In Hamiltonian Monte Carlo sampling, the shape of the potential and the choice of the momentum distribution jointly give rise to the Hamiltonian dynamics of the sampler. An efficient sampler propagates quickly in all regions of the parameter space, so that the chain has a low autocorrelation length and the sampler has a high acceptance rate, with the goal of optimising the number of near-independent samples for given computational cost. Standard Gaussian momentum distributions allow arbitrarily large velocities, which can lead to inefficient exploration in posteriors with ridges or funnel-like geometries. We investigate alternative momentum distributions based on relativistic and Student's t kinetic energies, which naturally limit particle velocities and may improve robustness. Using Almanac, a sampler for cosmological posterior distributions of sky maps and power spectra on the sphere, we test these alternatives in both low- and high-dimensional settings. We find that the choice of parameterization and momentum distribution can improve convergence and effective sample rate, though the achievable gains are generally modest and strongly problem-dependent, reaching up to an order of magnitude in favorable cases. Among the momentum distributions that we tested, those with moderately heavy tails achieved the best balance between efficiency and stability. These results highlight the importance of sampler design and encourage future work on adaptive and self-tuning strategies for kinetic energy parameter optimization in high-dimensional settings.

翻译：在哈密顿蒙特卡洛采样中，势函数的形态与动量分布的选取共同决定了采样器的哈密顿动力学。一个高效的采样器能在参数空间的所有区域快速传播，从而使马尔可夫链具有较低的自相关长度，且采样器具有较高的接受率，其目标是在给定计算成本下优化近独立样本的数量。标准的高斯动量分布允许任意大的速度，这可能导致在具有脊状或漏斗状几何结构的后验分布中进行低效探索。我们研究了基于相对论性和学生t分布动能的替代动量分布，这些分布天然限制粒子速度并可能提升鲁棒性。使用Almanac——一个针对球面上天图与功率谱宇宙学后验分布的采样器，我们在低维与高维场景中测试了这些替代方案。我们发现参数化与动量分布的选择能够改善收敛性与有效采样率，尽管可实现的增益通常较为有限且高度依赖于具体问题，在有利情况下可达一个数量级。在我们测试的动量分布中，具有适度重尾的分布在效率与稳定性之间取得了最佳平衡。这些结果凸显了采样器设计的重要性，并鼓励未来在高维场景中针对动能参数优化的自适应与自调谐策略开展进一步研究。