  We consider the problem of sampling a high dimensional multimodal target
probability measure. We assume that a good proposal kernel to move only a
subset of the degrees of freedoms (also known as collective variables) is known
a priori. This proposal kernel can for example be built using normalizing
flows. We show how to extend the move from the collective variable space to the
full space and how to implement an accept-reject step in order to get a
reversible chain with respect to a target probability measure. The
accept-reject step does not require to know the marginal of the original
measure in the collective variable (namely to know the free energy). The
obtained algorithm admits several variants, some of them being very close to
methods which have been proposed previously in the literature. We show how the
obtained acceptance ratio can be expressed in terms of the work which appears
in the Jarzynski-Crooks equality, at least for some variants. Numerical
illustrations demonstrate the efficiency of the approach on various simple test
cases, and allow us to compare the variants of the algorithm.
