Do-notation expresses the distribution produced by externally setting a variable, separating causal intervention from ordinary observation. P(I|E="high") describes players observed with high engagement; P(I|do(E="high")) describes purchases when engagement is set to high. section 7.3.1
An ideal intervention targets a variable, fixes its value, and removes causal influence from that variable’s parents. In a DAG this is graph surgery; in an SCM it replaces the variable’s assignment function. section 8.2.3
The notation supports several causal queries:
- Joint distributions under intervention, such as P(I, G|do(E="high")). section 7.3.1
- Multiple interventions, such as setting engagement to high and guild membership to nonmember. section 7.3.1
- Average treatment effects, computed as E(Y|do(X=1)) − E(Y|do(X=0)) for binary causes. section 7.4.1
Engagement and in-game purchases
Asking what purchases would be for players who happen to have high engagement uses conditional probability. Asking what purchases would result if engagement were made high uses do-notation, excluding the non-causal association through guild membership. section 8.2.3
Table 7.11 Examples of do-notation and counterfactual notation
|
Literal
|
Do-notation
|
Counterfactual notation
|
|---|---|---|
|
The probability distribution of
In-Game Purchases across all players
|
P(
I
)
|
P(
I
)
|
|
The probability distribution of
In-Game Purchases for players with “high” engagement
|
P(
I
|
E=“high”)
|
P(
I
|
E=“high”)
|
|
The probability distribution of
In-Game Purchases when a player’s engagement level is set (by intervention) to “high”
|
P(
I
|do(
E= “high”))
|
P(
I
E
= “high”)
|
|
The joint probability distribution of
In-Game Purchases and
Guild Membership when engagement is set to “high”
|
P(
I,
G|do(
E=“high”))
|
P(
I
E=“high”,
G
E=“high”)
|
|
The probability distribution of In-Game Purchases when engagement is set to “high” and membership is set to “nonmember”
|
P(
I
|do(
E
=“high”,
G=“nonmember”))
|
P(
I
E
= “high”,
G
=“nonmember”)
|
|
The probability distribution of
In-Game Purchases for guild members when engagement is set to “high”
|
P(
I
|do(
E
=“high”),
G
=
“member”)
|
P(
I
E
=
“high”|
G
=
“member”)
|
Compare observational, interventional, joint, and multiple-intervention expressions with their counterfactual equivalents. section 7.3.1
