101 research outputs found
Poisson Latent Feature Calculus for Generalized Indian Buffet Processes
The purpose of this work is to describe a unified, and indeed simple,
mechanism for non-parametric Bayesian analysis, construction and generative
sampling of a large class of latent feature models which one can describe as
generalized notions of Indian Buffet Processes(IBP). This is done via the
Poisson Process Calculus as it now relates to latent feature models. The IBP
was ingeniously devised by Griffiths and Ghahramani in (2005) and its
generative scheme is cast in terms of customers entering sequentially an Indian
Buffet restaurant and selecting previously sampled dishes as well as new
dishes. In this metaphor dishes corresponds to latent features, attributes,
preferences shared by individuals. The IBP, and its generalizations, represent
an exciting class of models well suited to handle high dimensional statistical
problems now common in this information age. The IBP is based on the usage of
conditionally independent Bernoulli random variables, coupled with completely
random measures acting as Bayesian priors, that are used to create sparse
binary matrices. This Bayesian non-parametric view was a key insight due to
Thibaux and Jordan (2007). One way to think of generalizations is to to use
more general random variables. Of note in the current literature are models
employing Poisson and Negative-Binomial random variables. However, unlike their
closely related counterparts, generalized Chinese restaurant processes, the
ability to analyze IBP models in a systematic and general manner is not yet
available. The limitations are both in terms of knowledge about the effects of
different priors and in terms of models based on a wider choice of random
variables. This work will not only provide a thorough description of the
properties of existing models but also provide a simple template to devise and
analyze new models.Comment: This version provides more details for the multivariate extensions in
section 5. We highlight the case of a simple multinomial distribution and
showcase a multivariate Levy process prior we call a stable-Beta Dirichlet
process. Section 4.1.1 expande
Poisson calculus for spatial neutral to the right processes
Neutral to the right (NTR) processes were introduced by Doksum in 1974 as
Bayesian priors on the class of distributions on the real line. Since that time
there have been numerous applications to models that arise in survival analysis
subject to possible right censoring. However, unlike the Dirichlet process, the
larger class of NTR processes has not been used in a wider range of more
complex statistical applications. Here, to circumvent some of these
limitations, we describe a natural extension of NTR processes to arbitrary
Polish spaces, which we call spatial neutral to the right processes. Our
construction also leads to a new rich class of random probability measures,
which we call NTR species sampling models. We show that this class contains the
important two parameter extension of the Dirichlet process. We provide a
posterior analysis, which yields tractable NTR analogues of the
Blackwell--MacQueen distribution. Our analysis turns out to be closely related
to the study of regenerative composition structures. A new computational
scheme, which is an ordered variant of the general Chinese restaurant
processes, is developed. This can be used to approximate complex posterior
quantities. We also discuss some relationships to results that appear outside
of Bayesian nonparametrics.Comment: Published at http://dx.doi.org/10.1214/009053605000000732 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bayesian Poisson process partition calculus with an application to Bayesian L\'evy moving averages
This article develops, and describes how to use, results concerning
disintegrations of Poisson random measures. These results are fashioned as
simple tools that can be tailor-made to address inferential questions arising
in a wide range of Bayesian nonparametric and spatial statistical models. The
Poisson disintegration method is based on the formal statement of two results
concerning a Laplace functional change of measure and a Poisson Palm/Fubini
calculus in terms of random partitions of the integers {1,...,n}. The
techniques are analogous to, but much more general than, techniques for the
Dirichlet process and weighted gamma process developed in [Ann. Statist. 12
(1984) 351-357] and [Ann. Inst. Statist. Math. 41 (1989) 227-245]. In order to
illustrate the flexibility of the approach, large classes of random probability
measures and random hazards or intensities which can be expressed as
functionals of Poisson random measures are described. We describe a unified
posterior analysis of classes of discrete random probability which identifies
and exploits features common to all these models. The analysis circumvents many
of the difficult issues involved in Bayesian nonparametric calculus, including
a combinatorial component. This allows one to focus on the unique features of
each process which are characterized via real valued functions h. The
applicability of the technique is further illustrated by obtaining explicit
posterior expressions for L\'evy-Cox moving average processes within the
general setting of multiplicative intensity models.Comment: Published at http://dx.doi.org/10.1214/009053605000000336 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
- …