  On any denumerable product of probability spaces, we extend the discrete
Malliavin structure for conditionally independent random variables. As a
consequence, we obtain the chaos decomposition for functionals of conditionally
independent random variables. We also show how to derive some concentration
results in that framework. The Malliavin-Stein method yields Berry-Esseen
bounds for U-Statistics of such random variables. It leads to quantitative
statements of conditional limit theorems: Lyapunov's central limit theorem, De
Jong's limit theorem for multilinear forms. The latter is related to the fourth
moment phenomenon. The final application consists of obtaining the rates of
normal approximation for subhypergraph counts in random exchangeable
hypergraphs including the Erd{\"o}s-R{\'e}nyi hypergraph model. The estimator
of subhypergraph counts is an example of homogeneous sums for which we derive a
new decomposition that extends the Hoeffding decomposition.
