[Paper Review] A Deep Probabilistic Model for Customer Lifetime Value Prediction

Xiaojing Wang, Tianqi Liu+1 more|arXiv (Cornell University)|Dec 16, 2019

Customer churn and segmentation29 references11 citations

TL;DR

This paper proposes a deep neural network (DNN) model with a zero-inflated lognormal (ZILN) loss function to predict customer lifetime value (LTV) for new customers, addressing the challenges of zero-inflated and heavy-tailed LTV distributions. The method jointly models churn probability and LTV prediction with uncertainty quantification, outperforming mean squared error (MSE) loss in discrimination and calibration on two real-world datasets.

ABSTRACT

Accurate predictions of customers' future lifetime value (LTV) given their attributes and past purchase behavior enables a more customer-centric marketing strategy. Marketers can segment customers into various buckets based on the predicted LTV and, in turn, customize marketing messages or advertising copies to serve customers in different segments better. Furthermore, LTV predictions can directly inform marketing budget allocations and improve real-time targeting and bidding of ad impressions. One challenge of LTV modeling is that some customers never come back, and the distribution of LTV can be heavy-tailed. The commonly used mean squared error (MSE) loss does not accommodate the significant fraction of zero value LTV from one-time purchasers and can be sensitive to extremely large LTV's from top spenders. In this article, we model the distribution of LTV given associated features as a mixture of zero point mass and lognormal distribution, which we refer to as the zero-inflated lognormal (ZILN) distribution. This modeling approach allows us to capture the churn probability and account for the heavy-tailedness nature of LTV at the same time. It also yields straightforward uncertainty quantification of the point prediction. The ZILN loss can be used in both linear models and deep neural networks (DNN). For model evaluation, we recommend the normalized Gini coefficient to quantify model discrimination and decile charts to assess model calibration. Empirically, we demonstrate the predictive performance of our proposed model on two real-world public datasets.

Motivation & Objective

• To improve LTV prediction for new customers, who lack historical purchase frequency and recency data, making traditional BTYD models inapplicable.
• To address the dual challenges of zero-inflated labels (one-time purchasers) and heavy-tailed distributions (high-spenders) in LTV regression.
• To develop a unified deep learning framework that simultaneously predicts churn probability and LTV with uncertainty quantification.
• To recommend normalized Gini coefficient and decile charts as superior evaluation metrics over MSE for LTV modeling.
• To demonstrate the effectiveness of the ZILN loss in real-world marketing scenarios, particularly in optimizing ad targeting and campaign profitability.

Proposed method

• Models the LTV distribution as a mixture of a point mass at zero (for churners) and a lognormal distribution (for returning customers), forming the zero-inflated lognormal (ZILN) distribution.
• Uses a differentiable ZILN loss function in a deep neural network to jointly optimize for both the probability of zero LTV and the expected positive LTV.
• Enables end-to-end training that captures complex, nonlinear relationships between customer features and LTV without requiring a two-stage modeling pipeline.
• Provides full predictive distributions, allowing for uncertainty quantification of point estimates via predictive variance.
• Employs a DNN architecture with four fully connected layers, trained via stochastic gradient descent with the ZILN loss.
• Evaluates model performance using normalized Gini coefficient for discrimination and decile-level MAPE for calibration, with precision-recall AUC for churn prediction.

Experimental results

Research questions

• RQ1 Can a deep neural network with a ZILN loss function outperform standard MSE-based regression in predicting customer lifetime value for new customers?
• RQ2 Does the ZILN model better capture the zero-inflated and heavy-tailed nature of real-world LTV distributions compared to MSE?
• RQ3 To what extent does the ZILN model improve model discrimination and calibration in LTV prediction tasks?
• RQ4 Can the ZILN model enhance real-world marketing outcomes, such as profit from direct mail campaigns?
• RQ5 Is the ZILN loss effective in jointly modeling churn probability and LTV prediction within a single deep learning framework?

Key findings

• On the Kaggle Acquire Valued Shoppers Challenge dataset, the ZILN loss achieved a normalized Gini coefficient of 0.911, outperforming MSE (0.906) in model discrimination.
• On the KDD Cup 1998 dataset, the ZILN loss achieved a Spearman’s rank correlation of 0.027, compared to 0.020 for MSE, indicating better ranking of high-value donors.
• The ZILN model achieved a normalized Gini coefficient of 0.190 on KDD Cup 1998, compared to 0.184 for MSE, showing superior ability to differentiate high- from low-value customers.
• The ZILN model reduced decile-level MAPE to 0.176, compared to 0.210 for MSE, indicating better model calibration.
• The ZILN-based DNN achieved a total profit of $15,498.24 in the KDD Cup 1998 campaign, a 5% relative improvement over the competition winner’s $14,712.24.
• The ZILN model eliminated the need for a two-stage pipeline by jointly modeling churn and LTV prediction in a single end-to-end framework.