A Simplified and Numerically Stable Approach to the BG/NBD Churn Prediction model

TL;DR

The paper addresses churn prediction in seasonally driven purchase domains by redefining churn as no purchases within a window $M$ and deriving a simplified zero-purchase probability under the BG/NBD framework. It introduces a numerically stable transformation that rewrites the computation in log-space and uses a max-based reparameterization to avoid overflow/underflow. Contributions include a closed-form expression for $Pr(Y(t)=0|\cdot)$, a robust numerical scheme, and an open-source implementation in $pymc_nmarketing$. The approach enables reliable churn forecasting in irregular activity patterns and extends BG/NBD applicability to the iGaming sector.

Abstract

This study extends the BG/NBD churn probability model, addressing its limitations in industries where customer behaviour is often influenced by seasonal events and possibly high purchase counts. We propose a modified definition of churn, considering a customer to have churned if they make no purchases within M days. Our contribution is twofold: First, we simplify the general equation for the specific case of zero purchases within M days. Second, we derive an alternative expression using numerical techniques to mitigate numerical overflow or underflow issues. This approach provides a more practical and robust method for predicting customer churn in industries with irregular purchase patterns.