A BayesianNon-Parametric Pareto/NBD Model

Fernando A. Quintana and Pablo Marshall[1]

Pontificia Universidad Católica de Chile

Extended Abstract

The need to understand customer behavior and the interest of managers to focus on customers who can deliver long-term profits has changed the main purpose of marketers activities from acquisition to retention. Many companies, nowadays, face a database containing information on the frequency and timing of transactions for a list of customers, for the purpose of forecasting their future purchasing at the individual level. Manager decisions oriented to the growth and retention of customers are usually based on these predictions.

In an influential study, Schmittlein et al. (1987) proposed the Pareto/Negative Binomial Distribution (P/NBD) model to analyze and predict the purchase behavior of customers at the individual level in a non-contractual scenario. Under a non-contractual setting (e.g., retail store, supermarket, mail order) consumers do not declare that they have become inactive, but simply stop transactions with the firm. Thus, the time at which a customer becomes inactive is unobserved by the firm and the single evidence for this is an unusually long interval since the last recorded purchase. The challenge facing the analyst is how to differentiate between those customers who have ended their relationship with the firm versus those who are in the middle of a long interval between transactions.

The P/NBD model estimates the probability of being active and the expectednumber of transactions in the future for each customer. The model assumes that at time t=0 the individuals are active and: (i) while active, the number of transactions made by a customer follows a Poisson probability distribution, (ii) the lifetime of a customer has an Exponential probability distribution, (iii) customers are heterogeneous in both their activity levels and their lifetimes, (iv) transaction and the drop-out rates across individuals follow a Gamma distributions and (iv) transactions and the drop-out rates are independent at the individual level.

Despite its recognized relevance for academics and practitioners, the P/NBD model has led, in recent years, to (i) several simplifications or alternative estimation procedures in order to avoid complicated computation (Fader, Hardie and Lee, 2007 and Ma and Liu, 2007), and (ii) some extensions in order to incorporate more realistic assumptions for the probability distribution of the transactions and drop-out rates (Singh, Borle and Jain, 2009 and Bemmaor and Glady, 2012).

The purpose of this study is to propose a bayesian non-parametric estimation procedure of the P/NBD model that relaxes the assumption that purchases and drop-out rates are distributed according to a Gamma probability distribution. In our specification, we do not assume any probability distribution for these coefficients and adopt a non-parametric distribution. Hence, the prior probability distribution across customers is more flexible than previous models and is specified as a mix of probability distributions determined by a Dirichlet Process as introduced by Ferguson(1973) and Antoniak (1974). In line with the observation that in many datasets a large proportion of customers have no transactions during the estimation period, we also consider a mass probability at zero for the purchase intensity in line with the extension proposed by Batislam, Denizel and Filiztekin (2007).We maintain the other assumptions of the P/NBD model: at the individual level, transactions and lifetimes are independent and follow Poisson and Exponential distributions with customer specific coefficients.

We expect a-priori our model to fit the data best in scenarios for customers with very low or very large purchase rates since the Gamma model, which has one mode and does not allow a zero value, might have a conflict between these groups of customers and hence a poor fit.We explore the performance of the model using a dataset on the purchasing of CDs at the online retailer CDNOW; see Fader and Hardie (2001) and a dataset that comes from a specific store of a large grocery retail that offers a broad assortment of grocery products; seeBatislam, Denizel and Filiztekin(2007).

The results show that at the aggregated level, our model fits the data slightly better. The improvements in our model are more significant for customers with 0 transactions and for customers with a relatively large number of transactions. The implication for practitioners is an improvement on the prediction performance of the buying behavior of those active customers who do not buy at all in the estimation period and those who buy very frequently; that is, precisely the ones who are more important for marketers.

References

Antoniak, C. E. (1974). Mixtures of Dirichlet processes with applicationsto Bayesian nonparametric problems, The Annals of Statistics 2: 1152-1174.

Batislam, E. P., M. Denizel, A. Filiztekin (2007) Empirical validation and comparison of models for customer base analysis. Internat. J. Res. Marketing 24(3): 201-209.

Bemmaor, A.C. and N.Glady (2012) Modeling Purchasing Behavior with Sudden Death: A Flexible Customer Lifetime Model. Management Science 58(5): 1012-102.

Fader, Peter S., Bruce G. S. Hardie (2001) Forecasting repeat sales at CDNOW: A case study. Part 2 of 2.Interfaces 31(May–June) S94–S107.

Fader, P., B. Hardie and K.L. Lee (2007) Counting Your Customers the Easy Way: An Alternative to the Pareto/NBD Model,Marketing Science 24(2): 275-284.

Ferguson, T. S. (1973) A Bayesian analysis of some nonparametric problems, The Annals of Statistics 1: 209-230.

Ma, S.H. and Liu, J.L. (2007).The MCMC Approach for Solving the Pareto/NBD Model and Possible Extensions, Proceedings of the Third International Conference on Natural.

Singh, S. S., S. Borle, D. C. Jain(2009) A generalized framework for estimating customer lifetime value when customer lifetimes are not observed.Quantitative Marketing Economics 7(2): 181-205.

Schmittlein, D.C., D.G. Morrison and R.Colombo (1987) Counting your Customers: Who Are They and What Will They Do Next? Management Science 33(1):1-24.

[1] Author for correspondence: