What is a hierarchical model that can estimated via the Metropolis-Hastings Algorithm but not the Gibbs Sampler?

Question

My understanding of the differences between MH and Gibbs Samplers is that a Gibbs Sampler is usually used when the full conditionals are present to us. In other words, it is a known distribution, so that sampling from it is just as easy as calling a function in R.

I would usually use MH if the conditionals are not well known. However, I am failing to think of an example which cannot be done by Gibbs and must be handled by MH. Is there a prototypical hierarchical model? Thanks.

Answer 1

Take for instance a hierarchical logit model $$\mathbb{P}(Y_{ij}=1|\boldsymbol{\theta},\mathbf{X})=\dfrac{\exp\{\theta_i^\prime x_{ij}\}}{1+\exp\{\theta_i^\prime x_{ij}\}}$$ with a non-standard prior on the $\theta_i$'s, itself parameterised to achieve a hierarchical structure.