  We present a method for identification of models with good predictive
performances in the family of Bayesian log-linear mixed models with Dirichlet
process random effects. Such a problem arises in many different applications;
here we consider it in the context of disclosure risk estimation, an
increasingly relevant issue raised by the increasing demand for data collected
under a pledge of confidentiality. Two different criteria are proposed and
jointly used via a two-stage selection procedure, in a M-open view. The first
stage is devoted to identifying a path of search; then, at the second, a small
number of nonparametric models is evaluated through an application-specific
score based Bayesian information criterion. We test our method on a variety of
contingency tables based on microdata samples from the US Census Bureau and the
Italian National Security Administration, treated here as populations, and
carefully discuss its features. This leads us to a journey around different
forms and sources of bias along which we show that (i) while based on the so
called "score+search" paradigm, our method is by construction well protected
from the selection-induced bias, and (ii) models with good performances are
invariably characterized by an extraordinarily simple structure of fixed
effects. The complexity of model selection - a very challenging and difficult
task in a strictly parametric context with large and sparse tables - is
therefore significantly defused by our approach. An attractive collateral
result of our analysis are fruitful new ideas about modeling in small area
estimation problems, where interest is in total counts over cells with a small
number of observations.
