  Generative models on discrete state-spaces have a wide range of potential
applications, particularly in the domain of natural sciences. In continuous
state-spaces, controllable and flexible generation of samples with desired
properties has been realized using guidance on diffusion and flow models.
However, these guidance approaches are not readily amenable to discrete
state-space models. Consequently, we introduce a general and principled method
for applying guidance on such models. Our method depends on leveraging
continuous-time Markov processes on discrete state-spaces, which unlocks
computational tractability for sampling from a desired guided distribution. We
demonstrate the utility of our approach, Discrete Guidance, on a range of
applications including guided generation of images, small-molecules, DNA
sequences and protein sequences.
