Metropolis Sampler

The original Metropolis algorithm (1953) is different from the Metropolis-Hasting algorithm (1970), where the proposal distribution is symmetric (i.e. normal distribution). Thus,
\[q(Y|X)=q(X|Y)\]
and yield
\[\alpha(X_t,Y)=min \left\{1,\frac{\pi(Y)}{\pi(X_t)}\right\}\]
when the proposal distribution is such that $q(X|Y)=q(Y|X)=q(|X-Y|)$, then it is also called "random walk Metropolis"