Direct and Indirect Effects

文章目录

概
主要内容
- CDE
- NDE
- NIE
- TDE, TIE, PDE, PIE

Judea Pearl. Direct and indirect effects. In Proceedings of the 17th conference on uncertainty in artificial intelligence. Morgan Kaufmann Publishers Inc., 2001.

概

CDE: Controlled Direct Effect;

NDE: Natural Direct Effect;

NIE: Natural Indirect Effect.

TDE: Total Direct Effect;

TIE: Total Indirect Effect;

PDE: Pure Direct Effect;

PIE: Pure Indirect Effect.

主要内容

#mermaid-svg-NowsqWGSEyihWL04 .label{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);fill:#333;color:#333}#mermaid-svg-NowsqWGSEyihWL04 .label text{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 .node rect,#mermaid-svg-NowsqWGSEyihWL04 .node circle,#mermaid-svg-NowsqWGSEyihWL04 .node ellipse,#mermaid-svg-NowsqWGSEyihWL04 .node polygon,#mermaid-svg-NowsqWGSEyihWL04 .node path{fill:#ECECFF;stroke:#9370db;stroke-width:1px}#mermaid-svg-NowsqWGSEyihWL04 .node .label{text-align:center;fill:#333}#mermaid-svg-NowsqWGSEyihWL04 .node.clickable{cursor:pointer}#mermaid-svg-NowsqWGSEyihWL04 .arrowheadPath{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 .edgePath .path{stroke:#333;stroke-width:1.5px}#mermaid-svg-NowsqWGSEyihWL04 .flowchart-link{stroke:#333;fill:none}#mermaid-svg-NowsqWGSEyihWL04 .edgeLabel{background-color:#e8e8e8;text-align:center}#mermaid-svg-NowsqWGSEyihWL04 .edgeLabel rect{opacity:0.9}#mermaid-svg-NowsqWGSEyihWL04 .edgeLabel span{color:#333}#mermaid-svg-NowsqWGSEyihWL04 .cluster rect{fill:#ffffde;stroke:#aa3;stroke-width:1px}#mermaid-svg-NowsqWGSEyihWL04 .cluster text{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 div.mermaidTooltip{position:absolute;text-align:center;max-width:200px;padding:2px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);font-size:12px;background:#ffffde;border:1px solid #aa3;border-radius:2px;pointer-events:none;z-index:100}#mermaid-svg-NowsqWGSEyihWL04 .actor{stroke:#ccf;fill:#ECECFF}#mermaid-svg-NowsqWGSEyihWL04 text.actor>tspan{fill:#000;stroke:none}#mermaid-svg-NowsqWGSEyihWL04 .actor-line{stroke:grey}#mermaid-svg-NowsqWGSEyihWL04 .messageLine0{stroke-width:1.5;stroke-dasharray:none;stroke:#333}#mermaid-svg-NowsqWGSEyihWL04 .messageLine1{stroke-width:1.5;stroke-dasharray:2, 2;stroke:#333}#mermaid-svg-NowsqWGSEyihWL04 #arrowhead path{fill:#333;stroke:#333}#mermaid-svg-NowsqWGSEyihWL04 .sequenceNumber{fill:#fff}#mermaid-svg-NowsqWGSEyihWL04 #sequencenumber{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 #crosshead path{fill:#333;stroke:#333}#mermaid-svg-NowsqWGSEyihWL04 .messageText{fill:#333;stroke:#333}#mermaid-svg-NowsqWGSEyihWL04 .labelBox{stroke:#ccf;fill:#ECECFF}#mermaid-svg-NowsqWGSEyihWL04 .labelText,#mermaid-svg-NowsqWGSEyihWL04 .labelText>tspan{fill:#000;stroke:none}#mermaid-svg-NowsqWGSEyihWL04 .loopText,#mermaid-svg-NowsqWGSEyihWL04 .loopText>tspan{fill:#000;stroke:none}#mermaid-svg-NowsqWGSEyihWL04 .loopLine{stroke-width:2px;stroke-dasharray:2, 2;stroke:#ccf;fill:#ccf}#mermaid-svg-NowsqWGSEyihWL04 .note{stroke:#aa3;fill:#fff5ad}#mermaid-svg-NowsqWGSEyihWL04 .noteText,#mermaid-svg-NowsqWGSEyihWL04 .noteText>tspan{fill:#000;stroke:none}#mermaid-svg-NowsqWGSEyihWL04 .activation0{fill:#f4f4f4;stroke:#666}#mermaid-svg-NowsqWGSEyihWL04 .activation1{fill:#f4f4f4;stroke:#666}#mermaid-svg-NowsqWGSEyihWL04 .activation2{fill:#f4f4f4;stroke:#666}#mermaid-svg-NowsqWGSEyihWL04 .mermaid-main-font{font-family:"trebuchet ms", verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 .section{stroke:none;opacity:0.2}#mermaid-svg-NowsqWGSEyihWL04 .section0{fill:rgba(102,102,255,0.49)}#mermaid-svg-NowsqWGSEyihWL04 .section2{fill:#fff400}#mermaid-svg-NowsqWGSEyihWL04 .section1,#mermaid-svg-NowsqWGSEyihWL04 .section3{fill:#fff;opacity:0.2}#mermaid-svg-NowsqWGSEyihWL04 .sectionTitle0{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 .sectionTitle1{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 .sectionTitle2{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 .sectionTitle3{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 .sectionTitle{text-anchor:start;font-size:11px;text-height:14px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 .grid .tick{stroke:#d3d3d3;opacity:0.8;shape-rendering:crispEdges}#mermaid-svg-NowsqWGSEyihWL04 .grid .tick text{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 .grid path{stroke-width:0}#mermaid-svg-NowsqWGSEyihWL04 .today{fill:none;stroke:red;stroke-width:2px}#mermaid-svg-NowsqWGSEyihWL04 .task{stroke-width:2}#mermaid-svg-NowsqWGSEyihWL04 .taskText{text-anchor:middle;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 .taskText:not([font-size]){font-size:11px}#mermaid-svg-NowsqWGSEyihWL04 .taskTextOutsideRight{fill:#000;text-anchor:start;font-size:11px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 .taskTextOutsideLeft{fill:#000;text-anchor:end;font-size:11px}#mermaid-svg-NowsqWGSEyihWL04 .task.clickable{cursor:pointer}#mermaid-svg-NowsqWGSEyihWL04 .taskText.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-NowsqWGSEyihWL04 .taskTextOutsideLeft.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-NowsqWGSEyihWL04 .taskTextOutsideRight.clickable{cursor:pointer;fill:#003163 !important;font-weight:bold}#mermaid-svg-NowsqWGSEyihWL04 .taskText0,#mermaid-svg-NowsqWGSEyihWL04 .taskText1,#mermaid-svg-NowsqWGSEyihWL04 .taskText2,#mermaid-svg-NowsqWGSEyihWL04 .taskText3{fill:#fff}#mermaid-svg-NowsqWGSEyihWL04 .task0,#mermaid-svg-NowsqWGSEyihWL04 .task1,#mermaid-svg-NowsqWGSEyihWL04 .task2,#mermaid-svg-NowsqWGSEyihWL04 .task3{fill:#8a90dd;stroke:#534fbc}#mermaid-svg-NowsqWGSEyihWL04 .taskTextOutside0,#mermaid-svg-NowsqWGSEyihWL04 .taskTextOutside2{fill:#000}#mermaid-svg-NowsqWGSEyihWL04 .taskTextOutside1,#mermaid-svg-NowsqWGSEyihWL04 .taskTextOutside3{fill:#000}#mermaid-svg-NowsqWGSEyihWL04 .active0,#mermaid-svg-NowsqWGSEyihWL04 .active1,#mermaid-svg-NowsqWGSEyihWL04 .active2,#mermaid-svg-NowsqWGSEyihWL04 .active3{fill:#bfc7ff;stroke:#534fbc}#mermaid-svg-NowsqWGSEyihWL04 .activeText0,#mermaid-svg-NowsqWGSEyihWL04 .activeText1,#mermaid-svg-NowsqWGSEyihWL04 .activeText2,#mermaid-svg-NowsqWGSEyihWL04 .activeText3{fill:#000 !important}#mermaid-svg-NowsqWGSEyihWL04 .done0,#mermaid-svg-NowsqWGSEyihWL04 .done1,#mermaid-svg-NowsqWGSEyihWL04 .done2,#mermaid-svg-NowsqWGSEyihWL04 .done3{stroke:grey;fill:#d3d3d3;stroke-width:2}#mermaid-svg-NowsqWGSEyihWL04 .doneText0,#mermaid-svg-NowsqWGSEyihWL04 .doneText1,#mermaid-svg-NowsqWGSEyihWL04 .doneText2,#mermaid-svg-NowsqWGSEyihWL04 .doneText3{fill:#000 !important}#mermaid-svg-NowsqWGSEyihWL04 .crit0,#mermaid-svg-NowsqWGSEyihWL04 .crit1,#mermaid-svg-NowsqWGSEyihWL04 .crit2,#mermaid-svg-NowsqWGSEyihWL04 .crit3{stroke:#f88;fill:red;stroke-width:2}#mermaid-svg-NowsqWGSEyihWL04 .activeCrit0,#mermaid-svg-NowsqWGSEyihWL04 .activeCrit1,#mermaid-svg-NowsqWGSEyihWL04 .activeCrit2,#mermaid-svg-NowsqWGSEyihWL04 .activeCrit3{stroke:#f88;fill:#bfc7ff;stroke-width:2}#mermaid-svg-NowsqWGSEyihWL04 .doneCrit0,#mermaid-svg-NowsqWGSEyihWL04 .doneCrit1,#mermaid-svg-NowsqWGSEyihWL04 .doneCrit2,#mermaid-svg-NowsqWGSEyihWL04 .doneCrit3{stroke:#f88;fill:#d3d3d3;stroke-width:2;cursor:pointer;shape-rendering:crispEdges}#mermaid-svg-NowsqWGSEyihWL04 .milestone{transform:rotate(45deg) scale(0.8, 0.8)}#mermaid-svg-NowsqWGSEyihWL04 .milestoneText{font-style:italic}#mermaid-svg-NowsqWGSEyihWL04 .doneCritText0,#mermaid-svg-NowsqWGSEyihWL04 .doneCritText1,#mermaid-svg-NowsqWGSEyihWL04 .doneCritText2,#mermaid-svg-NowsqWGSEyihWL04 .doneCritText3{fill:#000 !important}#mermaid-svg-NowsqWGSEyihWL04 .activeCritText0,#mermaid-svg-NowsqWGSEyihWL04 .activeCritText1,#mermaid-svg-NowsqWGSEyihWL04 .activeCritText2,#mermaid-svg-NowsqWGSEyihWL04 .activeCritText3{fill:#000 !important}#mermaid-svg-NowsqWGSEyihWL04 .titleText{text-anchor:middle;font-size:18px;fill:#000;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 g.classGroup text{fill:#9370db;stroke:none;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family);font-size:10px}#mermaid-svg-NowsqWGSEyihWL04 g.classGroup text .title{font-weight:bolder}#mermaid-svg-NowsqWGSEyihWL04 g.clickable{cursor:pointer}#mermaid-svg-NowsqWGSEyihWL04 g.classGroup rect{fill:#ECECFF;stroke:#9370db}#mermaid-svg-NowsqWGSEyihWL04 g.classGroup line{stroke:#9370db;stroke-width:1}#mermaid-svg-NowsqWGSEyihWL04 .classLabel .box{stroke:none;stroke-width:0;fill:#ECECFF;opacity:0.5}#mermaid-svg-NowsqWGSEyihWL04 .classLabel .label{fill:#9370db;font-size:10px}#mermaid-svg-NowsqWGSEyihWL04 .relation{stroke:#9370db;stroke-width:1;fill:none}#mermaid-svg-NowsqWGSEyihWL04 .dashed-line{stroke-dasharray:3}#mermaid-svg-NowsqWGSEyihWL04 #compositionStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-NowsqWGSEyihWL04 #compositionEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-NowsqWGSEyihWL04 #aggregationStart{fill:#ECECFF;stroke:#9370db;stroke-width:1}#mermaid-svg-NowsqWGSEyihWL04 #aggregationEnd{fill:#ECECFF;stroke:#9370db;stroke-width:1}#mermaid-svg-NowsqWGSEyihWL04 #dependencyStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-NowsqWGSEyihWL04 #dependencyEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-NowsqWGSEyihWL04 #extensionStart{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-NowsqWGSEyihWL04 #extensionEnd{fill:#9370db;stroke:#9370db;stroke-width:1}#mermaid-svg-NowsqWGSEyihWL04 .commit-id,#mermaid-svg-NowsqWGSEyihWL04 .commit-msg,#mermaid-svg-NowsqWGSEyihWL04 .branch-label{fill:lightgrey;color:lightgrey;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 .pieTitleText{text-anchor:middle;font-size:25px;fill:#000;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 .slice{font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 g.stateGroup