Can unimodal prior and unimodal sampling distributions lead to a multimodal posterior distribution?
The Bayes rule tells us that $$ f(y|x) = f(x|y) f(y) / f(x). $$ which, I think, implies a unimodal prior and unimodal sampling distributions can only lead to a unimodal posterior distribution.
If $y$ is multidimensional, can the marginal posterior distribution of some component of $y$ be multimodal? (Related question: can a unimodal multivariate distribution has some multimodal marginal distribution? If all marginal distributions are unimodal, can the multivariate distribution be multimodal?)