I'm very confused with the use of MCMC to estimate distributions that have a complex shape, like multiple peaks, or that aren't generated from a known distribution. In particular, when calculating the acceptance probabilities (e.g. Metropolis acceptance ratio) how would you evaluate the likelihood of your proposal, if you don't have a parametric form to evaluate the likelihood?
I'm confused because in many examples of MCMC, the distribution to be estimated looks complicated, so I'm not sure how the likelihood of proposals is evaluated if you aren't assuming a particular generative distribution.. like, if you don't know if your distribution is unimodal or multimodal. do you just estimate a gaussian with the mean as your proposal? but what is the standard deviation?