  The usage of positive definite metric tensors derived from second derivative
information in the context of the simplified manifold Metropolis adjusted
Langevin algorithm (MALA) is explored. A new adaptive step length procedure
that resolves the shortcomings of such metric tensors in regions where the
log-target has near zero curvature in some direction is proposed. The adaptive
step length selection also appears to alleviate the need for different tuning
parameters in transient and stationary regimes that is typical of MALA. The
combination of metric tensors derived from second derivative information and
adaptive step length selection constitute a large step towards developing
reliable manifold MCMC methods that can be implemented automatically for models
with unknown or intractable Fisher information, and even for target
distributions that do not admit factorization into prior and likelihood.
Through examples of low to moderate dimension, it is shown that proposed
methodology performs very well relative to alternative MCMC methods.
