A novel classical and Bayesian adaptation of the Pólya urn with reinforcement dynamics: methodologies & applications

The Pólya urn model is a classical stochastic process that illustrates the effects of reinforcement in various fields when an event’s probability increases with its occurrence frequency. This article delves into the Pólya urn probability scheme, offering theoretical insights and practical applications regarding reinforcement techniques imposed via frequentist and Bayesian approaches. We show how initial conditions and reinforcement factors shape long-term outcomes by analyzing the model’s convergence properties, particularly its tendency toward a beta distribution or a mixture of betas. A novel Bayesian extension of the Pólya urn model is proposed, incorporating prior beliefs and dynamic updating mechanisms, enabling more robust and flexible modeling of real-world processes. We validate these theoretical predictions through extensive simulations and real data applications.