I wish to do inference in a joint space $P(x,y)$ that is particularly tricky, because I don't have access to an unnormalised density $f(x,y)\propto P(x,y)$. However, I do have:
An unnormalised marginal density for $x$ $$f(x) \propto P(x)$$
and an unnormalised conditional density for $y$ $$g(y;x) \propto P(y|x)$$
Is there any way for me to set up Metropolis Hastings in this space, so that the Markov chain converges in distribution to $P(x,y)$?