I'd like to implement a version of Metropolis-adjusted Langevin sampling, but I'm unsure how to go about tuning the parameters of the proposal density.
My understanding is that in MALA, a proposal is generated as: $$ X^*=X+\tau\nabla \log\pi(X)+\sqrt{2\tau}\xi $$ where $\pi(X)$ is the target density and $\xi\sim Normal\left(0, \Sigma\right)$.
Now there seem to be two things to be "tuned" about this proposal density. The first is the step-size $\tau$, which it seems to me would mostly control the influence of the gradient (i.e. how much you drift in the direction of the derivatives). How do I determine a good value for this step-size (and can this be done independently of the other term in the proposal)?
The second is the covariance $\Sigma$ of the "noise" around the "Langevin drift". I assume this can typically be diagonal (since the gradient already gives the desired directionality to the proposals, but please correct me on this), but I'm trying to figure out how to set its general scale, and also whether it's necessary to tune the variances for different parameters (i.e. elements along the diagonal) separately? That is, if my target density is relatively narrow in certain dimensions, do I need to adjust for that, or is that also somehow handled by the inclusion of the gradient term?
So far I haven't been able to find any practical references on how to tune these parameters (mostly papers on more complicated variants on MALA and the like), so I'd be grateful for any pointers or advice!
