We review and extend results related to optimal scaling of Metropolis–Hastings algorithms. We present various theoretical results for the high-dimensional limit. We also present simulation studies which confirm the theoretical results in finite-dimensional contexts.
Gareth O. Roberts. Jeffrey S. Rosenthal. "Optimal scaling for various Metropolis-Hastings algorithms." Statist. Sci. 16 (4) 351 - 367, November 2001. https://doi.org/10.1214/ss/1015346320