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Point process-based Monte Carlo estimation

机译：基于点过程的蒙特卡洛估计

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摘要

This paper addresses the issue of estimating the expectation of a real-valued random variable of the form where g is a deterministic function and can be a random finite- or infinite-dimensional vector. Using recent results on rare event simulation, we propose a unified framework for dealing with both probability and mean estimation for such random variables, i.e. linking algorithms such as Tootsie Pop Algorithm or Last Particle Algorithm with nested sampling. Especially, it extends nested sampling as follows: first the random variable X does not need to be bounded any more: it gives the principle of an ideal estimator with an infinite number of terms that is unbiased and always better than a classical Monte Carlo estimator-in particular it has a finite variance as soon as there exists such that . Moreover we address the issue of nested sampling termination and show that a random truncation of the sum can preserve unbiasedness while increasing the variance only by a factor up to 2 compared to the ideal case. We also build an unbiased estimator with fixed computational budget which supports a Central Limit Theorem and discuss parallel implementation of nested sampling, which can dramatically reduce its running time. Finally we extensively study the case where X is heavy-tailed.
机译：本文讨论了估计形式为g的确定性随机变量的期望的问题，其中g是确定性函数，可以是随机的有限维或无限维向量。利用罕见事件模拟的最新结果，我们提出了一个统一的框架来处理此类随机变量的概率和均值估计，即将诸如Tootsie Pop算法或Last粒子算法之类的算法与嵌套采样相链接。尤其是，它扩展了嵌套采样，如下所示：首先，随机变量X不再需要界定：它给出了具有无穷数量项的理想估计量的原理，该估计量是无偏的，并且总是比经典的Monte Carlo估计量更好-特别是，一旦存在使得它具有有限的方差。此外，我们解决了嵌套采样终止的问题，并表明，与理想情况相比，总和的随机截断可以保留无偏差，而方差仅增加2倍。我们还建立了一个具有固定计算预算的无偏估计器，该估计器支持一个中心极限定理，并讨论了嵌套采样的并行实现，这可以大大减少其运行时间。最后，我们广泛研究了X重尾的情况。

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