  The proposal and study of dependent prior processes has been a major research
focus in the recent Bayesian nonparametric literature. In this paper, we
introduce a flexible class of dependent nonparametric priors, investigate their
properties and derive a suitable sampling scheme which allows their concrete
implementation. The proposed class is obtained by normalizing dependent
completely random measures, where the dependence arises by virtue of a suitable
construction of the Poisson random measures underlying the completely random
measures. We first provide general distributional results for the whole class
of dependent completely random measures and then we specialize them to two
specific priors, which represent the natural candidates for concrete
implementation due to their analytic tractability: the bivariate Dirichlet and
normalized $\sigma$-stable processes. Our analytical results, and in particular
the partially exchangeable partition probability function, form also the basis
for the determination of a Markov Chain Monte Carlo algorithm for drawing
posterior inferences, which reduces to the well-known Blackwell--MacQueen
P\'{o}lya urn scheme in the univariate case. Such an algorithm can be used for
density estimation and for analyzing the clustering structure of the data and
is illustrated through a real two-sample dataset example.