text{fill:#9370db;stroke:none;font-size:10px;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 g.stateGroup text{fill:#9370db;fill:#333;stroke:none;font-size:10px}#mermaid-svg-NowsqWGSEyihWL04 g.statediagram-cluster .cluster-label text{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 g.stateGroup .state-title{font-weight:bolder;fill:#000}#mermaid-svg-NowsqWGSEyihWL04 g.stateGroup rect{fill:#ECECFF;stroke:#9370db}#mermaid-svg-NowsqWGSEyihWL04 g.stateGroup line{stroke:#9370db;stroke-width:1}#mermaid-svg-NowsqWGSEyihWL04 .transition{stroke:#9370db;stroke-width:1;fill:none}#mermaid-svg-NowsqWGSEyihWL04 .stateGroup .composit{fill:white;border-bottom:1px}#mermaid-svg-NowsqWGSEyihWL04 .stateGroup .alt-composit{fill:#e0e0e0;border-bottom:1px}#mermaid-svg-NowsqWGSEyihWL04 .state-note{stroke:#aa3;fill:#fff5ad}#mermaid-svg-NowsqWGSEyihWL04 .state-note text{fill:black;stroke:none;font-size:10px}#mermaid-svg-NowsqWGSEyihWL04 .stateLabel .box{stroke:none;stroke-width:0;fill:#ECECFF;opacity:0.7}#mermaid-svg-NowsqWGSEyihWL04 .edgeLabel text{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 .stateLabel text{fill:#000;font-size:10px;font-weight:bold;font-family:'trebuchet ms', verdana, arial;font-family:var(--mermaid-font-family)}#mermaid-svg-NowsqWGSEyihWL04 .node circle.state-start{fill:black;stroke:black}#mermaid-svg-NowsqWGSEyihWL04 .node circle.state-end{fill:black;stroke:white;stroke-width:1.5}#mermaid-svg-NowsqWGSEyihWL04 #statediagram-barbEnd{fill:#9370db}#mermaid-svg-NowsqWGSEyihWL04 .statediagram-cluster rect{fill:#ECECFF;stroke:#9370db;stroke-width:1px}#mermaid-svg-NowsqWGSEyihWL04 .statediagram-cluster rect.outer{rx:5px;ry:5px}#mermaid-svg-NowsqWGSEyihWL04 .statediagram-state .divider{stroke:#9370db}#mermaid-svg-NowsqWGSEyihWL04 .statediagram-state .title-state{rx:5px;ry:5px}#mermaid-svg-NowsqWGSEyihWL04 .statediagram-cluster.statediagram-cluster .inner{fill:white}#mermaid-svg-NowsqWGSEyihWL04 .statediagram-cluster.statediagram-cluster-alt .inner{fill:#e0e0e0}#mermaid-svg-NowsqWGSEyihWL04 .statediagram-cluster .inner{rx:0;ry:0}#mermaid-svg-NowsqWGSEyihWL04 .statediagram-state rect.basic{rx:5px;ry:5px}#mermaid-svg-NowsqWGSEyihWL04 .statediagram-state rect.divider{stroke-dasharray:10,10;fill:#efefef}#mermaid-svg-NowsqWGSEyihWL04 .note-edge{stroke-dasharray:5}#mermaid-svg-NowsqWGSEyihWL04 .statediagram-note rect{fill:#fff5ad;stroke:#aa3;stroke-width:1px;rx:0;ry:0}:root{--mermaid-font-family: '"trebuchet ms", verdana, arial';--mermaid-font-family: "Comic Sans MS", "Comic Sans", cursive}#mermaid-svg-NowsqWGSEyihWL04 .error-icon{fill:#522}#mermaid-svg-NowsqWGSEyihWL04 .error-text{fill:#522;stroke:#522}#mermaid-svg-NowsqWGSEyihWL04 .edge-thickness-normal{stroke-width:2px}#mermaid-svg-NowsqWGSEyihWL04 .edge-thickness-thick{stroke-width:3.5px}#mermaid-svg-NowsqWGSEyihWL04 .edge-pattern-solid{stroke-dasharray:0}#mermaid-svg-NowsqWGSEyihWL04 .edge-pattern-dashed{stroke-dasharray:3}#mermaid-svg-NowsqWGSEyihWL04 .edge-pattern-dotted{stroke-dasharray:2}#mermaid-svg-NowsqWGSEyihWL04 .marker{fill:#333}#mermaid-svg-NowsqWGSEyihWL04 .marker.cross{stroke:#333}:root { --mermaid-font-family: "trebuchet ms", verdana, arial;}#mermaid-svg-NowsqWGSEyihWL04 {color: rgba(0, 0, 0, 0.75);font: ;}

