TL;DR

# 正文

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 cali- bration. Empirically, we demonstrate the predictive performance of our proposed model on two real-world public datasets.

LTV建模的一个挑战是一些用户在来访之后就不会回访了，同时其它用户的LTV分布是长尾的形状。通用的MSE损失并不能适应这些单次购买后就流失的零值用户，同时对于特别头部的高购用户也会变得十分敏感。

ZILN损失既可以用于线性模型也可以用于深度神经网络。在模型评估上，我们推荐采用标准化的Gini系数来度量模型的区分度，以及用分位数图表来评估模型的校准程度。我们在两个真实的开放数据集上评估了我们模型的表现。

ZILN loss完整如下 $\ell_\text{ZLIN}(x;p,\mu,\sigma)=\ell_\text{CrossEntropy}(\mathbb{I}_{x>0};p)+\mathbb{I}_{x>0}\ell_\text{LogNormal}(x;\mu,\sigma)$ 网络结构如下图所示，模型有三个输出，分别输出购买概率$p$、均值$\mu$和方差$\sigma$

ZILN损失另一个优势是它提供了完整的分布预测，不仅可以得到客户流失的概率，同时也能对留存用户的商业价值进行预测。