Prediction of Individual Buying Behavior: A Poisson-Bernoulli Model with Arbitrary Heterogeneity

Abstract

Patterns of customer purchases are often modeled as discrete behavioral events following some probability law. These stochastic models have been applied separately to the phenomena of product class purchase incidence and brand choice (conditional on the products being purchased). Most of these models are probability mixture models where the parameters of an individual customer model vary across the population according to some distribution. This paper proposes a simple model of purchase behavior at the individual level, Poisson (λ) product purchases and Bernoulli (p) brand choices, so that λ and p characterise each customer. However, λ and p may vary jointly across the population in an arbitrary fashion. Therefore, purchase incidence and brand choice are treated together making no assumptions about the dependence between them. This generality is obtained at the cost of efficiency but enables us to focus on the individual behavior. Estimators (with known asymptotic properties) are developed for predicting future purchasing behavior of individuals conditional on their past purchasing behavior. These predictions of future purchases, at brand and product levels, are of interest to marketing managers. For example, what is the expected number of brand (or product) purchases in period 2 for the group of customers that made x brand and n product purchases in period 1? Apart from the basic insights provided by such predictions, they are particularly useful in evaluating the impact of marketing efforts. The model and the estimation results are applied to a frequently purchased product and compared with other stochastic approaches. While this paper focuses on buying behavior, the model is applicable to accidents, direct mail purchases and other random events.