The Joint Distribution of Generalized Dirichlet Process Samples with Applications to Nonparametric Bayesian Modeling
Hassan Akell, Farkhondeh-Alsadat Sajadi, Iraj Kazemi · Research Square · 2024
Abstract The Dirichlet process (DP) mixtures offer a flexible approach to nonparametric Bayesian modeling by specifying a unique joint distribution. The Generalized Dirichlet Process (GDP) expands upon the original DP’s stick-breaking process , influencing posterior distributions significantly and enhancing the ability to model intricate processes. This study’s objective is to determine the joint distribution of a random sample X1,. .. , Xn drawn from the GDP, satisfying X1 ∈ A1,. .. , Xn ∈ An for any measurable sets Ai, where i = 1,. .. , n. The GDP distribution is represented as a countable mixture with Bell number, Bn, components. Key properties such as moments, skewness, and kurtosis are discussed. Our innovative distribution encompasses all possible samples, whether they are distinct or identical variables. Specifically, the results for the Dirich-let process align with the Blackwell-MacQueen joint distribution. We explore the clustering behavior of samples from this mixture, evaluate the posterior distributions, and present two illustrative examples to highlight our findings. MSC Classification: 60E05 , 62E15 , 97K60