Back-Door Criterion (p150)

Authors: Aymen Merrouche and Pierre-Henri Wuillemin.

This notebook follows the example from “The Book Of Why” (Pearl, 2018) chapter 4 page 150

Back-Door Criterion

from IPython.display import display, Math, Latex,HTML

import pyAgrum as gum
import pyAgrum.lib.notebook as gnb
import pyAgrum.causal as csl
import pyAgrum.causal.notebook as cslnb
import os

In a causal diagram, confounding bias is due to the flow of non-causal information between treatment \(X\) and outcome \(Y\) through back-door paths. To neutralize this bias, we need to block these paths. To block a non-causal path, we must perform an adjustment operation for a variable or a set of variables that would block the flow of information on that path. Such a set of variables satisfies what we call the “back-door” criterion. A set of variables \(Z\) satisfies the back-door criterion for \((X, Y)\) if and only if: * \(Z\) blocks all back-door paths between \(X\) and \(Y\). A “back-door path” is any path in the causal diagram between \(X\) and \(Y\) starting with an arrow pointing towards \(X\). * No variable in \(Z\) is a descendant of \(X\) on a causal path, if we adjust for such a variable we would block a path that carries causal information hence the causal effect of \(X\) on \(Y\) would be biased.

If a set of $ Z $ variable satisfies the back-door criterion for \((X,Y)\), the causal effect of \(X\) on \(Y\) is given by the formula:

\[P(y \mid do(x)) = \sum_{z}{P(y \mid x,z) \times P(z)}\]

Example 1:

e1 = gum.fastBN("X->A->Y;A->B")
e1

m1 = csl.CausalModel(e1)
cslnb.showCausalImpact(m1, "Y", doing="X",values={})

Causal Model

$$\begin{equation*}P( Y \mid \text{do}(X)) = \sum_{A}{P\left(A\mid X\right) \cdot \left(\sum_{X'}{P\left(Y\mid A\right) \cdot P\left(X'\right)}\right)}\end{equation*}$$
Explanation : frontdoor ['A'] found.

	Y
X	0	1
0	0.3753	0.6247
1	0.3149	0.6851

Impact

# This function returns the set of variables which satisfies the back-door criterion for (X, Y)
# None if there are no back-door paths.
setOfVars = m1.backDoor("X","Y")
print("The set of variables which satisfies the back-door criterion for (X, Y) is :", setOfVars)

The set of variables which satisfies the back-door criterion for (X, Y) is : None

No incoming arrows into X, therefore there are no back-door paths between \(X\) and \(Y\) (as if we did a graph surgery according to the do operator), direct causal path \(X \rightarrow A \rightarrow Y\).

Example 2:

e2 = gum.fastBN("A->B->C;A->X->E->Y;B<-D->E")
e2

m2 = csl.CausalModel(e2)
gnb.show(m2)

cslnb.showCausalImpact(m2, "Y", doing="X",values={})

Causal Model

$$\begin{equation*}P( Y \mid \text{do}(X)) = \sum_{E}{P\left(E\mid X\right) \cdot \left(\sum_{X'}{P\left(Y\mid E\right) \cdot P\left(X'\right)}\right)}\end{equation*}$$
Explanation : frontdoor ['E'] found.

	Y
X	0	1
0	0.4867	0.5133
1	0.5983	0.4017

Impact

# This function returns the set of variables which satisfies the back-door criterion for (X, Y)
# None if there are no back-door paths.
setOfVars = m2.backDoor("X","Y")
print("The set of variables which satisfies the back-door criterion for (X, Y) is :", setOfVars)

The set of variables which satisfies the back-door criterion for (X, Y) is : None

There is one back-door path from \(X\) to \(Y\) :

\[X \leftarrow A \rightarrow B \leftarrow D \rightarrow E \rightarrow Y\]

We don’t need to control for any set of variables; this back-door path is blocked by collider node \(B\) (two incoming arrows)

\[A \rightarrow B \leftarrow D\]

Controlling for collider node \(B\) would open this causal path (controlling for colliders increases bias), direct causal path \(X \rightarrow E \rightarrow Y\).

Example 3:

e3 = gum.fastBN("B->X->Y;X->A<-B->Y")
e3

m3 = csl.CausalModel(e3)
cslnb.showCausalImpact(m3, "Y", doing="X",values={})

Causal Model

$$\begin{equation*}P( Y \mid \text{do}(X)) = \sum_{B}{P\left(Y\mid B,X\right) \cdot P\left(B\right)}\end{equation*}$$
Explanation : backdoor ['B'] found.

	Y
X	0	1
0	0.7753	0.2247
1	0.4314	0.5686

Impact

# This function returns the set of variables which satisfies the back-door criterion for (X, Y)
# None if there are no back-door paths.
setOfVars = m3.backDoor("X","Y")
print("The set of variables which satisfies the back-door criterion for (X, Y) is :", setOfVars)

The set of variables which satisfies the back-door criterion for (X, Y) is : {'B'}

There is one back-door path from \(X\) to \(Y\) :

\[Y \leftarrow B \rightarrow X\]

We need to block it by controlling for \(B\) wich satisfies the back-door criterion.

