We present a Bayesian methodology for extracting correlation lengths from small-angle neutron scattering (SANS) experiments. For demonstration, we apply the technique to data from a previous paper, which investigated the presence of dipolar ferromagnetism in assemblies of ferromagnetic Co nanoparticles. Bayesian analysis confirms the presence of multiparticle dipolar domains even at zero magnetic field, but higher-field correlation lengths were found to be much smaller than previously believed, yielding new information on the maximum lengthscale which the instrument can reliably probe. We use two complementary types of graph to visualize the results. Plots of standardized residual distributions show quality of fit, and guide model refinement. These principles can be applied to other types of sample, and even to other small-angle scattering techniques.

The infinite mixture of normals model has become a popular method for density estimation problems. This paper proposes an alternative hierarchical model that leads to hyperparameters that can be interpreted as the location, scale and smoothness of the density. The priors on other parts of the model have little effect on the density estimates and can be given default choices. Automatic Bayesian density estimation can be implemented by using uninformative priors for location and scale and default priors for the smoothness. The performance of these methods for density estimation are compared to previously proposed default priors for four data sets.

Computationally efficient simulation methods for hierarchical Bayesian analysis of the seemingly unrelated regression (SUR) and simultaneous equations models (SEM) are proposed and applied. These methods combine a direct Monte Carlo (DMC) approach and an importance sampling procedure to calculate Bayesian estimation and prediction results, namely, Bayesian posterior densities for parameters, predictive densities for future values of variables and associated moments, intervals and other quantities. The results obtained by our approach are compared to those yielded by use of MCMC techniques. Finally, we show that our algorithm can be applied to the Bayesian analysis of state space models.

We address the issue of inference for a noisy point pattern. The unobserved true point process is modelled as a nonhomogeneous Poisson process. For modeling the underlying intensity surface we use a scaled Gaussian mixture distribution. The noise that creeps in during the measurement procedure causes random displacement of the true locations. We consider two settings. With a bounded region of interest, (i) this displacement may cause a true location within the boundary to be associated with an ‘observed’ location outside of the region and thus missed and (ii) we have the possibility in (i) but also vice versa; the displacement may bring in an observed location whose true location lies outside the region. Under (i), we can only lose points and, depending on the variability in the measurement error as well as the number of true locations close to boundary, this can cause a significant number of locations to be lost from our recorded set of data. Estimation of the intensity surface from the observed data can be misleading especially near the boundary of our domain of interest. Under (ii), the modeling problem is more difficult; points can be both lost and gained and it is challenging to characterize how we may gain points with no data on the underlying intensity outside the domain of interest. In both cases, we work within a hierarchical Bayes framework, modeling the latent point pattern using a Cox process and, given the process realization, introducing a suitable measurement error model. Hence, the specification includes the true number of points as an unknown. We discuss choice of measurement error model as well as identifiability problems which arise. Models are fitted using an markov chain Monte Carlo implementation. After validating our method against several synthetic datasets we illustrate its application for two ecological datasets.

Functional analysis of variance (ANOVA) models partition a functional response according to the main effects and interactions of various factors. This article develops a general framework for functional ANOVA modeling from a Bayesian viewpoint, assigning Gaussian process prior distributions to each batch of functional effects. We discuss the choices to be made in specifying such a model, advocating the treatment of levels within a given factor as dependent but exchangeable quantities, and we suggest weakly informative prior distributions for higher level parameters that may be appropriate in many situations. We discuss computationally efficient strategies for posterior sampling using Markov Chain Monte Carlo algorithms, and we emphasize useful graphical summaries based on the posterior distribution of model-based analogues of traditional ANOVA decompositions of variance. We illustrate this process of model specification, posterior sampling, and graphical posterior summaries in two examples. The first considers the effect of geographic region on the temperature profiles at weather stations in Canada. The second example examines sources of variability in the output of regional climate models from a designed experiment.

Elastic net Zou and Hastie (2005) is a flexible regularization and variable selection method that uses a mixture of $L_1$ and $L_2$ penalties. It is particularly useful when there are much more predictors than the sample size. This paper proposes a Bayesian method to solve the elastic net model using a Gibbs sampler. While the marginal posterior mode of the regression coefficients is equivalent to estimates given by the non-Bayesian elastic net, the Bayesian elastic net has two major advantages. Firstly, as a Bayesian method, the distributional results on the estimates are straightforward, making the statistical inference easier. Secondly, it chooses the two penalty parameters simultaneously, avoiding the "double shrinkage problem" in the elastic net method. Real data examples and simulation studies show that the Bayesian elastic net behaves comparably in prediction accuracy but performs better in variable selection.

This paper considers the effects of placing an absolutely continuous prior distribution on the regression coefficients of a linear model. We show that the posterior expectation is a matrix-shrunken version of the least squares estimate where the shrinkage matrix depends on the derivatives of the prior predictive density of the least squares estimate. The special case of the normal-gamma prior, which generalizes the Bayesian Lasso (Park and Casella 2008), is studied in depth. We discuss the prior interpretation and the posterior effects of hyperparameter choice and suggest a data-dependent default prior. Simulations and a chemometric example are used to compare the performance of the normal-gamma and the Bayesian Lasso in terms of out-of-sample predictive performance.

This work is motivated by a quantitative Magnetic Resonance Imaging study of the relative change in tumor vascular permeability during the course of radiation therapy. The differences in tumor and healthy brain tissue physiology and pathology constitute a notable feature of the image data---spatial heterogeneity with respect to its contrast uptake profile (a surrogate for permeability) and radiation induced changes in this profile. To account for these spatial aspects of the data, we employ a Gaussian hidden Markov random field (MRF) model. The model incorporates a latent set of discrete labels from the MRF governed by a spatial regularization parameter. We estimate the MRF regularization parameter and treat the number of MRF states as a random variable and estimate it via a reversible jump Markov chain Monte Carlo algorithm. We conduct simulation studies to examine the performance of the model and compare it with a recently proposed method using the Expectation-Maximization (EM) algorithm. Simulation results show that the Bayesian algorithm performs as well, if not slightly better than the EM based algorithm. Results on real data suggest that the tumor "core" vascular permeability increases relative to healthy tissue three weeks after starting radiotherapy, which may be an opportune time to initiate chemotherapy and warrants further investigation.