陡势下随机游走Metropolis算法的鲁棒性
Robustness of random-walk Metropolis for steep potentials
中文总结 AI 辅助
本研究针对轻尾目标分布的采样问题，分析无梯度的随机游走Metropolis算法的鲁棒性，证明其接受概率稳定并可推导混合时间估计。
中文摘要 AI 辅助
在马尔可夫链蒙特卡罗采样中，轻尾目标分布是一把双刃剑：其轻尾特性能提供良好的约束，通常对自然连续时间动力学也意味着良好的混合性质，但尾部衰减的陡峭程度使其常超出现代定量收敛理论的适用范围。对于常规基于梯度的采样器，这体现为真正的不稳定性问题，即Metropolis接受率会严重下降。本研究探讨无梯度的随机游走Metropolis采样器，证明对于广泛的轻尾目标分布，在选择合理的提议方差时，接受概率保持稳定，由此可推导出有效且有利的混合时间估计。该分析依赖于目标分布对数密度的一阶与二阶导数之间的简单关系。
英文摘要
In Markov chain Monte Carlo sampling, light-tailed target distributions present something of a poisoned chalice: their light tails offer good confinement, and tend to imply good mixing properties for natural continuous-time dynamics, but the steepness of their tail decay means that they often fall outside of the scope of modern quantitative convergence theory. For usual gradient-based samplers, this reflects a genuine instability issue, whereby Metropolis acceptance rates can degrade badly. In this work, we study the gradient-free random-walk Metropolis sampler, and show that for a wide range of light-tailed targets, the acceptance probability remains stable for reasonable choices of proposal variance, from which effective and favourable mixing time estimates can be deduced. The analysis relies on a simple relationship between the first and second derivatives of the log-density of the target distribution.
