Early urn schemes with innovation: Hoppe’s model and the Dirichlet process

Abstract This chapter explores the first urn models that explicitly incorporate innovation at the level of the urn itself, allowing the space of possibilities to expand. It focuses on the Hoppe’s model, which was originally developed to explain Ewens’ sampling formula in population genetics. The chapter demonstrates how this model is equivalent to other fundamental stochastic processes, such as the Chinese restaurant process and the Kingman’s coalescent. While this framework successfully introduces an evolving space of possibilities, it is shown to predict logarithmic vocabulary growth (as described by Heaps’ law) and an exponential frequency distribution. However, it fails to capture the power laws observed in empirical data.