Exponential ergodicity of the bouncy particle sampler
弹跳粒子采样器的指数遍历性AI 解读 ↓
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弹跳粒子采样器的指数遍历性
- 研究问题
- 对于基于分段确定性过程的非可逆弹跳粒子采样器，哪些目标分布条件可以保证几何遍历性与遍历平均的中心极限定理？
- 方法与思路
- 作者给出涉及曲率及负对数密度增长的可核验条件，并分别通过修改采样机制和变换目标分布处理更薄或更厚的尾部。
- 材料陈述的贡献
- 摘要称，论文建立弹跳粒子采样器的几何遍历性和相关中心极限定理，并为不同尾部情形构造具有几何遍历性的方案。
- 阅读关注点
- 应核对曲率与尾部条件的精确定义、采样机制参数，以及目标变换后的过程如何映回原参数空间并保留所需性质。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；摘要仅概述条件，未给出收敛常数或计算比较，遍历性不能直接说明实际采样效率。
原文摘要
Nonreversible Markov chain Monte Carlo schemes based on piecewise deterministic Markov processes have been recently introduced in applied probability, automatic control, physics and statistics. Although these algorithms demonstrate experimentally good performance and are accordingly increasingly used in a wide range of applications, geometric ergodicity results for such schemes have only been established so far under very restrictive assumptions. We give here verifiable conditions on the target distribution under which the Bouncy Particle Sampler algorithm introduced in [Phys. Rev. E 85 (2012) 026703, 1671–1691] is geometrically ergodic and we provide a central limit theorem for the associated ergodic averages. This holds essentially whenever the target satisfies a curvature condition and the growth of the negative logarithm of the target is at least linear and at most quadratic. For target distributions with thinner tails, we propose an original modification of this scheme that is geometrically ergodic. For targets with thicker tails, we extend the idea pioneered in [Ann. Statist. 40 (2012) 3050–3076] in a random walk Metropolis context. We establish geometric ergodicity of the Bouncy Particle Sampler with respect to an appropriate transformation of the target. Mapping the resulting process back to the original parameterization, we obtain a geometrically ergodic piecewise deterministic Markov process.