Pólya urn models in the presence of additional information

Abstract

Pólya urn models are of the well-known probabilistic concepts and stochastic processes. The proportion of the colors is the question of the most interest in these models. Always, the structure (The number of all the balls in the urn and the proportion of the colors) of the urn at the beginning is given. Also, in the studies where the outcome of certain withdrawals is known, these withdrawals are connected to the first structure, and there is no gap between the known structure and the additional information about the color of the withdrawn balls. The existing gap between the first structure and any relevant information motivates this paper. In the present article, we consider the Pólya-Eggenberger urn model. We prove that an urn with an initial known structure after some unknown withdrawals – conditioned to meet an assumption – behaves like an urn with a known structure. By the results of the present paper, we overcome the complexity and ambiguity caused by the unknown withdrawals, and in spite of the lack of information concerning some withdrawals, we deal with the urn as a process that behaves like a completely determined structure. To illustrate the truth of the obtained results, we simulate several urns and survey the variation in the probabilities of the colors.