  Discrete Bayesian nonparametric models whose expectation is a convex linear
combination of a point mass at some point of the support and a diffuse
probability distribution allow to incorporate strong prior information, while
still being extremely flexible. Recent contributions in the statistical
literature have successfully implemented such a modelling strategy in a variety
of applications, including density estimation, nonparametric regression and
model-based clustering. We provide a thorough study of a large class of
nonparametric models we call inner spike and slab hNRMI models, which are
obtained by considering homogeneous normalized random measures with independent
increments (hNRMI) with base measure given by a convex linear combination of a
point mass and a diffuse probability distribution. In this paper we investigate
the distributional properties of these models and our results include: i) the
exchangeable partition probability function they induce, ii) the distribution
of the number of distinct values in an exchangeable sample, iii) the posterior
predictive distribution, and iv) the distribution of the number of elements
that coincide with the only point of the support with positive probability. Our
findings are the main building block for an actual implementation of Bayesian
inner spike and slab hNRMI models by means of a generalized P\'olya urn scheme.
