Step size control for Newton type MCMC samplers Jonathan Goodman

ABSTRACT: MCMC sampling can use ideas from the optimization community.  Optimization via Newton’s method can fail without line search, even for smooth strictly convex problems.  Affine invariant Newton based MCMC sampling uses a Gaussian proposal based on a quadratic model of the potential using the local gradient and Hessian.  This can fail (conjecture: give a transient Markov chain) even for smooth strictly convex potentials.  We describe a criterion that allows a sequence of proposal distributions from X_n with decreasing “step sizes” until (with probability 1) a proposal is accepted.  “Very detailed balance” allows the whole process to preserve the target distribution.  The method works in experiments but the theory is missing.