使用基于静默集的概率抽样估计标量对数凹随机变量
Estimating a scalar log-concave random variable, using a silence set based probabilistic sampling
摘要：
研究了随机变量的概率抽样问题，即只有当随机变量落在一个给定的集（称为沉默集）之外时，才对其进行抽样。这有助于我们理解IID随机过程的特殊情况下的最佳基于事件的采样，也有助于我们理解其他情况下的次优方案的设计。我们考虑这种概率抽样的标量，对数凹随机变量的设计，以最小化均方估计误差，或平均绝对估计误差。我们表明，最佳的沉默时间间隔：（i）本质上是唯一的，和（ii）是一个迭代过程的中心的限制。此外，我们通过数值实验表明，超级间隔似乎是显着接近最佳的均方估计。最后，我们使用高斯不等式的标量单峰密度，表明概率采样给出了一个均方失真，是小于三分之一的失真所引起的周期性采样，如果平均采样率是0.3和0.9之间的样本每滴答。
英文摘要：
We study the probabilistic sampling of a random variable, in which the variable is sampled only if it falls outside a given set, which is called the silence set. This helps us to understand optimal event-based sampling for the special case of IID random processes, and also to understand the design of a sub-optimal scheme for other cases. We consider the design of this probabilistic sampling for a scalar, log-concave random variable, to minimize either the mean square estimation error, or the mean absolute estimation error. We show that the optimal silence interval: (i) is essentially unique, and (ii) is the limit of an iterative procedure of centering. Further we show through numerical experiments that super-level intervals seem to be remarkably near-optimal for mean square estimation. Finally we use the Gauss inequality for scalar unimodal densities, to show that probabilistic sampling gives a mean square distortion that is less than a third of the distortion incurred by periodic sampling, if the average sampling rate is between 0.3 and 0.9 samples per tick.
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