| Abstract: | In this paper, I explore the usage of positive definite metric tensors derived from the second derivative information in the context of the simplified manifold Metropolis adjusted Langevin algorithm. I propose a new adaptive step size procedure that resolves the shortcomings of such metric tensors in regions where the log‐target has near zero curvature in some direction. The adaptive step size selection also appears to alleviate the need for different tuning parameters in transient and stationary regimes that is typical of Metropolis adjusted Langevin algorithm. The combination of metric tensors derived from the second derivative information and the adaptive step size selection constitute a large step towards developing reliable manifold Markov chain Monte Carlo 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, I show that the proposed methodology performs very well relative to alternative Markov chain Monte Carlo methods. |
