Computation of normalizing constants is a fundamental mathematical problem in various disciplines, particularly in Bayesian model selection problems. A sampling-based technique known as bridge sampling (Meng and Wong in Stat Sin 6(4):831–860, 1996) has been found to produce accurate estimates of normalizing constants and is shown to possess good asymptotic properties. For small to moderate sample sizes (as in situations with limited computational resources), we demonstrate that the (optimal) bridge sampler produces biased estimates. Specifically, when one density (we denote as \(p_2\)) is constructed to be close to the target density (we denote as \(p_1\)) using method of moments, our simulation-based results indicate that the correlation-induced bias through the moment-matching procedure is non-negligible. More crucially, the bias amplifies as the dimensionality of the problem increases. Thus, a series of theoretical as well as empirical investigations is carried out to identify the nature and origin of the bias. We then examine the effect of sample size allocation on the accuracy of bridge sampling estimates and discovered that one possibility of reducing both the bias and standard error with a small increase in computational effort is by drawing extra samples from the moment-matched density \(p_2\) (which we assume easy to sample from), provided that the evaluation of \(p_1\) is not too expensive. We proceed to show how the simple adaptive approach we termed “splitting” manages to alleviate the correlation-induced bias at the expense of a higher standard error, irrespective of the dimensionality involved. We also slightly modified the strategy suggested by Wang et al. (Warp bridge sampling: the next generation, Preprint, 2019. arXiv:1609.07690) to address the issue of the increase in standard error due to splitting, which is later generalized to further improve the efficiency. We conclude the paper by offering our insights of the application of a combination of these adaptive methods to improve the accuracy of bridge sampling estimates in Bayesian applications (where posterior samples are typically expensive to generate) based on the preceding investigations, with an application to a practical example.

中文翻译：

---

桥采样器的属性，重点是拆分MCMC样本

在各种学科中，尤其是在贝叶斯模型选择问题中，归一化常数的计算是一个基本的数学问题。已经发现了一种基于采样的技术，称为桥梁采样（Meng and Wong in Stat Sin 6（4）：831–860，1996），可以产生标准化常数的准确估计值，并被证明具有良好的渐近性质。对于中小样本量（例如，在计算资源有限的情况下），我们证明了（最佳）桥梁采样器会产生有偏差的估计。具体来说，当一个密度（我们表示为\（p_2 \））构造为接近目标密度（我们表示为\（p_1 \）时））使用矩量法，我们基于仿真的结果表明，通过矩量匹配过程的相关性引起的偏差是不可忽略的。更重要的是，随着问题规模的增加，偏见也会放大。因此，进行了一系列理论和实证研究，以确定偏差的性质和来源。然后，我们研究了样本数量分配对桥梁抽样估计准确度的影响，并发现通过少量增加计算量来减少偏差和标准误差的一种可能性是通过从矩匹配密度\（p_2 \）（我们假定很容易从中取样），前提是对\（p_1 \）的求值不太贵。我们将继续展示简单的自适应方法（称为“分离”）如何以较高的标准误差为代价来减轻相关性引起的偏差，而与所涉及的维数无关。我们还略微修改了Wang等人提出的策略。（Warp Bridge采样：下一代，Preprint，2019年。arXiv：1609.07690）来解决因拆分导致的标准误差增加的问题，后来对其进行了概括以进一步提高效率。通过总结我们对上述适应性方法的应用的见解，以提高贝叶斯应用中桥梁采样估计的准确性（其中后验样本通常会产生高昂的成本），该研究基于先前的研究，并应用于实际的例子。