Tuning the jump distribution

I mentioned in class that the optimal acceptance rate for MCMC jump proposals is 30-40% based on what others had recommended to me when teaching me the Way of the Markov Chain. It turns out, however, that for higher-order problems the optimal rate is 0.234 (Corolary 1.2 of Roberts, Gelman and Gilks 1997). If you want to know exactly why this is so, you should read the full paper. It's full of lemma-ny goodness! According to the abstract:

The limiting diffusion approximation admits a straightforward effi-
ciency maximization problem, and the resulting asymptotically optimal
policy is related to the asymptotic acceptance rate of proposed moves for
the algorithm. The asymptotically optimal acceptance rate is 0.234 under
quite general conditions.

No one breaks it down for the layperson like a statistician!

Supposedly ~50% is optimal for problems with 1 or 2 parameters (Section 4.2 of Gregory 2005). But if you have only 1 or 2 parameters, I don't know why you'd want to do MCMC, unless there is a nonanalytic likelihood to evaluate through a simulation or something. Gregory cites Roberts et al. above, but I didn't see anything about 50% for simpler problems. If you decide to read Roberts et al., let me know if I missed the note about 50%.

That said, 30-40% has treated me well over the years. Let's say you should aim for 25-30% acceptance of your jump proposals for the full line profile fit (Problem 3 of CA6).