I have a target distribution $\mu$ which I would like to investigate using, for instance Metropolis-Hastings-Green (MHG). So, given a Gaussian prior, $\pi$, and a likelihood $L$ such that $\mu(dx) \propto L(x) \pi(dx)$, I can proceed with the standard MHG with random walk proposals from the prior. But in the problems I'm interested in, I know I have multimodality in $\mu$, so I would like to use a multimodal proposal distribution, such as a mixture of two Gaussians, $\pi_1$ and $\pi_2$. Assuming I have likelihoods for each of these, $L_1$ and $L_2$, and I have chosen mixture weights, $w_1 + w_2 =1$, I can certainly sample from $w_1 \pi_1 + w_2\pi_2$. My question is that now I want to run MHG on top of this, and it's unclear to me how to set up the Metropolis ratio for accepting/rejecting the proposal. I've looked a bit in the literature, but haven't seen anything.