- Can Metropolis sampling be used in conjunction with Gibbs sampling? So for example, if I have three parameters of interest, but only two of them have full conditionals that are known distributions, can I sample using Gibbs for those two parameters and Metropolis using the other parameter for each iteration?
- If the parameter of interest only takes positive values, would it be wrong (in terms of producing an answer, not efficiency) to use a normal proposal distribution centered on the previous iteration's value of the parameter?
Yes. In fact, you can consider a Gibbs sampler to be a special case Metropolis Hasting sampler.
If the parameter of interest is strictly positive, using a normal proposal will still work (any negative proposals will be automatically rejected). However, there is a very good chance that using the normal distribution for the proposed log of the parameter will actually be more efficient, as the posterior of the log-parameter is often approximately normal.