Bayesian Inference of Arrival Rate and Substitution Behavior from Sales Transaction Data with Stockouts

When an item goes out of stock, sales transaction data no longer reflect the original customer demand, since some customers leave with no purchase while others substitute alternative products for the one that was out of stock. Here we develop a Bayesian hierarchical model for inferring the underlying customer arrival rate and choice model from sales transaction data and the corresponding stock levels. The model uses a nonhomogeneous Poisson process to allow the arrival rate to vary throughout the day, and allows for a variety of choice models. Model parameters are inferred using a stochastic gradient MCMC algorithm that can scale to large transaction databases. We fit the model to data from a local bakery and show that it is able to make accurate out-of-sample predictions, and to provide actionable insight into lost cookie sales.