A Bayesian approach to a logistic regression model with incomplete information.

We consider a set of independent Bernoulli trials with possibly different success probabilities that depend on covariate values. However, the available data consist only of aggregate numbers of successes among subsets of the trials along with all of the covariate values. We still wish to estimate the parameters of a modeled relationship between the covariates and the success probabilities, e.g., a logistic regression model. In this article, estimation of the parameters is made from a Bayesian perspective by using a Markov chain Monte Carlo algorithm based only on the available data. The proposed methodology is applied to both simulation studies and real data from a dose-response study of a toxic chemical, perchlorate.

Spatial Modelling Using a New Class of Nonstationary Covariance Functions.

We introduce a new class of nonstationary covariance functions for spatial modelling. Nonstationary covariance functions allow the model to adapt to spatial surfaces whose variability changes with location. The class includes a nonstationary version of the Matérn stationary covariance, in which the differentiability of the spatial surface is controlled by a parameter, freeing one from fixing the differentiability in advance. The class allows one to knit together local covariance parameters into a valid global nonstationary covariance, regardless of how the local covariance structure is estimated. We employ this new nonstationary covariance in a fully Bayesian model in which the unknown spatial process has a Gaussian process (GP) prior distribution with a nonstationary covariance function from the class. We model the nonstationary structure in a computationally efficient way that creates nearly stationary local behavior and for which stationarity is a special case. We also suggest non-Bayesian approaches to nonstationary kriging.To assess the method, we use real climate data to compare the Bayesian nonstationary GP model with a Bayesian stationary GP model, various standard spatial smoothing approaches, and nonstationary models that can adapt to function heterogeneity. The GP models outperform the competitors, but while the nonstationary GP gives qualitatively more sensible results, it shows little advantage over the stationary GP on held-out data, illustrating the difficulty in fitting complicated spatial data.