A Robbins-Munro stochastic approximation update is used to
adapt the tuning parameter of the proposal kernel. The idea
is to update the tuning parameter at each iteration of the
sampler: $$h^{(i+1)} = h^{(i)} +
\eta^{(i+1)}(\alpha^{(i)} - \alpha_{opt}),$$ where
$h^{(i)}$ and $\alpha^{(i)}$
are the tuning parameter and acceptance probability at
iteration $i$ and $\alpha_{opt}$
is a target acceptance probability. For Gaussian targets,
and in the limit as the dimension of the problem tends to
infinity, an appropriate target acceptance probability for
MALA algorithms is 0.574. The sequence
${\eta^{(i)}}$ is chosen so that
$\sum_{i=0}^\infty\eta^{(i)}$
is infinite whilst
$\sum_{i=0}^\infty\left(\eta^{(i)}\right)^{1+\epsilon}$
is finite for $\epsilon>0$. These two
conditions ensure that any value of $h$ can be
reached, but in a way that maintains the ergodic behaviour
of the chain. One class of sequences with this property is,
$$\eta^{(i)} = \frac{C}{i^\alpha},$$ where
$\alpha\in(0,1]$ and $C>0$.The
scheme is set via the mcmcpars function.