## Doudou Tang (Durham University): Hamiltonian Monte Carlo with Local Stochastic Step-Size

Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo (MCMC) algorithm that tries to avoid the slow exploration of the state space caused by random walk behaviour and correlated parameters by taking several steps according to gradient information of target distribution. This distinguishing feature makes HMC converge quicker than traditional MCMC algorithms like Metropolis random walk. HMC’s performance, however, is very sensitive to one parameter, the step-size $\varepsilon$ in approximating Hamiltonian dynamics, which the user must specify. In particular, too large a step-size leads to a low acceptance rate and a chain which gets stuck while too small a step-size causes random walk behaviour and slow exploration. We find that the upper boundary of a suitable step-size is related to the gradient information and curvature of target distributions and thus such a boundary might change across the state space. Therefore, we propose a device which uses stochastic step-size for HMC. This allows the step-size to take small values to get out of sticky points and large values to avoid small MCMC steps. The proposed method exploits the geometric structure of the log-posterior for a statistical model to generate step-size and thus step-size automatically adapts to the local structure at each MCMC iteration according to the parameter value.

Keywords: MCMC, Hamiltonian Monte Carlo, stochastic step-size