Fix interventional/counterfactual terminology conflation in `interventional_distribution.ipynb`

[drbenvincent]

Summary

interventional_distribution.ipynb is explicitly about interventional distributions and the do-operator (Pearl's Level 2). The notebook uses interventional terminology correctly throughout, with one notable exception: a sentence in the "Statistical versus interventional distributions" section parenthetically equates interventional questions with counterfactual questions, which is the same L2/L3 conflation flagged in #849.

There are also two minor typos.

Problems

1. L2/L3 conflation in "Statistical versus interventional distributions" section

The current text reads:

Interventional distributions are cool because they allow us to ask what-if (or counterfactual questions). For example, with a causal DAG we could ask questions of the form, "What do I think will happen in the future if I do X?" or "What do I think would have happened in the past if X had happened?"

Two issues here:

• The parenthetical "(or counterfactual questions)" equates interventional queries with counterfactual queries. In Pearl's causal hierarchy, these are distinct levels: interventional distributions ($P(Y|\operatorname{do}(X=x))$) are Level 2, while counterfactuals ($P(Y_x | X=x', Y=y')$, requiring abduction of unit-specific exogenous terms) are Level 3. The notebook only ever computes L2 quantities.
• The second example question — "What do I think would have happened in the past if X had happened?" — slides into L3 counterfactual territory (reasoning about a specific past that didn't occur), but the notebook never actually performs this kind of reasoning. It only computes forward-looking interventional distributions via pm.do.

2. Typo in notation

In the "Joint distributions" section:

how they compare to the interventional distribution P(y|\operatorname{do}=2)

Should be P(y|\operatorname{do}(x=2)) — the argument of do is missing.

3. Grammar

In the comparison section after computing interventional distributions:

So in DAGs 2 and 3, our intervention \operatorname{do}(x=2) have no influence on $y$.

Should be "has no influence" (singular subject).

Proposed changes

1. Fix the L2/L3 conflation: Replace "(or counterfactual questions)" with "(interventional) questions" or similar, and reframe the second example question to stay at L2 (e.g., "What would happen if we were to set X to a particular value?") rather than implying unit-level retrospective counterfactual reasoning.
2. Fix the notation typo: Change $P(y|\operatorname{do}=2)$ to $P(y|\operatorname{do}(x=2))$.
3. Fix the grammar: Change "have no influence" to "has no influence".

No callout box is needed here — unlike the quasi-experimental notebooks addressed in #852, this notebook is explicitly about interventional distributions and doesn't use the term "counterfactual" in a Rubin/potential-outcomes sense. It just needs the one sentence corrected to avoid claiming the do-operator produces counterfactuals.

Context

This follows the same terminology review that produced #849 (reframing counterfactuals_do_operator.ipynb) and #852 (adding callouts to the quasi-experimental notebooks).