Example 4 (M-bias):

e4 = gum.fastBN("X<-A->B<-C->Y")
e4

m4 = csl.CausalModel(e4)
cslnb.showCausalImpact(m4, "Y", doing="X",values={})

Causal Model

$$\begin{equation*}P( Y \mid \text{do}(X)) = P\left(Y\right)\end{equation*}$$
Explanation : No causal effect of X on Y, because they are d-separated (conditioning on the observed variables if any).

Y
0	1
0.3612	0.6388

Impact

# This function returns the set of variables which satisfies the back-door criterion for (X, Y)
# None if there are no back-door paths.
setOfVars = m4.backDoor("X","Y")
print("The set of variables which satisfies the back-door criterion for (X, Y) is :", setOfVars)

The set of variables which satisfies the back-door criterion for (X, Y) is : None

There is one back-door path from \(X\) to \(Y\) :

\[X \leftarrow A \rightarrow B \leftarrow C \rightarrow Y\]

We don’t need to control for any set of variables, this back-door path is blocked by collider node \(B\), the two variables are d-separated, deconfounded, independent. Controlling for collider node \(B\) would make them dependant (introducing the M-bias).

Example 5:

e5 = gum.fastBN("X<-B<-A->X->Y<-C->B")
e5

m5 = csl.CausalModel(e5)
cslnb.showCausalImpact(m5, "Y", doing="X",values={})

Causal Model

$$\begin{equation*}P( Y \mid \text{do}(X)) = \sum_{C}{P\left(Y\mid C,X\right) \cdot P\left(C\right)}\end{equation*}$$
Explanation : backdoor ['C'] found.

	Y
X	0	1
0	0.4043	0.5957
1	0.5770	0.4230

Impact

# This function returns the set of variables which satisfies the back-door criterion for (X, Y)
# None if there are no back-door paths.
setOfVars = m5.backDoor("X","Y")
print("The set of variables which satisfies the back-door criterion for (X, Y) is :", setOfVars)

The set of variables which satisfies the back-door criterion for (X, Y) is : {'C'}

The difference between this example and the previous one is that we added an arrow between \(B\) and \(X\) ( \(B \rightarrow X\) ), this opens a new back-door path between \(X\) and \(Y\) that isn’t blocked by any colliders

\[X \leftarrow B \leftarrow C \rightarrow Y\]

We need to block the non-causal information that flows through it, controlling for \(B\) closes this backdoor path (it prevents information from getting from \(X\) to \(C\)). However, this action will open the back-door path that was formerly blocked by collider node \(B\) that we are adjusting for now:

\[X \leftarrow A \rightarrow B \leftarrow C \rightarrow Y\]

And, in this case, in addition to \(B\) we would also control for \(C\) or for \(A\) to reblock the path we opened and to block the new path. Another solution is to control for \(C\) (it prevents information from getting from \(B\) to \(Y\)) which satisfies the back-door criterion, it blocks the new path without reopening the one that is blocked by \(B\).

Example 6:

e6 = gum.fastBN("A->X;A->B;D->A;B->X;C->B;C->E;C->Y;D->C;E->Y;E->X;F->C;F->X;F->Y;G->X;G->Y;X->Y")
e6

m6 = csl.CausalModel(e6)
cslnb.showCausalImpact(m6, "Y", doing="X",values={})

Causal Model

$$\begin{equation*}P( Y \mid \text{do}(X)) = \sum_{C,E,F,G}{P\left(Y\mid C,E,F,G,X\right) \cdot P\left(C,E,F,G\right)}\end{equation*}$$
Explanation : backdoor ['C', 'E', 'F', 'G'] found.

	Y
X	0	1
0	0.5787	0.4213
1	0.4705	0.5295

Impact

# This function returns the set of variables which satisfies the back-door criterion for (X, Y)
# None if there are no back-door paths.
setOfVars = m6.backDoor("X","Y")
print("The set of variables which satisfies the back-door criterion for (X, Y) is :", setOfVars)

The set of variables which satisfies the back-door criterion for (X, Y) is : {'F', 'E', 'C', 'G'}

Back-door paths are: 1) - \(X \leftarrow G \rightarrow Y\) 2) - \(X \leftarrow E \rightarrow Y\) and any other back-door paths that go through \(E\) 3) - \(X \leftarrow F \rightarrow Y\) and any other back-door paths that go through \(F\) 4) - Blocked by collider \(B\) : \(X \leftarrow A \rightarrow B \leftarrow C \rightarrow Y\) and any other back-door paths that go through $ A$ will go through \(C\) 5) - \(X \leftarrow B \leftarrow C \rightarrow Y\) and any other back-door paths that go through \(B\) will go through \(C\) Two sets of variables that satisfy the back-door criterion are: * {\(C\),\(E\),\(F\),\(G\)} blocking (1), (2), (3) and (5) * {\(A\),\(B\),\(E\),\(F\),\(G\)} blocking (1), (2), (3), (5), opening (4) and reblocking it.