Bayesian analysis is frequently confused with conjugate Bayesian ana-
lysis. This is particularly the case in the analysis of clinical trial data. Even
though conjugate analysis is perceived to be simpler computationally (but see
below, Berger's prior), the price to be paid is high: such analysis is not robust with
respect to the prior, i.e. changing the prior may a®ect the conclusions without
bound. Furthermore, conjugate Bayesian analysis is blind with respect to the
potential con°ict between the prior and the data. Robust priors, however, have
bounded in°uence. The prior is discounted automatically when there are con°icts
between prior information and data. In other words, conjugate priors may lead
to a dogmatic analysis while robust priors promote self-criticism since prior and
sample information are not on equal footing. The original proposal of robust priors
was made by de-Finetti in the 1960's. However, the practice has not taken hold
in important areas where the Bayesian approach is making de¯nite advances such
as in clinical trials where conjugate priors are ubiquitous.

Counts or averages over arbitrary regions are often analyzed using con-
ditionally autoregressive (CAR) models. The neighborhoods within CAR models
are generally determined using only the inter-distances or boundaries between the
sub-regions. To accommodate spatial variations that may depend on directions,
a new class of models is developed using di®erent weights given to neighbors in
di®erent directions. By accounting for such spatial anisotropy, the proposed model
generalizes the usual CAR model that assigns equal weight to all directions. Within
a fully hierarchical Bayesian framework, the posterior distributions of the param-
eters are derived using conjugate and non-informative priors. E±cient Markov
chain Monte Carlo (MCMC) sampling algorithms are provided to generate sam-
ples from the marginal posterior distribution of the parameters. Simulation studies
are presented to evaluate the performance of the estimators and are used to com-
pare results with traditional CAR models. Finally the method is illustrated using
data sets on local crime frequencies in Columbus, OH and on the elevated blood
lead levels of children under the age of 72 months observed in Virginia counties
for the year of 2000.

We have developed a sophisticated statistical model for predicting the
hitting performance of Major League baseball players. The Bayesian paradigm
provides a principled method for balancing past performance with crucial covari-
ates, such as player age and position. We share information across time and across
players by using mixture distributions to control shrinkage for improved accuracy.
We compare the performance of our model to current sabermetric methods on a
held-out season (2006), and discuss both successes and limitations.

A simple example is presented using standard continuous distributions
with a real valued parameter in which the posterior mean is inconsistent on a dense
subset of the real line

Markov switching models can be used to study heterogeneous pop-
ulations that are observed over time. This paper explores modeling the group
characteristics nonparametrically, under both homogeneous and nonhomogeneous
Markov switching for group probabilities. The model formulation involves a nite
mixture of conditionally independent Dirichlet process mixtures, with a Markov
chain dening the mixing distribution. The proposed methodology focuses on
settings where the number of subpopulations is small and can be assumed to be
known, and
exible modeling is required for group regressions. We develop Dirich-
let process mixture prior probability models for the joint distribution of individual
group responses and covariates. The implied conditional distribution of the re-
sponse given the covariates is then used for inference. The modeling framework al-
lows for both non-linearities in the resulting regression functions and non-standard
shapes in the response distributions. We design a simulation-based model tting
method for full posterior inference. Furthermore, we propose a general approach
for inclusion of external covariates dependent on the Markov chain but condition-
ally independent from the response. The methodology is applied to a problem
from sheries research involving analysis of stock-recruitment data under shifts in
the ecosystem state

This paper demonstrates the use and value of stochastic di®erential
equations for modeling space-time data in two common settings. The ¯rst consists
of point-referenced or geostatistical data where observations are collected at ¯xed
locations and times. The second considers random point pattern data where the
emergence of locations and times is random. For both cases, we employ stochas-
tic di®erential equations to describe a latent process within a hierarchical model
for the data. The intent is to view this latent process mechanistically and endow
it with appropriate simple features and interpretable parameters. A motivating
problem for the second setting is to model urban development through observed
locations and times of new home construction; this gives rise to a space-time point
pattern. We show that a spatio-temporal Cox process whose intensity is driven
by a stochastic logistic equation is a viable mechanistic model that a®ords mean-
ingful interpretation for the results of statistical inference. Other applications of
stochastic logistic di®erential equations with space-time varying parameters in-
clude modeling population growth and product di®usion, which motivate our ¯rst,
point-referenced data application. We propose a method to discretize both time
and space in order to ¯t the model. We demonstrate the inference for the geosta-
tistical model through a simulated dataset. Then, we ¯t the Cox process model
to a real dataset taken from the greater Dallas metropolitan area.

Estimation of the number of species extant in a geographic region has
been discussed in the statistical literature for more than sixty years. The focus
of this work is on the use of pilot data to design future studies in this context.
A Dirichlet-multinomial probability model for species frequency data is used to
obtain a posterior distribution on the number of species and to learn about the dis-
tribution of species frequencies. A geometric distribution is proposed as the prior
distribution for the number of species. Simulations demonstrate that this prior dis-
tribution can handle a wide range of species frequency distributions including the
problematic case with many rare species and a few exceptionally abundant species.
Monte Carlo methods are used along with the Dirichlet-multinomial model to per-
form sample size calculations from pilot data, e.g., to determine the number of
additional samples required to collect a certain proportion of all the species with
a pre-speci¯ed coverage probability. Simulations and real data applications are
discussed

We propose a Bayesian method for multiple hypothesis testing in ran-
dom e®ects models that uses Dirichlet process (DP) priors for a nonparametric
treatment of the random e®ects distribution. We consider a general model for-
mulation which accommodates a variety of multiple treatment conditions. A key
feature of our method is the use of a product of spiked distributions, i.e., mixtures
of a point-mass and continuous distributions, as the centering distribution for the
DP prior. Adopting these spiked centering priors readily accommodates sharp
null hypotheses and allows for the estimation of the posterior probabilities of such
hypotheses. Dirichlet process mixture models naturally borrow information across
objects through model-based clustering while inference on single hypotheses aver-
ages over clustering uncertainty. We demonstrate via a simulation study that our
method yields increased sensitivity in multiple hypothesis testing and produces a
lower proportion of false discoveries than other competitive methods. While our
modeling framework is general, here we present an application in the context of
gene expression from microarray experiments. In our application, the modeling
framework allows simultaneous inference on the parameters governing di®erential
expression and inference on the clustering of genes. We use experimental data on
the transcriptional response to oxidative stress in mouse heart muscle and compare
the results from our procedure with existing nonparametric Bayesian methods that
provide only a ranking of the genes by their evidence for di®erential expression.