I am using MCMC (Metropolis-Hastings) to simulate values of $\theta$: I have a Log-likelihood (using 10 inputs $x_i$) $$L=-\frac{n}{2}\ln(2\pi)-\frac{1}{2}\sum_{i=1}^n(x_i-\theta)^2$$ The variation over the outputs of my Likelihood is limited, even though the extreme outputs are in the range of 0 to 1, the range of the value being about 0.2-0.5.
If a Uniform prior is used as proposal, the Metropolis-Hastings acceptance ratio is generally high (0.8 or higher) no matter what the chosen value for the parameter $\theta$ is. This results in (almost) any selected point being accepted regardless of the $\sigma$ value being used for selecting new points. There is little to no guidance being provided by the use of Metropolis Hastings algorithm as all points are accepted.
So when the output of a likelihood shows little variation when changing the parameter value how can you ensure that MCMC will still be able to simulate the posterior distribution?
