Which sampling methods (MCMC or otherwise) can be used if the posterior distribution is unknown?

Question

The goal is to sample the posterior distribution of parameters describing some model (fairly low dimensional, generally no more than 10 parameters at the absolute most, usually around 5), but I don't necessarily want to assume the shape of the posterior/proposal, in case there is some complicated, nonlinear correlation between model parameters, for example. Which methods can I use for this task?

I haven't used Gibbs sampling in practice, only adaptive Metropolis-Hastings, but from the literature it seems that an advantage of Gibbs is that you don't need to supply a posterior/proposal. Are there other "non-parametric" methods that could prove useful?

Answer 1

"I don't necessarily want to assume the shape of the posterior/proposal"

1. If the posterior distribution is not completely defined, it is obviously impossible to sample from this distribution. Hence MCMC does not apply.
2. If "proposal" is understood to be the distribution from which new values are proposed in a Metropolis-Hastings algorithm and hence differs from the posterior distribution, there is no constraint in its choice, which means it can be chosen to be available in closed form. If instead an intractable proposal is chosen in a Metropolis-Hastings scheme, assuming it is symmetric, $q(x,y)=q(y,x)$, it does not need to be computed in the Metropolis-Hastings acceptance probability.

"...an advantage of Gibbs is that you don't need to supply a posterior/proposal"

The Gibbs sampler is associated with a single and well-defined posterior distribution, which makes the statement incorrect. The Hammersley-Clifford theorem furthermore demonstrates that the joint posterior distribution can be analytically reconstructed from the posterior conditionals that are used in the Gibbs sampler.