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 $Y$ and $X$ conditionally on $Z$. 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 $Y$, $X$ and $Z$. The framework allows $Y$, $X$ and $Z$ 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