Random-walk Metropolis algorithm

Random-walk Metropolis proposes a symmetric increment from the current state and accepts a proposal $y$ from $x$ with probability $\min \{1,\pi (y)/\pi (x)\}$.

To update a positive parameter, a symmetric additive proposal distribution can be generated on the whole real line. Reject nonpositive proposals; accept positive ones using the target density ratio. The off-diagonal proposal remains symmetric. Repeatedly redrawing until the proposal is positive instead truncates and renormalizes the proposal differently at different current states, so a Hastings correction is then required. A symmetric walk in $\log v$ is another valid method, but its transformed target includes the Jacobian determinant factor $v$.

A small proposal scale increases acceptance while making successive states almost equal. A large scale often proposes low-density states and creates long rejection runs. A pilot comparison can optimize effective sample size of a Markov chain per unit computing time, or expected accepted squared displacement, rather than acceptance alone. Tune on representative functions and then fix the scale for production; an arbitrarily history-dependent adaptation is not automatically a chain with the desired invariant distribution.