Algorithm 1 Hamiltonian Monte Carlo (HMC)

Require: initial states ${\mathit{\xi }}^0$, time step size $\delta t$, leapfrog steps I, total running steps J     for  $j=0,1,\dots ,(J-1)$ do         Sample $r^j$ from $\mathcal{N}(0,\mathit{M})$         ${\mathit{\xi }}_0\leftarrow {\mathit{\xi }}^j$         $r_0\leftarrow r^j$         for $i=0,1,\dots ,(I-1)$ do             ${\mathit{r}}_i\leftarrow {\mathit{r}}_i-{\displaystyle \frac{\delta t}{2}}\nabla V\left({\mathit{\xi }}_i\right)$             ${\mathit{\xi }}_{i+1}\leftarrow {\mathit{\xi }}_i+\delta t{M}^{-1}{\mathit{r}}_i$             ${\mathit{r}}_{i+1}\leftarrow {\mathit{r}}_i-{\displaystyle \frac{\delta t}{2}}\nabla V\left({\mathit{\xi }}_{i+1}\right)$         end for         Sample $\mathit{p}$ from $\mathcal{U}(0,1)$         $\alpha \leftarrow min\left[1,exp\left(H\left({\mathit{\xi }}_I,{\mathit{r}}_I\right)-H\left({\mathit{\xi }}^j,{\mathit{r}}^j\right)\right)\right]$         if $p\ge \alpha$ then             ${\mathit{\xi }}^{j+1}\leftarrow {\mathit{\xi }}_I$         else             ${\mathit{\xi }}^{j+1}\leftarrow {\mathit{\xi }}^j$         end if     end for     Return:  ${\mathit{\xi }}^1,\dots ,{\mathit{\xi }}^J$