  Connecting optimal transport and variational inference, we present a
principled and systematic framework for sampling and generative modelling
centred around divergences on path space. Our work culminates in the
development of the \emph{Controlled Monte Carlo Diffusion} sampler (CMCD) for
Bayesian computation, a score-based annealing technique that crucially adapts
both forward and backward dynamics in a diffusion model. On the way, we clarify
the relationship between the EM-algorithm and iterative proportional fitting
(IPF) for Schr{\"o}dinger bridges, deriving as well a regularised objective
that bypasses the iterative bottleneck of standard IPF-updates. Finally, we
show that CMCD has a strong foundation in the Jarzinsky and Crooks identities
from statistical physics, and that it convincingly outperforms competing
approaches across a wide array of experiments.
