On simulations from the two-parameter Poisson-Dirichlet process and the normalized inverse-Gaussian process

Abstract

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.

Stochastic Approximations to the Pitman–Yor Process

“…the center panel of Figure 1. It also underlines that the approximation deteriorates for fixed n and increasing α, which is coherent with the findings in Al Labadi and Zarepour (2014). A fair comparison with their approach can only be done for a given nominal approximation error, but unfortunately the authors did not provide a precise assessment of it.…”

“…It would be also interesting to compare the accuracy of our finite dimensional approximation of the Pitman-Yor process to the one proposed in Al Labadi and Zarepour (2014). The latter is based on a representation of the frequencies in decreasing order, cf.…”

“…Pitman and Yor (1997, Proposition 22). Al Labadi and Zarepour (2014) compare the accuracy of their approximation scheme to a stick-breaking truncation at a number n of stick-breaking frequencies that matches the number of frequencies used in their scheme. Not surprisingly, their approximation is superior since it generates weights in decreasing order, specially when α is large.…”

“…A fair comparison with their approach can only be done for a given nominal approximation error, but unfortunately the authors did not provide a precise assessment of it. The number of stick-breaking frequencies needed to match the approximation accuracy of Al Labadi and Zarepour (2014) would be de facto larger due to the non monotonicity. However, since the stopping rule (4) adapts to the size of α, we do not expect the accuracy of our approximation scheme to deteriorate for α large.…”

“…However, since the stopping rule (4) adapts to the size of α, we do not expect the accuracy of our approximation scheme to deteriorate for α large. As for computation time, the techniques used by Al Labadi and Zarepour (2014) in order to obtain decreasing frequencies are computational heavy. Their average computing time for α = 0.5 is about 2.30 seconds/iteration with 10 4 locations.…”