\[ \newcommand{\Exg}{\operatorname{\mathbb{E}}} \newcommand{\Ex}{\mathbb{E}} \newcommand{\Ind}{\mathbb{I}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Corr}{\operatorname{Corr}} \newcommand{\ee}{\mathrm{e}} \]

21  Metropolis–Hastings in continuous space

Summary:

• The Metropolis–Hastings algorithm in continuous space works essentially the same as in discrete space: propose a move according to the transition density \(r(x,y)\), and accept it with probability \[\alpha(x,y) = \min \left\{ \frac{\pi(y)\,r(y,x)}{\pi(x)\,r(x,y)} , \, 1\right\} . \]

• When the proposal distribution is that of the symmetric Gaussian random walk, this is the random walk Metropolis algorithm.

• Choosing the step size \(\sigma\) in the Metropolis random walk is important: too small and the Markov chain takes only very small steps; too large and most proposals are rejected.

• A burn-in period can be used to find a good initial point for MCMC.

Read more: Voss, An Introduction to Statistical Computing, Subsection ?.?.?.