实用多层马尔可夫链蒙特卡洛的高效子采样率选择

中文摘要 AI 辅助

多层马尔可夫链蒙特卡洛算法通过利用跨模型分辨率层次上离散化感兴趣量之间的相关性来加速贝叶斯反演。一个关键的实际参数是用于为更细层链生成粗层提议的子采样率。现有实践将该比率设置为等于积分自相关时间，作为提议去相关性的代理，但这种选择可能过于保守，并导致大量计算开销。在这项工作中，我们将有限子采样视为一个误差预算问题。我们推导出一个成本误差准则，用于使用样本方差、积分自相关时间和每层成本的估计值来选择计算高效的子采样率。该准则平衡了跨层的多层采样误差，而不是强制执行粗层提议的近似独立性。我们通过一个理想的复位参考核阐明了独立性的理论作用，对于该参考核，独立的粗层提议可以被由可逆粗层核生成的提议所替代。我们推导出有限子采样引起的扰动偏差的泊松方程表示。在状态平均的较低层遗忘假设下，这产生了一个条件几何扰动偏差界；在计算中，相应的投影诊断被用作经验误差预算检查。在一个高度自相关的弹性梁基准上的数值实验表明，与基于IAT的子采样相比，计算成本大幅降低。第二个达西流反问题提供了另一个椭圆基准中定性相似有限子采样行为的支持性证据。

英文摘要

The Multilevel Markov chain Monte Carlo algorithm accelerates Bayesian inversion by exploiting correlations between discretized quantities of interest across a hierarchy of model resolutions. A key practical parameter is the subsampling rate used to generate coarse-level proposals for the finer-level chains. Existing practice sets this rate equal to an integrated autocorrelation time as a proxy for proposal decorrelation, but this choice can be overly conservative and lead to substantial computational overhead. In this work we treat finite subsampling as an error-budget question. We derive a cost-error criterion for selecting a computationally efficient subsampling rate using estimates of sample variances, integrated autocorrelation times, and per-level costs. The criterion balances multilevel sampling errors across levels rather than enforcing approximate independence of the coarse proposals. We clarify the theoretical role of independence through an ideal reset-reference kernel, for which independent coarse proposals may be replaced by proposals generated from a reversible coarse-level kernel. We derive a Poisson-equation representation of the perturbation bias resulting from finite subsampling. Under a state-averaged lower-level forgetting assumption, this yields a conditional geometric perturbation-bias bound; in computation, the corresponding projected diagnostic is used as an empirical error-budget check. Numerical experiments on a highly autocorrelated elasticity beam benchmark show substantial reductions in computational cost compared with IAT-based subsampling. A second Darcy-flow inverse problem gives supporting evidence of qualitatively similar finite-subsampling behaviour in another elliptic benchmark.

发表机构

• KU Leuven（荷语鲁汶大学）

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