Introduction to Uplift Modeling

Dr. Juan Orduz

Mathematician & Data Scientist

PyConDE & PyData Berlin 2022

Motivation

How can we optimally select customers to be treated by marketing incentives?

We can not send and not send incentives to the same customers at the same time

What is Uplift Modeling?

From Gutierrez, P., & Gérardy, J. Y. (2017). "Causal Inference and Uplift Modelling: A Review of the Literature"

• Uplift modeling refers to the set of techniques used to model the incremental impact of an action or treatment on a customer outcome.

• Uplift modeling is therefore both a Causal Inference problem and a Machine Learning one.

Conditional Average Treatment Effect

• Let $Y^{1}_{i}$ denote person $i$'s outcome when it receives the treatment and $Y^{0}_{i}$ when it does not receive the treatment.

• We are interested in:

  • The causal effect $\tau_{i} \coloneqq Y^{1}_{i} - Y^{0}_{i}$
  • Given a feature vector $X_{i}$ of the $i$-th person, we would like to estimate the conditional average treatment effect

• However, we can not observe them!

CATE Estimation

Let $W_{i}$ is a binary variable indicating whether person $i$ received the treatment, so that

Unconfoundedness Assumption

If we assume that the treatment assignment $W_{i}$ is independent of $Y^{1}_{i}$ and $Y^{0}_{i}$ conditional on $X_i$, then we can estimate the $CATE$ from observational data by computing the empirical counterpart:

Data Collection

Estimating Uplift

• Meta Algorithms $\longleftarrow$ Today

• The Class Transformation Method

• Direct measurements (e.g. trees)

S-Learner

Step 1: Training

Step 2: Uplift Prediction

T-Learner

Step 1: Training

T-Learner

Step 2: Uplift Prediction

X-Learner

Step 1: Training: Same as T-Learner

Step 2: Compute imputed treatment effects

X-Learner

Step 3: Train with different targets

X-Learner

Step 4: Uplift Prediction

where $g(X') \in [0, 1]$ is a weight function.

Remark: A common choice for $g(X)$ is an estimator of the propensity score, which is defined as the probability of treatment given the covariates $X$, i.e. $p(W_{i}=1|X_i)$.

Intuition behind the X-Learner

We study an simulated example where we know the uplift is exactly $\tau=1$.

X-Learner Step 1 (same as T-Learner):

Model fit for control (red) and treatment (blue) groups.

T-Learner Estimation:

• The solid line represents the difference between the model fit for the control group and the treatment groups.
• The estimation is not good as the treatment group is very small.

Imputed Treatment Effects:

X-Learner Estimation:

• The dashed line represents the X-Learner estimation.
• It combines the fit from the imputed effects by using and estimator of the propensity score, i.e. $g(x)=\hat{e}(x)$. In this example $\hat{e}(x)$ will be small as we have much more observations in the control group. Hence the estimated uplift will be close to $\hat{\tau}^{T}$.

Some Python Implementations

• causalml

• EconML

• scikit-uplift

Python Code: Example

Uplift Model Evaluation?

Uplift Evaluation: Uplift by Percentile

1. Sort uplift predictions by decreasing order.
2. Compute percentiles.
3. Predict uplift for both treated and control observations per percentile.
4. The difference between those averages is taken for each percentile.

Uplift Evaluation: Uplift by Percentile

A well performing model would have large values in the first percentiles and decreasing values for larger ones

Uplift Evaluation: Cumulative Gain Chart

Predict uplift for both treated and control observations and compute the average prediction per decile (bins) in both groups. Then, the difference between those averages is taken for each decile.

• $Y^{T} / Y^{C}$: sum of the treated / control individual outcomes in the bin.
• $N^{T} / N^{C}$: number of treated / control observations in the bin.

How to compute it ? 🫠

Uplift Evaluation: Cumulative Gain Chart

• We can assess whether the treatment has a global positive or negative effect and if one can expect a better gain by targeting part of the population.
• We can thus choose the decile that maximizes the gain as the limit of the population to be targeted.

Uplift Metrics: Uplift Curve

We can generalize the cumulative gain chart for each observation of the test set:

where the $t$ subscript indicates that the quantity is calculated for the first $t$ observations, sorted by inferred uplift value.

Uplift Metrics: Uplift Curve & AUC

Best uplift model?

A perfect model assigns higher scores to all treated individuals with positive outcomes than any individuals with negative outcomes.

Perfect Uplift Curve

Random Uplift Curves

Model Comparison

Compute AUC on a test set.

Other metrics: Qini Curve

Corrects uplifts of selected individuals with respect to the number of individuals in treatment/control using the $N^{T}_{t} / N^{C}_{t}$ factor.

Demo

Notebook Link

References

• Diemert, Eustache, et.al. (2020). "A Large Scale Benchmark for Uplift Modeling"

• Sävje, F. (2021). "Randomization does not imply unconfoundedness"

• Gutierrez, P., & Gérardy, J. Y. (2017). "Causal Inference and Uplift Modelling: A Review of the Literature"

• Karlsson, H. (2019) "Uplift Modeling: Identifying Optimal Treatment Group Allocation and Whom to Contact to Maximize Return on Investment"

• Oostra, G. (2019). "Preventing churn like a bandit" (PyData Eindhoven 2019)

• Sören, R, et.al. (2019). "Meta-learners for Estimating Heterogeneous Treatment Effects using Machine Learning"

Thank you!

More Info: juanitorduz.github.io