Causal Inference 01: Confounding & Adjustment

In Causal Inference 00, Veranda Home's dashboard — and a genuinely good Bayesian MMM — over-credited the two channels whose budgets chase demand. The disease was an open back-door: a common cause driving both spend and sales. The classic prescription is adjustment: find a variable that measures the common cause, condition on it, and the spurious correlation is blocked. This page mirrors nbs/causal/causal_01_confounding_and_adjustment.ipynb, which tests that prescription the only way that settles anything — against sealed answer keys — across three possible hidden histories of the same brand.

Three rules to take away, each demonstrated below with measured numbers: (1) conditioning on a real confounder closes the door — most of the bias disappears; (2) the door only closes as far as the proxy is good — a noisy measurement of demand leaves residual bias, and the model won't warn you; (3) adjustment is powerless against simultaneity — when spend responds to sales, there is no third variable to condition on.

Three DAGs, one dashboard

The dashboards these worlds generate are statistically similar — positive spend–sales correlations everywhere. The graphs are what differ, and the graph is what decides whether adjustment can work:

Adjustment's textbook logic, for the world where it applies: condition on the common cause \( d \) and the interventional quantity becomes estimable from observational strata,

\[ P\bigl(y \mid \operatorname{do}(s)\bigr) \;=\; \sum_{d} P(y \mid s, d)\, P(d). \]

The formula's fine print is the whole notebook: it requires \( d \) to be measured, measured well, and to actually be a common cause rather than the outcome wearing a disguise.

Fit all three histories

Same model family for each — Bayesian MMM with geometric adstock, saturating response, linear trend, NumPyro NUTS — graded against each world's sealed key. In History B the demand index enters marked as a CONFOUNDER: the framework routes it to a wide, un-shrunk coefficient prior, because shrinking a confounder's coefficient re-opens the very door you're trying to close.

from mmm_framework.config import CausalControlRole

# Declaring the control's causal role is part of the model spec:
# CONFOUNDER -> wide, un-shrunk prior (closing a back-door);
# unmarked   -> default precision-control prior (shrunk toward zero).
role = CausalControlRole.CONFOUNDER

Relative error of estimated counterfactual contribution vs each world's sealed key (nbs/artifacts/causal_01_adjustment_scorecard.json and the notebook's grade tables). Histories A and B are the same world — the only difference is one column in the analyst's dataset.

Grading: what the control bought

Because Histories A and B share the same latent demand, the same chasing budgets, and the same true ROAS, the comparison is surgical: any improvement is attributable to adjustment and nothing else.

Measured — per-channel bias before and after the demand-index control. The chasers' bias roughly halves: Search +43% → +25%, Social +38% → +18%; Social's 90% interval now covers the truth (it didn't in History A). But Search is still over-read by a quarter — the index is a noisy proxy for true demand, so residual confounding survives, and nothing inside the model announces it. A proxy closes the door only as far as it is a good proxy.

Aside — does the CONFOUNDER marking matter?

The framework asks you to declare a control's causal role, because unmarked controls get the default precision-control prior (narrower, shrunk toward zero), and shrinking a confounder can leave the door half-open. Here is the same History B fit with the index left unmarked:

The honest aside: in this world the index's signal is strong enough to survive the default shrinkage, so both variants close the door about equally (mean chaser bias 21.8% marked vs 21.5% unmarked). Marking the role is cheap insurance, not magic — it exists for the worlds where the proxy is weaker and shrinkage genuinely bites.

Measured — mean |relative error| over the demand-chasing channels (Search & Social), the channels confounding actually attacks: 40% with demand hidden, 22% with the index marked as a confounder, 21% unmarked. Adjustment is real, cheap, and the first thing to reach for — and it stops halfway because the proxy is noisy, not because the method is wrong.

History C: the disease adjustment can't touch

In History C there is no hidden demand at all. Budgets are paced: finance sets next week's spend partly from recent revenue, so spend and sales are jointly determined — the arrow into spend starts at the KPI itself.

No control variable can fix this. Conditioning works by blocking a path through a third variable; here the offending path runs through last week's outcome. And conditioning on lagged sales does not rescue you either — lagged sales carries the adstocked effect of last week's media, so it is a bad control sitting on the causal path. The honest fixes live further up the ladder: model the feedback structurally, or experiment — randomization severs the arrow into spend by construction.

Measured — Veranda paces gently (a fraction of each budget tracks revenue: TV 0.6, Search 1.0, Social 0.8, Display 0.5), and the media calendar keeps its identifying pulses, so the damage is moderate. But the pattern is diagnostic: the hardest-paced channel (Search, pacing 1.0) is the most over-read at +15% — the bias follows the pacing intensities, not the media plan.

What this rung bought — and what it can't

Bought: with a measured confounder, most of the chasers' bias disappears. Adjustment is real, cheap, and the first thing to reach for. Declare causal roles when you do it.

Can't:

• an unmeasured confounder (History A) leaves you exactly where the dashboard left you — and nothing inside the model tells you;
• a noisy proxy closes the door partway (Search +43% → +25%, not → 0%) and, again, doesn't tell you how far;
• simultaneity (History C) has no door to close.

The rest of the series is about earning confidence honestly — first by modeling more of the structure we believe in (03 · structural mediation and the latent-confounder notebook after it), then by buying ground truth with experiments (notebooks 05–07).