1,839 research outputs found
Asymptotic behavior of beta-integers
Beta-integers (``-integers'') are those numbers which are the
counterparts of integers when real numbers are expressed in irrational basis
. In quasicrystalline studies -integers supersede the
``crystallographic'' ordinary integers. When the number is a Parry
number, the corresponding -integers realize only a finite number of
distances between consecutive elements and somewhat appear like ordinary
integers, mainly in an asymptotic sense. In this letter we make precise this
asymptotic behavior by proving four theorems concerning Parry -integers.Comment: 17 page
Posterior analysis for some classes of nonparametric models
Recently, James [15, 16] has derived important results for various models in Bayesian nonparametric inference. In particular, he dened a spatial version of neutral to the right processes and derived their posterior distribution. Moreover, he obtained the posterior distribution for an intensity or hazard rate modeled as a mixture under a general multiplicative intensity model. His proofs rely on the so{called Bayesian Poisson partition calculus. Here we provide new proofs based on an alternative technique.Bayesian Nonparametrics; Completely random measure; Hazard rate; Neutral to the right prior; Multiplicative intensity model.
Consistency of Bayes estimators of a binary regression function
When do nonparametric Bayesian procedures ``overfit''? To shed light on this
question, we consider a binary regression problem in detail and establish
frequentist consistency for a certain class of Bayes procedures based on
hierarchical priors, called uniform mixture priors. These are defined as
follows: let be any probability distribution on the nonnegative integers.
To sample a function from the prior , first sample from
and then sample uniformly from the set of step functions from
into that have exactly jumps (i.e., sample all jump locations
and function values independently and uniformly). The main result states
that if a data-stream is generated according to any fixed, measurable
binary-regression function , then frequentist consistency
obtains: that is, for any with infinite support, the posterior of
concentrates on any neighborhood of . Solution of an
associated large-deviations problem is central to the consistency proof.Comment: Published at http://dx.doi.org/10.1214/009053606000000236 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
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