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37
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 ..."
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Cited by 32 (10 self)
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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
Hierarchical Bayesian Nonparametric Models with Applications ∗
, 2008
"... Hierarchical modeling is a fundamental concept in Bayesian statistics. The basic idea is that parameters are endowed with distributions which may themselves introduce new parameters, and this construction recurses. A common motif in hierarchical modeling is that of the conditionally independent hier ..."
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Cited by 19 (2 self)
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Hierarchical modeling is a fundamental concept in Bayesian statistics. The basic idea is that parameters are endowed with distributions which may themselves introduce new parameters, and this construction recurses. A common motif in hierarchical modeling is that of the conditionally independent hierarchy, in
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 ..."
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Cited by 19 (6 self)
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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
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 point ..."
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Cited by 14 (8 self)
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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.
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 ..."
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Cited by 12 (0 self)
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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
Heterogenuous ubiquitous systems and Hausdorff dimension
"... Abstract. Let {xn}n∈N be a sequence of [0,1] d, {λn}n∈N a sequence of positive real numbers converging to 0, and δ> 1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsupsets of the form S(δ) = ⋂ ⋃ N∈N n≥N B(xn, λδn). Let µ be a positive Borel me ..."
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Cited by 10 (8 self)
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Abstract. Let {xn}n∈N be a sequence of [0,1] d, {λn}n∈N a sequence of positive real numbers converging to 0, and δ> 1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsupsets of the form S(δ) = ⋂ ⋃ N∈N n≥N B(xn, λδn). Let µ be a positive Borel measure on [0,1] d, ρ ∈ (0, 1] and α> 0. Consider the finer limsupset Sµ(ρ, δ, α) = ⋂ ⋃ B(xn, λ δ n). N∈N n≥N: µ(B(xn,λ ρ n))∼λ ρα n We show that, under suitable assumptions on the measure µ, the Hausdorff dimension of the sets Sµ(ρ, δ, α) can be computed. Moreover, when ρ < 1, a yet unknown saturation phenomenon appears in the computation of the Hausdorff dimension of Sµ(ρ, δ, α). Our results apply to several classes of multifractal measures, and S(δ) corresponds to the special case where µ is a monofractal measure like the Lebesgue measure. The computation of the dimensions of such sets opens the way to the study of several new objects and phenomena. Applications are given for the Diophantine approximation conditioned by (or combined with) badic expansion properties, by averages of some Birkhoff sums and branching random walks, as well as by asymptotic behavior of random covering numbers. 1.
Moments, cumulants and diagram formulae for nonlinear functionals of random measures
, 2008
"... This survey provides a unified discussion of multiple integrals, moments, cumulants and diagram formulae associated with functionals of completely random measures. Our approach is combinatorial, as it is based on the algebraic formalism of partition lattices and Möbius functions. Gaussian and Poisso ..."
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Cited by 10 (7 self)
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This survey provides a unified discussion of multiple integrals, moments, cumulants and diagram formulae associated with functionals of completely random measures. Our approach is combinatorial, as it is based on the algebraic formalism of partition lattices and Möbius functions. Gaussian and Poisson measures are treated in great detail. We also present several combinatorial interpretations of some recent CLTs involving sequences of random variables belonging to a fixed Wiener chaos.
Distinguished properties of the gamma processes and related properties
, 2000
"... We study fundamental properties of the gamma process and their relation to various topics such as Poisson–Dirichlet measures and stable processes. We prove the quasiinvariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit ..."
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Cited by 7 (1 self)
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We study fundamental properties of the gamma process and their relation to various topics such as Poisson–Dirichlet measures and stable processes. We prove the quasiinvariance of the gamma process with respect to a large group of linear transformations. We also show that it is a renormalized limit of the stable processes and has an equivalent sigmafinite measure (quasiLebesgue) with important invariance properties. New properties of the gamma process can be applied to the Poisson—Dirichlet measures. We also emphasize the deep similarity between the gamma process and the Brownian motion. The connection of the above topics makes more transparent some old and new facts about stable and gamma processes, and the PoissonDirichlet measures.
Hierarchical Models, Nested Models and Completely Random Measures
, 2010
"... Statistics has both optimistic and pessimistic faces, with the Bayesian perspective often associated with the former and the frequentist perspective with the latter, but with foundational thinkers such as Jim Berger reminding us that statistics is fundamentally a Januslike creature with two faces. ..."
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Cited by 7 (1 self)
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Statistics has both optimistic and pessimistic faces, with the Bayesian perspective often associated with the former and the frequentist perspective with the latter, but with foundational thinkers such as Jim Berger reminding us that statistics is fundamentally a Januslike creature with two faces. In creating one field out of two perspectives, one of the unifying
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 ..."
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Cited by 7 (2 self)
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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