# Running two MCMC chains in parallel while minimizing Kullback-Leibler divergence between both sample distributions

I want to sample from a distribution $$p(X)$$ with $$X \in R^n$$. However, I can only evaluate the likelihoods of $$Z = AX$$ and $$Z = BX$$ with $$A,B \in R^{m \times n}$$ and $$m = n-1$$.

Now my idea is to run two Markov Chain Monte Carlo samplers in parallel while minimizing the Kullback-Leibler divergence of the sample-approximated distributions of both chains.

I was thinking of some kind of autoregressive proposal distribution that becomes tighter when the approximated distribution of a chain is wider than that of the other chain (and vice versa).

Is this feasible? Does anyone know an existing algorithm that does something similar? I would be very happy about any expert opinion on that.