Pólyaféle

Pólyaféle refers to a family of stochastic processes that exhibit self-reinforcement: the probability of future events increases with their past frequency. The best-known example is the Polya urn model.

In the classic two-color Polya urn, the urn starts with α red and β blue balls. At each step, a ball is drawn uniformly at random, returned to the urn, and one additional ball of the same color is added. This reinforcement causes early draws to influence later composition. After many draws, the proportion of red balls converges almost surely to a random limit distributed as Beta(α, β). The sequence is exchangeable, and the model provides a constructive representation of certain Bayesian posteriors.

Generalizations include more colors, different reinforcement schemes (adding more than one ball per draw), and time-varying rules. Pólyaféle processes have applications in Bayesian statistics (notably the Dirichlet process and the Blackwell–MacQueen urn scheme), population genetics, ecology, and network science, where reinforcement mechanisms model rich-get-richer dynamics.

Historically, the model was introduced by the Hungarian mathematician George Pólya in the early 20th century. The term Polya-type is now used in probability to denote this family of self-reinforcing processes.