I'm currently working on a project where my objective is to sample from a target distribution $p(x)$ using Markov Chain Monte Carlo (MCMC) techniques. The primary tool at my disposal is an unconditional proposal distribution $q(x)$ that is easy to draw independent samples from. The standard approach I'm considering involves drawing sample $x'$ from $q(x)$ and then accepting or rejecting this sample based on the acceptance ratio:
$\text{acceptance ratio} = \min\left(1, \frac{p(x')q(x)}{p(x)q(x')}\right)$
However, a challenge arises with this method: the sample $x'$ drawn from $q(x)$ might end up being quite far away from the current sample $x$, which could potentially lead to very low acceptance rate.
Question: Is there a technique or modification to the proposal mechanism that would allow me to propose new sample $x'$ that are close to the current sample $x$, while still maintaining the form of the acceptance ratio mentioned above?
Thank you in advance for your insights!