贝叶斯可信域的本质探测：物理系统的可行误差证明
Probing Bayesian credible regions intrinsically: a feasible error certification for physical systems
摘要：
计算贝叶斯可信区域的大小和可信度以证明未知参数（诸如量子态、信道、相位、\emph{etc.}）的任何点估计的可靠性依赖于来自整个参数空间的拒绝采样，这对于大数据集实际上是不可行的。我们重新制定的贝叶斯模糊区域理论表明，这两个属性可以单独获得的平均值的对数似然在该地区本身，这是可计算的直接区域抽样。既不拒绝采样也不需要关于整个参数空间的任何几何知识，因此一般误差认证现在变得可行。我们把这个区域平均理论推广到一个新的水平，将大小推广到一个随机区域点和估计器之间的平均$l_p$ -范数距离$(p>0)$，并给出了$p=2$的分析公式，以估计任何物理系统和大数据集的距离引起的大小和可信度，从而意味着渐近贝叶斯误差证明是可能的，而无需任何Monte~Carlo计算。所有结果进行了讨论的背景下，量子态层析成像。
英文摘要：
Computing size and credibility of Bayesian credible regions for certifying the reliability of any point estimator of an unknown parameter (such as a quantum state, channel, phase, \emph{etc.}) relies on rejection sampling from the entire parameter space that is practically infeasible for large datasets. We reformulate the Bayesian credible-region theory to show that both properties can be obtained solely from the average of log-likelihood over the region itself, which is computable with direct region sampling. Neither rejection sampling nor any geometrical knowledge about the whole parameter space is necessary, so that general error certification now becomes feasible. We take this region-average theory to the next level by generalizing size to the average $l_p$-norm distance $(p>0)$ between a random region point and the estimator, and present analytical formulas for $p=2$ to estimate distance-induced size and credibility for any physical system and large datasets, thus implying that asymptotic Bayesian error certification is possible without any Monte~Carlo computation. All results are discussed in the context of quantum-state tomography.
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