Results 1  10
of
80
Poisson process partition calculus with an application to Bayesian . . .
, 2005
"... 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 tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The P ..."
Abstract

Cited by 56 (14 self)
 Add to MetaCart
(Show Context)
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 tailormade 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
Sharing Features among Dynamical Systems with Beta Processes
"... We propose a Bayesian nonparametric approach to the problem of modeling related time series. Using a beta process prior, our approach is based on the discovery of a set of latent dynamical behaviors that are shared among multiple time series. The size of the set and the sharing pattern are both infe ..."
Abstract

Cited by 38 (11 self)
 Add to MetaCart
(Show Context)
We propose a Bayesian nonparametric approach to the problem of modeling related time series. Using a beta process prior, our approach is based on the discovery of a set of latent dynamical behaviors that are shared among multiple time series. The size of the set and the sharing pattern are both inferred from data. We develop an efficient Markov chain Monte Carlo inference method that is based on the Indian buffet process representation of the predictive distribution of the beta process. In particular, our approach uses the sumproduct algorithm to efficiently compute MetropolisHastings acceptance probabilities, and explores new dynamical behaviors via birth/death proposals. We validate our sampling algorithm using several synthetic datasets, and also demonstrate promising results on unsupervised segmentation of visual motion capture data. 1
Indian Buffet Processes with Powerlaw Behavior
"... The Indian buffet process (IBP) is an exchangeable distribution over binary matrices used in Bayesian nonparametric featural models. In this paper we propose a threeparameter generalization of the IBP exhibiting powerlaw behavior. We achieve this by generalizing the beta process (the de Finetti me ..."
Abstract

Cited by 24 (1 self)
 Add to MetaCart
(Show Context)
The Indian buffet process (IBP) is an exchangeable distribution over binary matrices used in Bayesian nonparametric featural models. In this paper we propose a threeparameter generalization of the IBP exhibiting powerlaw behavior. We achieve this by generalizing the beta process (the de Finetti measure of the IBP) to the stablebeta process and deriving the IBP corresponding to it. We find interesting relationships between the stablebeta process and the PitmanYor process (another stochastic process used in Bayesian nonparametric models with interesting powerlaw properties). We derive a stickbreaking construction for the stablebeta process, and find that our powerlaw IBP is a good model for word occurrences in document corpora. 1
Betanegative binomial process and Poisson factor analysis
 In AISTATS
, 2012
"... A betanegative binomial (BNB) process is proposed, leading to a betagammaPoisson process, which may be viewed as a “multiscoop” generalization of the betaBernoulli process. The BNB process is augmented into a betagammagammaPoisson hierarchical structure, and applied as a nonparametric Bayesia ..."
Abstract

Cited by 22 (9 self)
 Add to MetaCart
(Show Context)
A betanegative binomial (BNB) process is proposed, leading to a betagammaPoisson process, which may be viewed as a “multiscoop” generalization of the betaBernoulli process. The BNB process is augmented into a betagammagammaPoisson hierarchical structure, and applied as a nonparametric Bayesian prior for an infinite Poisson factor analysis model. A finite approximation for the beta process Lévy random measure is constructed for convenient implementation. Efficient MCMC computations are performed with data augmentation and marginalization techniques. Encouraging results are shown on document count matrix factorization. 1
MCMC for normalized random measure mixture models Statistical Science 28
, 2013
"... Abstract. This paper concerns the use of Markov chain Monte Carlo methods for posterior sampling in Bayesian nonparametric mixture models with normalized random measure priors. Making use of some recent posterior characterizations for the class of normalized random measures, we propose novel Markov ..."
Abstract

