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 di®erences in tumor and healthy brain tissue physiology
and pathology constitute a notable feature of the image data|spatial heterogene-
ity with respect to its contrast uptake pro¯le (a surrogate for permeability) and
radiation induced changes in this pro¯le. To account for these spatial aspects of
the data, we employ a Gaussian hidden Markov random ¯eld (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. Simula-
tion 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.


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 signi¯cant 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 di±cult; 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
speci¯cation includes the true number of points as an unknown. We discuss choice
of measurement error model as well as identi¯ability problems which arise. Models
are ¯tted 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 func-
tional response according to the main e®ects 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 e®ects. 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 e±cient 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 decom-
positions of variance. We illustrate this process of model speci¯cation, posterior
sampling, and graphical posterior summaries in two examples. The ¯rst consid-
ers the e®ect of geographic region on the temperature pro¯les at weather stations
in Canada. The second example examines sources of variability in the output of
regional climate models from a designed experiment.

The in¯nite 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 e®ect 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.

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 nanopar-
ticles. Bayesian analysis con¯rms the presence of multiparticle dipolar domains
even at zero magnetic ¯eld, but higher-¯eld 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 ¯t, and guide model re¯nement. These principles can be applied
to other types of sample, and even to other small-angle scattering techniques.

Computationally e±cient simulation methods for hierarchical Bayesian
analysis of the seemingly unrelated regression (SUR) and simultaneous equa-
tions 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.

This paper considers the e®ects of placing an absolutely continuous
prior distribution on the regression coe±cients 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 den-
sity 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 e®ects of hyperparameter
choice and suggest a data-dependent default prior. Simulations and a chemomet-
ric example are used to compare the performance of the normal-gamma and the
Bayesian Lasso in terms of out-of-sample predictive performance.

Elastic net (Zou and Hastie 2005) is a °exible regularization and vari-
able selection method that uses a mixture of L1 and L2 penalties. It is particularly
useful when there are much more predictors than the sample size. This paper pro-
poses a Bayesian method to solve the elastic net model using a Gibbs sampler.
While the marginal posterior mode of the regression coe±cients 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.