Efficiency of Markov chain Monte Carlo methods

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Random walk Metropolis (RWM) algorithms, an important class of Markov chain Monte Carlo (MCMC) algorithms, are methods to generate data from highly complex probability distributions (the target distribution). Researchers and practitioners in various fields of applications such as biostatistics, computer science, physics, finance, and applied statistics extensively use such methods. In applying RWM algorithms, it is necessary to choose a proposal density; due to its accessibility, the normal density is the most popular choice. The variance of this distribution must be selected carefully in order to have some level of optimality in the performance of the algorithm; this is called the optimal scaling problem. Results about this issue are available in the literature for high-dimensional target distributions with independent components.