1,169 research outputs found
Fuzzy sets in nonparametric Bayes regression
A simple Bayesian approach to nonparametric regression is described using
fuzzy sets and membership functions. Membership functions are interpreted as
likelihood functions for the unknown regression function, so that with the help
of a reference prior they can be transformed to prior density functions. The
unknown regression function is decomposed into wavelets and a hierarchical
Bayesian approach is employed for making inferences on the resulting wavelet
coefficients.Comment: Published in at http://dx.doi.org/10.1214/074921708000000084 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
Nonparametric Bayes modeling of count processes
Data on count processes arise in a variety of applications, including
longitudinal, spatial and imaging studies measuring count responses. The
literature on statistical models for dependent count data is dominated by
models built from hierarchical Poisson components. The Poisson assumption is
not warranted in many applications, and hierarchical Poisson models make
restrictive assumptions about over-dispersion in marginal distributions. This
article proposes a class of nonparametric Bayes count process models, which are
constructed through rounding real-valued underlying processes. The proposed
class of models accommodates applications in which one observes separate
count-valued functional data for each subject under study. Theoretical results
on large support and posterior consistency are established, and computational
algorithms are developed using Markov chain Monte Carlo. The methods are
evaluated via simulation studies and illustrated through application to
longitudinal tumor counts and asthma inhaler usage
Nonparametric Bayes inference on conditional independence
In broad applications, it is routinely of interest to assess whether there is
evidence in the data to refute the assumption of conditional independence of
and conditionally on . Such tests are well developed in parametric
models but are not straightforward in the nonparametric case. We propose a
general Bayesian approach, which relies on an encompassing nonparametric Bayes
model for the joint distribution of , and . The framework allows ,
and to be random variables on arbitrary spaces, and can accommodate
different dimensional vectors having a mixture of discrete and continuous
measurement scales. Using conditional mutual information as a scalar summary of
the strength of the conditional dependence relationship, we construct null and
alternative hypotheses. We provide conditions under which the correct
hypothesis will be consistently selected. Computational methods are developed,
which can be incorporated within MCMC algorithms for the encompassing model.
The methods are applied to variable selection and assessed through simulations
and criminology applications
Nonparametric Bayes Modeling of Populations of Networks
Replicated network data are increasingly available in many research fields.
In connectomic applications, inter-connections among brain regions are
collected for each patient under study, motivating statistical models which can
flexibly characterize the probabilistic generative mechanism underlying these
network-valued data. Available models for a single network are not designed
specifically for inference on the entire probability mass function of a
network-valued random variable and therefore lack flexibility in characterizing
the distribution of relevant topological structures. We propose a flexible
Bayesian nonparametric approach for modeling the population distribution of
network-valued data. The joint distribution of the edges is defined via a
mixture model which reduces dimensionality and efficiently incorporates network
information within each mixture component by leveraging latent space
representations. The formulation leads to an efficient Gibbs sampler and
provides simple and coherent strategies for inference and goodness-of-fit
assessments. We provide theoretical results on the flexibility of our model and
illustrate improved performance --- compared to state-of-the-art models --- in
simulations and application to human brain networks
Nonparametric Bayes dynamic modeling of relational data
Symmetric binary matrices representing relations among entities are commonly
collected in many areas. Our focus is on dynamically evolving binary relational
matrices, with interest being in inference on the relationship structure and
prediction. We propose a nonparametric Bayesian dynamic model, which reduces
dimensionality in characterizing the binary matrix through a lower-dimensional
latent space representation, with the latent coordinates evolving in continuous
time via Gaussian processes. By using a logistic mapping function from the
probability matrix space to the latent relational space, we obtain a flexible
and computational tractable formulation. Employing P\`olya-Gamma data
augmentation, an efficient Gibbs sampler is developed for posterior
computation, with the dimension of the latent space automatically inferred. We
provide some theoretical results on flexibility of the model, and illustrate
performance via simulation experiments. We also consider an application to
co-movements in world financial markets
Gaussian process methods for one-dimensional diffusions: optimal rates and adaptation
We study the performance of nonparametric Bayes procedures for
one-dimensional diffusions with periodic drift. We improve existing convergence
rate results for Gaussian process (GP) priors with fixed hyper parameters.
Moreover, we exhibit several possibilities to achieve adaptation to smoothness.
We achieve this by considering hierarchical procedures that involve either a
prior on a multiplicative scaling parameter, or a prior on the regularity
parameter of the GP
- …