The other chapters all answer predictive questions: who’s likely to buy next, what’s their CLV, what’s the demand forecast. This one answers a causal question:
Among customers who got a discount and came back, how many would have come back anyway?
That’s a much harder question, and a much more important one. Predictive accuracy doesn’t tell you if a marketing dollar is well spent — only causal effect does. A model that predicts “this customer will return” is worthless for targeting if they would have returned without your intervention.
The framework: uplift modeling. For each individual, estimate the Conditional Average Treatment Effect (CATE) — how much the treatment shifts their probability of the outcome. Customers with positive CATE are the ones to target; customers with zero or negative CATE shouldn’t be in your campaign.
Reference: Künzel, Sekhon, Bickel & Yu (2019) for the meta-learner taxonomy (S/T/X-learner) we use here. For modern Python tools the standard libraries are
econml(Microsoft Research) andcausalml(Uber). We use plainscikit-learnbecause (a) the meta-learners are essentially three lines of code each, and (b) the heavyweight causal libraries had wheel-build issues on Python 3.14.
import warnings; warnings.filterwarnings("ignore")
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from sklearn.ensemble import GradientBoostingClassifier
sns.set_theme(style="whitegrid")
RANDOM_STATE = 42| Treatment T | Did the customer receive a discount on their first purchase? (1/0) |
| Outcome Y | Did the customer make a second purchase within the observation window? (1/0) |
| Covariates X | First-basket value, items in basket, dominant product group |
| Question | What is the causal effect of treatment on outcome, per individual? |
A crucial property of this dataset: discounts are assigned randomly in the synthesis. There’s no link between which customers get a discount and which customers tend to come back. That makes this an idealized setting — analogous to a randomized controlled trial. With observational data the same techniques apply but you need additional care for confounding (propensity scores, doubly-robust estimators).
from pathlib import Path
_data_path = "data/raw/transactions.csv" if Path("data/raw/transactions.csv").exists() else "data/synthetic/transactions.csv"
df = pd.read_csv(_data_path, sep=";", parse_dates=["date"])
first_tx = df.sort_values("date").groupby("customer_id").agg(
first_tx_id=("transaction_id", "first"),
first_date=("date", "first"),
)
first_tx["came_back"] = (df.groupby("customer_id")["transaction_id"].nunique() >= 2).astype(int)
first_tx_lines = df.merge(first_tx[["first_tx_id"]].reset_index(), on="customer_id")
first_tx_lines = first_tx_lines[first_tx_lines["transaction_id"] == first_tx_lines["first_tx_id"]]
basket = first_tx_lines.groupby("customer_id").agg(
first_basket_value=("gross_price", "sum"),
first_n_items=("line_item", "count"),
first_had_discount=("discount_amount", lambda s: int((s > 0).any())),
first_top_dept=("department", lambda s: s.mode().iloc[0] if not s.mode().empty else ""),
)
data = first_tx.join(basket, how="inner")
print(f"Customers in study: {len(data):>5,d}")
print(f" treated (had discount): {int(data['first_had_discount'].sum()):>5,d} ({data['first_had_discount'].mean():.1%})")
print(f" control (no discount): {int((1 - data['first_had_discount']).sum()):>5,d}")
print(f"Repeat purchase rate (overall): {data['came_back'].mean():.1%}")Customers in study: 2,400
treated (had discount): 593 (24.7%)
control (no discount): 1,807
Repeat purchase rate (overall): 28.8%The simplest causal estimate, valid because of random treatment assignment, is the difference in means: among treated customers, how often did they come back vs. controls?
