The Construction of Generalized Dirichlet Process Distributions via Polya urn and Gibbs Sampling

Bayesian nonparametric inference is increasingly demanding in statistical modeling due to incorporating flexible prior processes in complex data analysis. This paper represents the Polya urn scheme for the generalized Dirichlet process (GDP). It utilizes the partition analysis to construct the joint distribution of a random sample from the GDP as a mixture prior distribution of countable components. Using permutation theory, we present the components' weights in a computationally accessible manner to make the resulting joint prior equation applicable. The advantages of our findings include tractable algebraic operations that lead to closed-form equations. The paper recommends the Polya urn Gibbs sampler algorithm, derive full conditional posterior distributions, and as an illustration, implement the algorithm for fitting some popular statistical models in nonparametric Bayesian settings.