用于贝叶斯变量选择的变复杂度加权回火Gibbs采样器
Variable-Complexity Weighted-Tempered Gibbs Samplers for Bayesian Variable Selection
AI总结：
针对高维贝叶斯变量选择中子集加权回火Gibbs采样器方差过高的问题，提出一种变复杂度采样器，在降低每次MCMC迭代计算量的同时减小了Rao-Blackwellized估计量的方差。
摘要：
Jankowiak最近引入了子集加权回火Gibbs采样器（wTGS），以降低高维应用中每次MCMC迭代的计算复杂度，在这些应用中，后验包含概率（PIP）的精确计算并非必需。然而，当信号维度与条件PIP估计数量之比较大时，与该采样器相关的Rao-Blackwellized估计量具有高方差。在本文中，我们设计了一种新的子集加权回火Gibbs采样器（wTGS），其中每次MCMC迭代中条件PIP的预期计算次数可以远小于信号维度。与子集wTGS和wTGS不同，我们的采样器在每次MCMC迭代中具有可变的复杂度。我们为该采样器在有限迭代次数$T$下相关的Rao-Blackwellized估计量提供了方差上限，并证明对于给定数据集，该方差为$O\big(\big(\frac{P}{S}\big)^2 \frac{\log T}{T}\big)$，其中$S$是每次MCMC迭代中条件PIP的预期计算次数。实验表明，我们的Rao-Blackwellized估计量可以具有比与子集wTGS相关的对应估计量更小的方差。
英文摘要：
Subset weighted-Tempered Gibbs Sampler (wTGS) has been recently introduced by Jankowiak to reduce the computation complexity per MCMC iteration in high-dimensional applications where the exact calculation of the posterior inclusion probabilities (PIP) is not essential. However, the Rao-Backwellized estimator associated with this sampler has a high variance as the ratio between the signal dimension and the number of conditional PIP estimations is large. In this paper, we design a new subset weighted-Tempered Gibbs Sampler (wTGS) where the expected number of computations of conditional PIPs per MCMC iteration can be much smaller than the signal dimension. Different from the subset wTGS and wTGS, our sampler has a variable complexity per MCMC iteration. We provide an upper bound on the variance of an associated Rao-Blackwellized estimator for this sampler at a finite number of iterations, $T$, and show that the variance is $O\big(\big(\frac{P}{S}\big)^2 \frac{\log T}{T}\big)$ for a given dataset where $S$ is the expected number of conditional PIP computations per MCMC iteration. Experiments show that our Rao-Blackwellized estimator can have a smaller variance than its counterpart associated with the subset wTGS.
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