Cited by 20 (9 self)
 Add to MetaCart
(Show Context)
Abstract. This paper concerns the use of Markov chain Monte Carlo methods for posterior sampling in Bayesian nonparametric mixture models with normalized random measure priors. Making use of some recent posterior characterizations for the class of normalized random measures, we propose novel Markov chain Monte Carlo methods of both marginal type and conditional type. The proposed marginal samplers are generalizations of Neal’s wellregarded Algorithm 8 for Dirichlet process mixture models, whereas the conditional sampler is a variation of those recently introduced in the literature. For both the marginal and conditional methods, we consider as a running example a mixture model with an underlying normalized generalized Gamma process prior, and describe comparative simulation results demonstrating the efficacies of the proposed methods. Key words and phrases: Bayesian nonparametrics, hierarchical mixture model, completely random measure, normalized random measure, Dirichlet process, normalized generalized Gamma process, MCMC posterior sampling method, marginalized sampler, Algorithm 8, conditional sampler, slice sampling. 1.
Spatial Normalized Gamma Processes
"... Dependent Dirichlet processes (DPs) are dependent sets of random measures, each being marginally DP distributed. They are used in Bayesian nonparametric models when the usual exchangeability assumption does not hold. We propose a simple and general framework to construct dependent DPs by marginalizi ..."
Abstract

Cited by 19 (3 self)
 Add to MetaCart
(Show Context)
Dependent Dirichlet processes (DPs) are dependent sets of random measures, each being marginally DP distributed. They are used in Bayesian nonparametric models when the usual exchangeability assumption does not hold. We propose a simple and general framework to construct dependent DPs by marginalizing and normalizing a single gamma process over an extended space. The result is a set of DPs, each associated with a point in a space such that neighbouring DPs are more dependent. We describe Markov chain Monte Carlo inference involving Gibbs sampling and three different MetropolisHastings proposals to speed up convergence. We report an empirical study of convergence on a synthetic dataset and demonstrate an application of the model to topic modeling through time. 1
Combinatorial clustering and the beta negative binomial process. arXiv preprint arXiv:1111.1802
, 2013
"... Abstract—We develop a Bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual. We introduce a combinatorial stochastic process known as the ne ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
(Show Context)
Abstract—We develop a Bayesian nonparametric approach to a general family of latent class problems in which individuals can belong simultaneously to multiple classes and where each class can be exhibited multiple times by an individual. We introduce a combinatorial stochastic process known as the negative binomial process (NBP) as an infinitedimensional prior appropriate for such problems. We show that the NBP is conjugate to the beta process, and we characterize the posterior distribution under the betanegative binomial process (BNBP) and hierarchical models based on the BNBP (the HBNBP). We study the asymptotic properties of the BNBP and develop a threeparameter extension of the BNBP that exhibits powerlaw behavior. We derive MCMC algorithms for posterior inference under the HBNBP, and we present experiments using these algorithms in the domains of image segmentation, object recognition, and document analysis.
The multifractal nature of heterogeneous sums of Dirac masses
 MATH. PROC. CAMBRIDGE PH. SOC
, 2008
"... This article investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masses ν = ∑ n≥0 wn δxn, where (xn)n≥0 is a sequence of points in [0, 1] d and (wn)n≥0 is a positive sequence of weights such that ∑ n≥0 wn < ∞. We consider the case where the po ..."
Abstract

Cited by 15 (9 self)
 Add to MetaCart
This article investigates the natural problem of performing the multifractal analysis of heterogeneous sums of Dirac masses ν = ∑ n≥0 wn δxn, where (xn)n≥0 is a sequence of points in [0, 1] d and (wn)n≥0 is a positive sequence of weights such that ∑ n≥0 wn < ∞. We consider the case where the points xn are roughly uniformly distributed in [0, 1] d, and the weights wn depend on a random selfsimilar measure µ, a parameter ρ ∈ (0, 1], and a sequence of positive radii (λn)n≥1 converging to 0 in the following way wn = λ d(1−ρ) n µ ( B(xn, λ ρ n) )  log λn  −2. The measure ν has a rich multiscale structure. The computation of its multifractal spectrum is related to heterogeneous ubiquity properties of the system {(xn, λn)}n with respect to µ.