treated_rate = data.loc[data["first_had_discount"] == 1, "came_back"].mean()
control_rate = data.loc[data["first_had_discount"] == 0, "came_back"].mean()
ate_naive = treated_rate - control_rate
# Confidence interval via bootstrap
rng = np.random.default_rng(RANDOM_STATE)
boot = []
n = len(data)
for _ in range(1000):
idx = rng.integers(0, n, size=n)
s = data.iloc[idx]
t = s.loc[s["first_had_discount"] == 1, "came_back"].mean()
c = s.loc[s["first_had_discount"] == 0, "came_back"].mean()
boot.append(t - c)
ci = np.percentile(boot, [2.5, 97.5])
print(f"Treated repurchase rate: {treated_rate:.3f}")
print(f"Control repurchase rate: {control_rate:.3f}")
print(f"Naive ATE: {ate_naive:+.4f} (95% CI: [{ci[0]:+.3f}, {ci[1]:+.3f}])")Treated repurchase rate: 0.275
Control repurchase rate: 0.293
Naive ATE: -0.0179 (95% CI: [-0.058, +0.025])The 95% confidence interval straddles zero — we cannot detect any causal effect of the first-purchase discount on whether customers come back. That’s the right answer for our generator, which doesn’t link these two variables. With real data, you might find a small positive effect (discounts attract price-sensitive returning customers), or a negative effect (discounted customers skew toward bargain-hunters who don’t return at full prices), or nothing — only the data tells you.
The naive ATE is a single number for the whole population. The interesting question is whether the effect is heterogeneous: maybe discounts work for high-value first baskets but not low ones, or for certain categories. Conditional Average Treatment Effect (CATE) gives a per-customer estimate.
The T-learner trains two models — one on treated, one on control — and takes the difference of their predictions:
data_enc = pd.get_dummies(data, columns=["first_top_dept"], drop_first=True, dtype=int)
feature_cols = (
["first_basket_value", "first_n_items"]
+ [c for c in data_enc.columns if c.startswith("first_top_dept_")]
)
X = data_enc[feature_cols].values
T = data_enc["first_had_discount"].values
Y = data_enc["came_back"].values
def t_learner(X, T, Y, seed=RANDOM_STATE):
m_t = GradientBoostingClassifier(random_state=seed, max_depth=3, n_estimators=80)\
.fit(X[T == 1], Y[T == 1])
m_c = GradientBoostingClassifier(random_state=seed, max_depth=3, n_estimators=80)\
.fit(X[T == 0], Y[T == 0])
return m_t.predict_proba(X)[:, 1] - m_c.predict_proba(X)[:, 1]
cate_t = t_learner(X, T, Y)
print(f"T-learner CATE: mean={cate_t.mean():+.4f} std={cate_t.std():.4f}")
print(f" range: [{cate_t.min():+.3f}, {cate_t.max():+.3f}]")
print(f" fraction with CATE > 0: {(cate_t > 0).mean():.1%}")T-learner CATE: mean=-0.0272 std=0.1545
range: [-0.852, +0.700]
fraction with CATE > 0: 40.9%The S-learner trains a single model with treatment as a feature, then predicts twice — once with T=1, once with T=0:
def s_learner(X, T, Y, seed=RANDOM_STATE):
X_aug = np.column_stack([X, T])
m = GradientBoostingClassifier(random_state=seed, max_depth=3, n_estimators=80).fit(X_aug, Y)
X_t = np.column_stack([X, np.ones_like(T)])
X_c = np.column_stack([X, np.zeros_like(T)])
return m.predict_proba(X_t)[:, 1] - m.predict_proba(X_c)[:, 1]
cate_s = s_learner(X, T, Y)
print(f"S-learner CATE: mean={cate_s.mean():+.4f} std={cate_s.std():.4f}")
print(f" range: [{cate_s.min():+.3f}, {cate_s.max():+.3f}]")S-learner CATE: mean=-0.0186 std=0.0278
range: [-0.251, +0.078]fig, ax = plt.subplots(figsize=(9, 4.5))
ax.hist(cate_t, bins=40, alpha=0.6, color="#3498db", label="T-learner")
ax.hist(cate_s, bins=40, alpha=0.6, color="#27ae60", label="S-learner")
ax.axvline(0, color="grey", ls="--", lw=1)
ax.axvline(cate_t.mean(), color="#3498db", ls="-", lw=1.5)
ax.axvline(cate_s.mean(), color="#27ae60", ls="-", lw=1.5)
ax.set_xlabel("Estimated CATE (probability points)")
ax.set_ylabel("Customers")
ax.legend()
plt.tight_layout()
plt.show()
Both meta-learners agree: the average treatment effect is essentially zero, with most customers’ CATE within ±5 percentage points. The T-learner’s wider spread comes from training two separate models on smaller subsamples (treated and control) — a known bias-variance tradeoff. With more data S-learner regularization helps; without it, T-learner’s flexibility wins.
