Prior and Posterior Domain Essentially the posterior distribution $p(\theta|x)$ and the prior distribution $p(\theta)$ are defined on the same domain $\Theta$, it is just the allocate their mass differently?

Let's say that you are estimating binomial proportion and you have prior knowledge that it is certainly higher than 0.5 but you have no other knowledge on its value. In such a case, you could use a uniform prior between 0.5 and 1. Multiplying likelihood by it would restrict the posterior probabilities to be non-zero only for the range as restricted by the prior. So technically the binomial proportion could range over the whole unit interval, but the choice of prior restricted the posterior. Posterior, by definition, would be non-zero everywhere where prior times likelihood is non-zero.
