A Scaled Stochastic Newton Algorithm for Markov Chain Monte Carlo Simulations

We propose a scaled stochastic Newton algorithm (sSN) for local Metropolis-Hastings Markov chain Monte Carlo (MCMC) simulations. The method can be considered as an EulerMaruyama discretization of the Langevin diffusion on a Riemann manifold with piecewise constant Hessian of the negative logarithm of the target density as the metric tensor. The sSN proposal consists of deterministic and stochastic parts. The former corresponds to a Newton step that attempts to move the current state to a region of higher probability, hence potentially increasing the acceptance probability. The latter is distributed by a Gaussian tailored to the local Hessian as the inverse covariance matrix. The proposal step is then corrected by the standard Metropolization to guarantee that the target density is the stationary distribution. We study asymptotic convergence and geometric ergodicity of sSN chains. At the heart of the paper is the optimal scaling analysis, in which we show that, for inhomogeneous product target distribution at stationarity, the sSN proposal variance scales like O ( n−1/3 ) for the average acceptance rate to be bounded away from zero, as the dimension n approaches infinity. As a result, a sSN chain explores the stationary distribution in O ( n1/3 ) steps, regardless of the variance of the target density. The optimal scaling behavior of sSN chains in the transient phase is also discussed for Gaussian target densities, and an extension to inverse problems using the Bayesian formulation is presented. The theoretical optimal scaling result is verified for two i.i.d. targets in high dimensions. We also compare the sSN approach with other similar Hessianaware methods on i.i.d. targets, Bayesian logistic regression, and log-Gaussian Cox process examples. Numerical results show that sSN outperforms the others by providing Markov chains with small burnin and small correlation length. Finally, we apply the sSN method to a Bayesian inverse thermal fin problem to predict the posterior mean and its uncertainty.