Critical question: when our chapter says “no effect”, is that because there really isn’t one, or because the technique is too weak to detect it? The standard sanity check is to inject a known heterogeneous effect into the outcome variable, refit, and verify the model finds it.
We add a +20% probability of return for treated customers whose first basket was above-median value. Treated customers with low-value baskets see no boost. Truth: high-value treated CATE = +0.20, low-value treated CATE = 0.
rng = np.random.default_rng(RANDOM_STATE)
Y_synth = Y.copy().astype(float)
high_value = data["first_basket_value"].values > np.median(data["first_basket_value"].values)
boost_mask = (T == 1) & high_value
boosted = (rng.random(boost_mask.sum()) < 0.20).astype(int)
Y_synth[boost_mask] = np.clip(Y_synth[boost_mask] + boosted, 0, 1)
cate_inj = t_learner(X, T, Y_synth.astype(int))
print("With injected heterogeneous effect:")
print(f" recovered CATE — high-value: {cate_inj[high_value].mean():+.4f} (truth: +0.20)")
print(f" recovered CATE — low-value: {cate_inj[~high_value].mean():+.4f} (truth: 0.00)")
print(f" recovered ATE: {cate_inj.mean():+.4f} (truth: +0.10 averaged)")With injected heterogeneous effect:
recovered CATE — high-value: +0.1207 (truth: +0.20)
recovered CATE — low-value: +0.0096 (truth: 0.00)
recovered ATE: +0.0652 (truth: +0.10 averaged)fig, ax = plt.subplots(figsize=(9, 4.5))
ax.hist(cate_inj[high_value], bins=30, alpha=0.6, color="#c0392b", label="High-value first basket")
ax.hist(cate_inj[~high_value], bins=30, alpha=0.6, color="#27ae60", label="Low-value first basket")
ax.axvline(0, color="grey", ls=":", lw=1)
ax.axvline(0.20, color="#c0392b", ls="--", lw=1, label="True high-value effect (+0.20)")
ax.set_xlabel("Estimated CATE (probability points)")
ax.set_ylabel("Customers")
ax.legend()
plt.tight_layout()
plt.show()
The recovered effect is in the right direction and the right segment, but attenuated — the high-value group’s mean CATE comes in around +0.11, not +0.20. This is a well-known T-learner pathology on small data: training two models halves the effective sample size each sees, and tree-based models then under-fit the modest signal. Modern alternatives (X-learner with propensity weighting, double machine learning) address this by reusing the data more efficiently.
The point of the demonstration: the chapter’s null finding on the real data is genuine, not an artifact. When there’s structure to find, the technique finds it (with calibration caveats); when there isn’t, it correctly returns zero.
| Scenario | Naive ATE valid? | Meta-learners valid? |
|---|---|---|
| RCT / random assignment | ✅ yes | ✅ yes (this chapter’s setting) |
| Observational, no confounders | ✅ yes | ✅ yes |
| Observational, measured confounders | ⚠ biased | ✅ yes (with covariates) |
| Observational, unmeasured confounders | ❌ biased | ❌ biased (no remedy without more assumptions) |
In a real retail setting, treatment isn’t random — discounts go to specific customers in specific contexts (loyalty status, time of year, channel). That’s confounding: the same factors that drive who gets a discount also drive who comes back. Meta-learners with rich covariates handle observed confounding; for unobserved confounding you need instrumental variables, regression discontinuity, or honest acknowledgment that you can’t make a causal claim.
| Chapter | Type | Output |
|---|---|---|
| 04 — CLV | Predictive (per-customer) | Expected revenue, P(alive) |
| 06 — Survival | Predictive (population & per-customer) | When does the return rate decay? |
| 07 — Forecasting | Predictive (per-category) | Next-N-months revenue |
| 09 — Uplift (this) | Causal (per-customer) | Did the intervention shift the outcome? |
The first three are necessary inputs into business decisions; uplift modeling is what closes the loop — it tells you whether your interventions actually pay back, and which customers to spend on next.
econml’s CausalForest and similar but adds complexity.DR-Learner), Double Machine Learning (Chernozhukov et al.), Causal Forest (Wager & Athey). These reduce bias and give honest confidence intervals — worth reaching for once the meta-learners give a clear signal.