  The problem of optimally scaling the proposal distribution in a Markov chain
Monte Carlo algorithm is critical to the quality of the generated samples. Much
work has gone into obtaining such results for various Metropolis-Hastings (MH)
algorithms. Recently, acceptance probabilities other than MH are being employed
in problems with intractable target distributions. There is little resource
available on tuning the Gaussian proposal distributions for this situation. We
obtain optimal scaling results for a general class of acceptance functions,
which includes Barker's and Lazy-MH. In particular, optimal values for the
Barker's algorithm are derived and found to be significantly different from
that obtained for the MH algorithm. Our theoretical conclusions are supported
by numerical simulations indicating that when the optimal proposal variance is
unknown, tuning to the optimal acceptance probability remains an effective
strategy.
