The Normalizing Constant in the BG/BB Model

This note provides a clarification regarding the conditional and marginal likelihood functions in the BG/BB model, as published in Marketing Science by Fader, Hardie, and Shang (2010). Their Equations 4 and 5 do not include normalizing constants which, if included, would equate these likelihood functions with their corresponding joint probability functions. While these expressions are valid, because likelihood functions need only be correct up to a constant of proportionality, they are not joint probability functions, which may be a source of potential confusion for users who mistakenly equate the one for the other. Assuming the likelihood functions in Equations 4 and 5 are equal to their respective joint probability functions will lead to an incorrect joint probability distribution over recency and frequency data, resulting in incorrect goodness-of-fit metrics and managerially relevant expressions. We provide formal derivations of the joint probability functions that correspond to the likelihood functions in Equations 4 and 5 to remove this potential source of confusion for users of the BG/BB model