Suppose an online game team asks whether setting side-quest engagement to “high” would change in-game purchases. Looking only at players whose engagement is already high answers a different question. Counterfactual notation marks the hypothetical intervention with a subscript, keeping it distinct from an observed condition. section 7.3.1
Let I denote in-game purchases and E denote engagement:
- P(I | E="high") describes purchases among players observed to have high engagement. section 7.3.1
- P(I_{E="high"}) describes purchases if engagement were set to high by intervention. I_{E="high"} is the version of I under that intervention; the equivalent do-notation is P(I | do(E="high")). section 7.3.1
The subscript describes an ideal intervention: set the target variable to a value and sever the influence of its usual causes. In a causal DAG, this cuts incoming edges; in a structural causal model, it replaces the target variable’s assignment function. The notation states the question, not whether available data can answer it. section 8.2.3 section 1.5.3
A hypothetical conditioned on facts
P(I_{E="high"} > 50 | E="low", I <= 50) asks about spending above $50 if engagement were set to high, among players who actually had low engagement and spent at most $50. The subscript gives the hypothetical condition; the terms after the bar select the factual group. Their disagreement is what makes this a counterfactual question. section 8.2.4
The same notation supports causal-effect comparisons. For images of a digit 4 at a given thickness, an expected image under one writer-type intervention can be subtracted pixel by pixel from the expected image under another. Age-related confounding means those expectations cannot simply be read from an observational comparison; a causal model is needed to connect them to estimable quantities. section 1.5.3
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”)
|
The notation table compares observed conditions, single and multiple interventions, and their do-notation counterparts. section 7.3.1
