28 research outputs found
Dynamic density estimation with diffusive Dirichlet mixtures
We introduce a new class of nonparametric prior distributions on the space of
continuously varying densities, induced by Dirichlet process mixtures which
diffuse in time. These select time-indexed random functions without jumps,
whose sections are continuous or discrete distributions depending on the choice
of kernel. The construction exploits the widely used stick-breaking
representation of the Dirichlet process and induces the time dependence by
replacing the stick-breaking components with one-dimensional Wright-Fisher
diffusions. These features combine appealing properties of the model, inherited
from the Wright-Fisher diffusions and the Dirichlet mixture structure, with
great flexibility and tractability for posterior computation. The construction
can be easily extended to multi-parameter GEM marginal states, which include,
for example, the Pitman--Yor process. A full inferential strategy is detailed
and illustrated on simulated and real data.Comment: Published at http://dx.doi.org/10.3150/14-BEJ681 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
A Bayesian Nonparametric Method for Prediction in EST Analysis
In this work we propose a Bayesian nonparametric approach for tackling statistical problems related to EST surveys. In particular, we provide estimates for: a) the coverage, defined as the proportion of unique genes in the library represented in the given sample of reads; b) the number of new unique genes to be observed in a future sample; c) the discovery rate of new genes as a function of the future sample size. The Bayesian nonparametric model we adopt conveys, in a statistically rigorous way, the available information into prediction. Our proposal has appealing properties over frequentist nonparametric methods, which become unstable when prediction is required for large future samples. EST libraries studied in Susko and Roger (2004), with frequentist methods, are analyzed in detail.
Exchangeable Claims Sizes in a Compound Poisson Type Proces
When dealing with risk models the typical assumption of independence among claim size distributions is not always satisfied. Here we consider the case when the claim sizes are exchangeable and study the implications when constructing aggregated claims through compound Poisson type processes. In par- ticular, exchangeability is achieved through conditional independence and using parametric and nonparametric measures for the conditioning distribution. A full Bayesian analysis of the proposed model is carried out to illustrate.Bayes nonparametrics, compound Poisson process, exchangeable claim process, exchangeable sequence, risk model.
On a flexible construction of a negative binomial model
This work presents a construction of stationary Markov models with
negative-binomial marginal distributions. A simple closed form expression for
the corresponding transition probabilities is given, linking the proposal to
well-known classes of birth and death processes and thus revealing interesting
characterizations. The advantage of having such closed form expressions is
tested on simulated and real data.Comment: Forthcoming in "Statistics & Probability Letters
Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process
Discrete random probability measures are a key ingredient of Bayesian
nonparametric inferential procedures. A sample generates ties with positive
probability and a fundamental object of both theoretical and applied interest
is the corresponding random number of distinct values. The growth rate can be
determined from the rate of decay of the small frequencies implying that, when
the decreasingly ordered frequencies admit a tractable form, the asymptotics of
the number of distinct values can be conveniently assessed. We focus on the
geometric stick-breaking process and we investigate the effect of the choice of
the distribution for the success probability on the asymptotic behavior of the
number of distinct values. We show that a whole range of logarithmic behaviors
are obtained by appropriately tuning the prior. We also derive a two-term
expansion and illustrate its use in a comparison with a larger family of
discrete random probability measures having an additional parameter given by
the scale of the negative binomial distribution.Comment: 20 page
Continuous-time Markov processes, orthogonal polynomials and Lancaster probabilities
This work links the conditional probability structure of Lancaster probabilities to a construction of reversible continuous-time Markov processes. Such a task is achieved by using the spectral expansion of the corresponding transition probabilities in order to introduce a continuous time dependence in the orthogonal representation inherent to Lancaster probabilities. This relationship provides a novel methodology to build continuous-time Markov processes via Lancaster probabilities. Particular cases of well-known models are seen to fall within this approach. As a byproduct, it also unveils new identities associated to well known orthogonal polynomials