  Diffusion models can be parameterised in terms of either a score or an energy
function. An energy parameterisation is appealing since it enables an extended
sampling procedure with a Metropolis--Hastings (MH) correction step, based on
the change in total energy in the proposed samples. Improved sampling is
important for model compositions, where off-the-shelf models are combined with
each other, in order to sample from new distributions. For model composition,
score-based diffusions have the advantages that they are popular and that many
pre-trained models are readily available. However, this parameterisation does
not, in general, define an energy, and the MH acceptance probability is
therefore unavailable, and generally ill-defined. We propose keeping the score
parameterisation and computing an acceptance probability inspired by
energy-based models through line integration of the score function. This allows
us to reuse existing diffusion models and still combine the reverse process
with various Markov-Chain Monte Carlo (MCMC) methods. We evaluate our method
using numerical experiments and find that score-parameterised versions of the
MCMC samplers can achieve similar improvements to the corresponding energy
parameterisation.
