We develop Microcanonical Hamiltonian Monte Carlo (MCHMC), a class of models which follow a fixed energy Hamiltonian dynamics, in contrast to Hamiltonian Monte Carlo (HMC), which follows canonical distribution with different energy levels. MCHMC tunes the Hamiltonian function such that the marginal of the uniform distribution on the constant-energy-surface over the momentum variables gives the desired target distribution. We show that MCHMC requires occasional energy conserving billiard-like momentum bounces for ergodicity, analogous to momentum resampling in HMC. We generalize the concept of bounces to a continuous version with partial direction preserving bounces at every step, which gives an energy conserving underdamped Langevin-like dynamics with non-Gaussian noise (MCLMC). MCHMC and MCLMC exhibit favorable scalings with condition number and dimensionality. We develop an efficient hyperparameter tuning scheme that achieves high performance and consistently outperforms NUTS HMC on several standard benchmark problems, in some cases by more than an order of magnitude.

翻译：我们提出了微正则哈密顿蒙特卡洛（MCHMC），这是一类遵循固定能量哈密顿动力学的模型，与遵循不同能量水平正则分布的哈密顿蒙特卡洛（HMC）形成对比。MCHMC通过调节哈密顿函数，使得动量变量在等能面上的均匀分布的边际分布与目标分布一致。我们证明，MCHMC需要偶尔进行能量守恒的台球式动量反弹来保证遍历性，这类似于HMC中的动量重采样。我们将反弹概念推广到连续版本，即在每一步中进行部分方向保持的反弹，从而得到一种具有非高斯噪声的能量守恒欠阻尼朗之万型动力学（MCLMC）。MCHMC和MCLMC在条件数和维度上展现出良好的缩放性能。我们开发了一种高效的超参数调优方案，该方案实现了高性能，并在多个标准基准问题上始终优于NUTS HMC，在某些情况下性能提升超过一个数量级。