Optimal design of the Barker proposal and other locally balanced Metropolis-Hastings algorithms
Optimal design of the Barker proposal and other locally balanced Metropolis-Hastings algorithms
复制标题
Barker提案和其他局部平衡Metropolis-Hastings算法的优化设计
DOI：
10.1093/biomet/asac056
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发表时间：
2023
期刊：
影响因子：
2.7
通讯作者：
Vogrinc J
中科院分区：
文献类型：
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作者：
Vogrinc J
我们研究了Livingstone & Zanella（2022）中引入的一类一阶局部平衡Metropolis-Hastings算法。要在类中选择特定的算法，用户必须为建议增量选择一个满足的平衡函数和一个噪声分布。类内的流行选择是大都会调整的Langevin算法和最近推出的巴克建议。我们首先建立了一个一般限制的最佳接受率为57和缩放，作为dimensiontends到无穷大类的所有成员之间的温和的光滑性假设上，当目标分布的算法是产品的形式。特别是，我们得到了一个明确的表达式，在类中的任意算法的渐近效率，预期平方跳跃距离测量。然后，我们考虑如何在各种约束条件下优化这个表达式。我们推导出一个最佳的选择噪声分布的巴克建议，一个最佳的选择平衡功能下的高斯噪声分布，和一个最佳的选择一阶局部平衡算法之间的整个类，这原来取决于特定的目标分布。数值模拟证实了我们的理论研究结果，特别是，表明在巴克建议的噪声分布的双峰选择产生了一个实用的算法，始终比原来的高斯版本更有效。
We study the class of first-order locally balanced Metropolis–Hastings algorithms introduced in Livingstone & Zanella (2022). To choose a specific algorithm within the class, the user must select a balancing functionsatisfyingand a noise distribution for the proposal increment. Popular choices within the class are the Metropolis-adjusted Langevin algorithm and the recently introduced Barker proposal. We first establish a general limiting optimal acceptance rate of 57and scaling of, as the dimensiontends to infinity among all members of the class under mild smoothness assumptions onand when the target distribution for the algorithm is of product form. In particular, we obtain an explicit expression for the asymptotic efficiency of an arbitrary algorithm in the class, as measured by expected squared jumping distance. We then consider how to optimize this expression under various constraints. We derive an optimal choice of noise distribution for the Barker proposal, an optimal choice of balancing function under a Gaussian noise distribution, and an optimal choice of first-order locally balanced algorithm among the entire class, which turns out to depend on the specific target distribution. Numerical simulations confirm our theoretical findings, and in particular, show that a bimodal choice of noise distribution in the Barker proposal gives rise to a practical algorithm that is consistently more efficient than the original Gaussian version.