  We show that the Hodge numbers of Sasakian manifolds are invariant under
arbitrary deformations of the Sasakian structure. We also present an upper semi
continuity Theorem for the dimensions of kernels of a smooth family of
transversely elliptic operators on manifolds with transversely Riemannian
foliations. We use this to prove that the $\partial\bar{\partial}$-lemma and
being transversely K\"{a}hler are rigid properties under small deformations of
the transversely holomorphic structure which preserve the foliation. Finally,
we study an example which shows that this is not the case for arbitrary
deformations of the transversely holomorphic foliation.
