  In this paper we propose Discretely Indexed flows (DIF) as a new tool for
solving variational estimation problems. Roughly speaking, DIF are built as an
extension of Normalizing Flows (NF), in which the deterministic transport
becomes stochastic, and more precisely discretely indexed. Due to the discrete
nature of the underlying additional latent variable, DIF inherit the good
computational behavior of NF: they benefit from both a tractable density as
well as a straightforward sampling scheme, and can thus be used for the dual
problems of Variational Inference (VI) and of Variational density estimation
(VDE). On the other hand, DIF can also be understood as an extension of mixture
density models, in which the constant mixture weights are replaced by flexible
functions. As a consequence, DIF are better suited for capturing distributions
with discontinuities, sharp edges and fine details, which is a main advantage
of this construction. Finally we propose a methodology for constructiong DIF in
practice, and see that DIF can be sequentially cascaded, and cascaded with NF.
