Causal Inference 04: Latent Confounders

Patio furniture is a discretionary purchase. When the economy hums — hiring up, confidence up, houses turning over — Veranda Home sells more and spends more, because performance budgets get topped up in good quarters. Economic health is a textbook confounder: it drives both sides of every spend–sales correlation. And nobody has a column called economic_health in their warehouse — what they have is four noisy indicators that each reflect it partially. This guide mirrors nbs/causal/causal_04_latent_confounders.ipynb: three rungs of adjustment on one synthetic world, same estimand, every rung graded against the sealed key. Notebook 03 modeled a mediator; this one models a measurement.

1 — The world on the desk

The economic_health world (make_economic_health in src/mmm_framework/synth/dgp.py): a latent macro factor — AR(1) cycle plus slow growth plus a seasonal wobble — drives spend and sales. Search and Social chase it hardest (their budgets are topped up in booms); TV and Display barely move with it. The factor is measured only through four noisy indicators, each of the form \( \mathrm{ind}_k = \lambda_k \cdot \mathrm{econ}_t + \epsilon_{kt} \), with the planted loadings sealed in the answer key:

from mmm_framework.synth.dgp import make_economic_health

world = make_economic_health()        # sealed answer key rides in world.notes
world.notes["true_loadings"]
# {'gdp_growth': 0.9, 'consumer_confidence': 0.8,
#  'unemployment': -0.7, 'retail_sales': 0.6}
world.notes["chasers"]                # ['Search', 'Social'] — the demand-chasers

The indicators visibly co-move (they share one driver), but none of them is the driver: each carries its own noise, and unemployment runs upside-down. The DAG has a classic back-door: spend ← economic health → sales, with the confounder itself unobservable and its four measurements dangling off it. One methodological note for everything below: all three rungs are fit without a linear trend, matching the framework's own recovery test, so the comparison isolates de-confounding cleanly — the world's only slow structure is the economic factor. (What a trend term accidentally does is measured in the aside at the end.)

2 — Rung A: ignore it

The naive fit — Price is the only control. The chasers absorb the boom-time correlation, and the regression rebalances the rest of the portfolio around the distortion. Graded against the sealed key:

• Search: +778% — credited roughly nine times its true contribution, truth far outside the 90% interval.
• Display: −81% — a genuinely working channel crushed to near-zero to compensate. Truth outside the interval.
• TV +81%, Social +33% — inflated, but their (wide) intervals still cover.

A budget that acted on this read would pour money into the channel that happened to surf the boom and cut one that was quietly earning.

3 — Rung B: indicators as controls

The practitioner's move: throw all four indicators into the control set (marked CONFOUNDER, per notebook 01's lesson). This works, mostly — the four series jointly span much of the economy's variance, and the chaser bias collapses by roughly a factor of four: Search +778% → +172%.

But look closely at what's left. The indicators each carry idiosyncratic noise, so they can never absorb all of the factor's variance — the classic errors-in-variables attenuation: regression adjusts for the proxy, not the confounder, and the unmeasured remainder keeps a back-door open. The Search estimate stays materially inflated (+172%, truth still outside its 90% interval), and nothing in the fit output announces it. The model converged; the controls "worked"; the residual bias is silent. (Marking the indicators as confounders vs leaving them unmarked barely matters here — +172% vs +175% — their signal survives default shrinkage either way.)

4 — Rung C: model the measurement

The structural move: instead of handing the noisy proxies to the regression, declare what they are — four measurements of one latent factor — and estimate the factor inside the same PyMC graph as the MMM. The framework's worked garden-model example examples/garden_models/latent_factor_mmm.py does exactly this: a latent AR(1) factor with a measurement block, entering the sales equation as the de-confounder, its uncertainty propagating into every media coefficient — no two-stage plug-in.

# illustrative — rung C rides the framework's worked garden-model example
# (examples/garden_models/latent_factor_mmm.py): a measurement block and the
# MMM estimated jointly in ONE PyMC graph.
latent_mmm = LatentFactorMMM(...)             # indicators tagged INDICATOR, not CONTROL
latent_mmm.fit(draws=400, tune=800, chains=4)  # NUTS only — MAP is unstable here
latent_mmm.factor_loadings_summary()           # graded below vs the planted loadings

Identification is delicate and instructive: the realized factor is standardized in-graph (otherwise the AR variance trades off against the loadings and the sampler collapses the loadings to zero), and the first loading is constrained positive (gdp_growth, the anchor) to pin the factor's orientation — every other loading is free to go negative, which is exactly what unemployment should do. The fit is 4-chain NUTS (max r-hat 1.017).

Measured — posterior loadings (mean, HDI) vs the planted values. All four signs recovered, including unemployment's negative loading — a model that forced positive loadings would have broken this world — and the planted ordering (gdp > confidence > retail in magnitude) comes through. Posterior means land near the planted values (0.90 vs 0.9, 0.87 vs 0.8, −0.83 vs −0.7, 0.75 vs 0.6); magnitudes run slightly hot for the smaller loadings, a known scale trade-off with the AR(1) factor. Numbers: nbs/artifacts/causal_04_latent_recovery.json.

Measured — the confounder, made visible: the recovered economic-health factor (posterior mean, 90% band) against the true series from the sealed key, aligned for display. Correlation: 0.981. This is a nameable weekly series with uncertainty — a thing a CFO can monitor, extend with new indicators, or compare against external indices.

5 — The ladder, graded honestly

Same estimand for all three rungs — counterfactual contribution vs the sealed key, chaser channels (Search & Social) averaged:

Measured — mean |relative error| on the chaser channels: A 406% → B 101% → C 112%. Search alone: +778% → +172% → +191%. Both structural rungs collapse the naive bias ~4×. And the honest part: on this world, B and C essentially tie on media point error — four indicators used as plain controls span about as much of the confounder's variance as the one-factor model extracts from them. Numbers: nbs/artifacts/causal_04_latent_recovery.json.

6 — What rung C actually buys

If B and C tie on point error, rung C is not an accuracy purchase — it's a knowledge purchase:

1. The measurement model is itself a graded causal claim. C recovered the loadings — all four signs, the planted ordering, the negative unemployment loading — and the factor's weekly path (corr 0.981). B's output is four regression coefficients on noisy proxies: nothing you could hand to an economist, reuse next year, or falsify.
2. A nameable series to monitor. C produces "economic health, weekly, with uncertainty." B produces nothing reusable.
3. Honest propagation. C's media intervals carry the factor's estimation uncertainty inside the same posterior. B silently treats noisy proxies as if they were the confounder itself.
4. A platform. The measurement block composes with everything else in this series — the funnel mediators of notebook 03, the experiment calibration the series turns to next.