If the variations of the target $\pi(\theta|x)\propto f(x|\theta)$ are limited,
- it means that the data is little informative about $\theta$; this is a poor situation from an inferential viewpoint.
- it means that an MCMC algorithm like the Metropolis-Hastings algorithm will converge very fast since most proposals (from the Uniform prior) will be accepted; this is an ideal situation from a simulation viewpoint.
However, I do not understand the part about $\sigma$, as this symbol does not appear in the likelihood. Is this the scale of the Uniform proposal? In which case this is no longer the prior and no longer a valid proposal.
