ~  S Xg@ 0   Here we state what we are after. Perhaps we can discuss more formally what  from experiments means (i.e. what the set P* means).H  0޽h ? 3380___PPT10.g{ni 0 `y(  X  C @     S ڥ@ 0   {gHere we went to some lengths to avoid actually introducing counterfactual logic, but perhaps we should.H  0޽h ? 3380___PPT10.g{po  0 *"p(  X  C @   "  S @ 0   Here we discuss the two graph transformations which we use in our proof. These rules do not change the path-specific effect. Dashed arrows represent directed _paths_.H  0޽h ? 3380___PPT10.g{>  0 (  X  C @     S @@ 0    H  0޽h ? 3380___PPT10.g{ UF 0 V(  X  C @     S @ 0   XDHere we can say that identifiable path-specific effects are related to natural direct and indirect effects. All of these effects only  cut root-emanating edges.H  0޽h ? 3380___PPT10.g{ 0 [S(  X  C @   S  S X:@ 0   This slide traces the application of the two rules for the two path-specific effects in the original example graph, showing how we obtain a root-emanating edge in the first case, and a kite pattern in the second case.H  0޽h ? 3380___PPT10.h{Ad  0 t(  X  C @     S @I@ 0   v@Give the intuition where is the problem with the kite structure.- H  0޽h ? 3380___PPT10.| `  0 (  X  C @     S  @ 0    H  0޽h ? 3380___PPT10.|Ոrt! 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