Session on

ORGANIZER AND DISCUSSANT: __F. Ruggeri__, CNR-IAMI, Milan, Italy

A key issue of Bayesian robustness is the optimization of posterior
functionals with respect to prior measures belonging to some
class. We consider here the class of priors defined by a number of
generalized moment conditions.
Constraints of this type are very general as they include for
instance, besides ordinary moment conditions, bounds on prior
quantiles or bounds on marginal probabilities of data. An
algorithm is presented for the numerical solution of the above
optimization problem based on ideas suggested by the interval
approach to numerical optimization as well as from semi-infinite
linear programming.

**ORTHOGONAL REGRESSIONS AND MODEL AVERAGING: A BAYESIAN PERSPECTIVE,**
__M. Clyde__, Duke University, Durham, USA

Bayesian model averaging provides a coherent method for making
inferences and predictions that captures uncertainty about the choice of
variables that should be included in the model. In this framework, one
would begin by assigning to each model a prior probability, and then use
Bayes theorem to update the model probability given the data. However in
many instances, it is often computationally infeasible to carry out the
Bayesian paradigm and calculate posterior probabilities of all possible
models. When the explanatory variables are orthogonal, we derive
approximations to the posterior model probabilities. These can be used
in deterministic sampling of models, importance sampling, and Markov
Chain Monte Carlo sampling. We show how the method can be used with
examples from wavelets and generalized linear models.

**BAYESIAN DENSITY ESTIMATION USING BERNSTEIN POLINOMIALS,**
__S. Petrone__, University of Pavia, Pavia, Italy

We discuss a Bayesian nonparametric procedure for density estimation, for data in a closed, bounded interval. To this aim, we use a prior based on Bernstein polynomials.

This corresponds to express the density of the data as a mixture of given beta densities, with random weights and a random number of components. The density estimate is then obtained as the corresponding predictive density function. Comparison with classical and Bayesian kernel estimates and with procedures based on a smoothing of the Dirichlet process is provided.

We also discuss a MCMC algorithm for approximating the estimate which
has some aspects of novelty since the problem in exam has a ``changing
dimension'' parameter space.

**DENSITY AND FUNCTION ESTIMATION: WAVELET APPROACH,**
__B. Vidakovic__, Duke University, Durham, USA

In the talk we discuss traditional and novel wavelet
methods in density estimation and in non-equally
spaced regression. Large sample properties and
algorithmic implementation are discussed. The case
of estimating a square root of density is elaborated in
more detail.
The methods are illustrated on Roeder's ``Galaxy'' data
and Silverman's ``Motorcycle'' data sets.