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.

翻译：我们研究随机变量的概率采样问题，即仅当变量落在给定集合（称为沉默集）之外时才进行采样。这有助于理解独立同分布随机过程这一特例下的最优事件驱动采样，也为其他情形下次优方案的设计提供了理论基础。针对标量对数凹随机变量，我们考虑该概率采样方案的设计，以最小化均方估计误差或平均绝对估计误差。研究表明，最优沉默区间：（i）本质上唯一，且（ii）是中心化迭代过程的极限。数值实验进一步表明，超水平区间在均方估计中表现出显著的近最优性。最后，利用标量单峰密度的高斯不等式证明：当平均采样率介于0.3至0.9采样点/时间单位之间时，概率采样的均方失真度不到周期性采样失真度的三分之一。