Metropolis-Hastings (MH) algorithm is a MCMC method to compute high-dimensional integrals. It is usually formulated in the continuum version, i.e. the target function to be integrated is known.
The question is, for some real problem, I can only get access to the target function in a lattice but the analytical form is not known. In that case, are there any adaption to MH for this? That basically means random walks are performed in the lattice, not in the continuum space.
I just natively put restrictions on the neighbour function so that the next point could be located in the lattice, i.e. capped with a ceiling function. But I am not sure if the detailed balance condition is violated or not in this naive way.