The Dirichlet process can be regarded as a random probability measure for which the authors examine various sum representations. They consider in particular the gamma process construction of Ferguson (1973) and the “stick‐breaking” construction of Sethuraman (1994). They propose a Dirichlet finite sum representation that strongly approximates the Dirichlet process. They assess the accuracy of this approximation and characterize the posterior that this new prior leads to in the context of Bayesian nonpara‐metric hierarchical models.

The beta process has recently been widely used as a nonparametric prior for different models in machine learning, including latent feature models. In this paper, we prove the asymptotic consistency of the finite dimensional approximation of the beta process due to Paisley & Carin (2009). In addition, we derive an almost sure approximation of the beta process. This approximation provides a direct method to efficiently simulate the beta process. A simulated example, illustrating the work of the method and comparing its performance to several existing algorithms, is also included.

In this paper, we develop simple, yet efficient, procedures for sampling approximations of the two-Parameter Poisson-Dirichlet Process and the normalized inverseGaussian process. We compare the efficiency of the new approximations to the corresponding stick-breaking approximations of the two-parameter Poisson-Dirichlet Process and the normalized inverse-Gaussian process, in which we demonstrate a substantial improvement.

In recent years, Bayesian nonparametric statistics has received extraordinary attention. The beta‐Stacy process, a generalization of the Dirichlet process, is a fundamental tool in studying Bayesian nonparametric statistics. In this article, we derive a simple, yet efficient, way to simulate the beta‐Stacy process. We compare the efficiency of the new approximation to several other well‐known approximations, and we demonstrate a significant improvement. Using the Kolmogorov distance and samples from the beta‐Stacy process, a Bayesian nonparametric goodness of fit test is proposed. The proposed test is very general in the sense that it can be applied to censored and non‐censored observations. Some illustrative examples are included. 41: 466–487; 2013 © 2013 Statistical Society of Canada

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