7 research outputs found
A moment-matching Ferguson and Klass algorithm
Completely random measures (CRM) represent the key building block of a wide
variety of popular stochastic models and play a pivotal role in modern Bayesian
Nonparametrics. A popular representation of CRMs as a random series with
decreasing jumps is due to Ferguson and Klass (1972). This can immediately be
turned into an algorithm for sampling realizations of CRMs or more elaborate
models involving transformed CRMs. However, concrete implementation requires to
truncate the random series at some threshold resulting in an approximation
error. The goal of this paper is to quantify the quality of the approximation
by a moment-matching criterion, which consists in evaluating a measure of
discrepancy between actual moments and moments based on the simulation output.
Seen as a function of the truncation level, the methodology can be used to
determine the truncation level needed to reach a certain level of precision.
The resulting moment-matching \FK algorithm is then implemented and illustrated
on several popular Bayesian nonparametric models.Comment: 24 pages, 6 figures, 5 table
A blocked Gibbs sampler for NGG-mixture models via a priori truncation.
We define a new class of random probability measures, approximating the well-known normalized generalized gamma (NGG) process. Our new process is defined from the representation of NGG processes as discrete measures where the weights are obtained by normalization of the jumps of Poisson processes and the support consists of independent identically distributed location points, however considering only jumps larger than a threshold TeX. Therefore, the number of jumps of the new process, called TeX-NGG process, is a.s. finite. A prior distribution for TeX can be elicited. We assume such a process as the mixing measure in a mixture model for density and cluster estimation, and build an efficient Gibbs sampler scheme to simulate from the posterior. Finally, we discuss applications and performance of the model to two popular datasets, as well as comparison with competitor algorithms, the slice sampler and a posteriori truncation
Truncated Random Measures
Completely random measures (CRMs) and their normalizations are a rich source
of Bayesian nonparametric priors. Examples include the beta, gamma, and
Dirichlet processes. In this paper we detail two major classes of sequential
CRM representations---series representations and superposition
representations---within which we organize both novel and existing sequential
representations that can be used for simulation and posterior inference. These
two classes and their constituent representations subsume existing ones that
have previously been developed in an ad hoc manner for specific processes.
Since a complete infinite-dimensional CRM cannot be used explicitly for
computation, sequential representations are often truncated for tractability.
We provide truncation error analyses for each type of sequential
representation, as well as their normalized versions, thereby generalizing and
improving upon existing truncation error bounds in the literature. We analyze
the computational complexity of the sequential representations, which in
conjunction with our error bounds allows us to directly compare representations
and discuss their relative efficiency. We include numerous applications of our
theoretical results to commonly-used (normalized) CRMs, demonstrating that our
results enable a straightforward representation and analysis of CRMs that has
not previously been available in a Bayesian nonparametric context.Comment: To appear in Bernoulli; 58 pages, 3 figure