设想, 药物XXX的影响通过俩种途径:

直接对身体产生的影响;
服用药物XXX会导致头疼, 故患者大概率会服用镇痛剂, 镇痛剂会利于(或者不利于)恢复.

如果我们直接计算causal effect, 则二者都会纳入其中, 但是往往我们所关心的只是单纯的X→YX \rightarrow YX→Y这一部分, 也即direct effect, 那么如何计算呢?

CDE

average CDE的计算是:
E[Y∣do(X=x),do(Z=z)]−E[Y∣do(X=x∗),do(Z=z)],\mathbb{E}[Y|do(X=x), do(Z=z)] -\mathbb{E}[Y|do(X=x^*), do(Z=z)], E[Y∣do(X=x),do(Z=z)]−E[Y∣do(X=x∗),do(Z=z)],
之所以被称之为controlled direct effect, 是因为我们认为的限定Z=zZ=zZ=z.
用上面的例子来说就是, 我们限定所有人服用的镇定剂为zzz.

NDE

average NDE的计算是:
E[Y∣do(X=x),do(Z=Zx∗)]−E[Y∣do(X=x∗)],\mathbb{E}[Y|do(X=x), do(Z=Z_{x*})] -\mathbb{E}[Y|do(X=x^*)], E[Y∣do(X=x),do(Z=Zx∗)]−E[Y∣do(X=x∗)],
相当于, 一个人服用了药物xxx, 但是我们骗他说服用了药物x∗x^*x∗, 导致其服用镇定剂的量是本应该服用药物x∗x^*x∗后的量.

不同于CDE, NDE的计算要略微复杂一点:

需要满足:
Yxz⨿Zx∗∣W,Y_{xz} \amalg Z_{x^*} | W, Yxz⨿Zx∗∣W,
这里WWW是confounder.
此时:
NDE(x,x∗,Y)=∑w,z[E[Yx,z∣w]−E[Yx∗,z∣w]]P(Zx∗=z∣w)P(w).NDE(x, x^*, Y) = \sum_{w, z}[\mathbb{E}[Y_{x,z}|w] - \mathbb{E}[Y_{x^*,z}|w]]P(Z_{x^*}=z|w)P(w). NDE(x,x∗,Y)=w,z∑[E[Yx,z∣w]−E[Yx∗,z∣w]]P(Zx∗=z∣w)P(w).

NIE

有些时候我们想要的是支线X→Z→YX \rightarrow Z \rightarrow YX→Z→Y, 此时我们需要计算NIE:

average NIE的计算是
E[Y∣do(X=x∗),do(Z=Zx)]−E[Y∣do(X=x∗)].\mathbb{E}[Y|do(X=x^*), do(Z=Z_{x})] -\mathbb{E}[Y|do(X=x^*)]. E[Y∣do(X=x∗),do(Z=Zx)]−E[Y∣do(X=x∗)].
类似的解释.

满足
Yx∗,z⨿Zx∣W,Y_{x^*, z} \amalg Z_x | W, Yx∗,z⨿Zx∣W,
可以得到
NIE(x,x∗,Y)=∑w,zE[Yx∗,z∣w][P(Zx=z∣w)−P(Zx∗=z∣w)]P(w).NIE(x, x^*, Y) = \sum_{w, z}\mathbb{E}[Y_{x^*,z}|w][P(Z_{x}=z|w) - P(Z_{x^*}=z|w)]P(w). NIE(x,x∗,Y)=w,z∑E[Yx∗,z∣w][P(Zx=z∣w)−P(Zx∗=z∣w)]P(w).

TDE, TIE, PDE, PIE

可以发现:

E[Yx]−E[Yx∗]=E[Yxz]−E[Yx∗z∗]=(E[Yxz]−E[Yxz∗])⏟TIE+(E[Yxz∗]−E[Yx∗z∗])⏟PDE=(E[Yxz]−E[Yx∗z]])⏟TDE+(E[Yx∗z]−E[Yx∗z∗])⏟PIE.\begin{array}{rl} \mathbb{E}[Y_{x}] -\mathbb{E}[Y_{x^*}] &=\mathbb{E}[Y_{xz}] -\mathbb{E}[Y_{x^*z^*}] \\ &=\underbrace{(\mathbb{E}[Y_{xz}]-\mathbb{E}[Y_{xz^*}])}_{TIE} + \underbrace{(\mathbb{E}[Y_{xz^*}]-\mathbb{E}[Y_{x^*z^*}])}_{PDE}\\ &=\underbrace{(\mathbb{E}[Y_{xz}]-\mathbb{E}[Y_{x^*z}]])}_{TDE} + \underbrace{(\mathbb{E}[Y_{x^*z}]-\mathbb{E}[Y_{x^*z^*}])}_{PIE}. \end{array} E[Yx]−E[Yx∗]=E[Yxz]−E[Yx∗z∗]=TIE(E[Yxz]−E[Yxz∗])+PDE(E[Yxz∗]−E[Yx∗z∗])=TDE(E[Yxz]−E[Yx∗z]])+PIE(E[Yx∗z]−E[Yx∗z∗]).