Download A Bayesian Non-Parametric Pareto/NBD Model

A Bayesian Non-Parametric Pareto/NBD Model
Fernando A. Quintana and Pablo Marshall1
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 noncontractual 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 expected number 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-
1
Author for correspondence: [email protected]
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; see Batislam, 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 applications to 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.