  A notion of conditionally identically distributed (c.i.d.) sequences has been
studied as a form of stochastic dependence that is weaker than exchangeability,
but is equivalent to exchangeability for stationary sequences. In this article
we extend this notion to families of sequences. Paralleling the extension from
exchangeability to partial exchangeability in the sense of de Finetti, we
propose a notion of partially c.i.d. dependence, that is equivalent to partial
exchangeability for stationary processes. Partially c.i.d. families of
sequences preserve attractive limit properties of partial exchangeability, and
are asymptotically partially exchangeable. Moreover, we provide strong laws of
large numbers and two central limit theorems. Our focus is on the asymptotic
agreement of predictions and empirical means, which lies in the foundations of
Bayesian statistics. Natural examples are interacting randomly reinforced
processes satisfying certain conditions on the reinforcement.
