A Dynamic Bayesian Network Model for Inventory Level Estimation in   Retail Marketing
  Many retailers today employ inventory management systems based on Re-Order Point Policies, most of which rely on the assumption that all decreases in product inventory levels result from product sales. Unfortunately, it usually happens that small but random quantities of the product get lost, stolen or broken without record as time passes, e.g., as a consequence of shoplifting. This is usual for retailers handling large varieties of inexpensive products, e.g., grocery stores. In turn, over time these discrepancies lead to stock freezing problems, i.e., situations where the system believes the stock is above the re-order point but the actual stock is at zero, and so no replenishments or sales occur. Motivated by these issues, we model the interaction between sales, losses, replenishments and inventory levels as a Dynamic Bayesian Network (DBN), where the inventory levels are unobserved (i.e., hidden) variables we wish to estimate. We present an Expectation-Maximization (EM) algorithm to estimate the parameters of the sale and loss distributions, which relies on solving a one-dimensional dynamic program for the E-step and on solving two separate one-dimensional nonlinear programs for the M-step. 
Luis I. Reyes-Castro and Andres G. Abad
stat.ML