基于定量Fisher信息下界的二元采样系统灵敏度分析
Sensitivity Analysis for Binary Sampling Systems via Quantitative Fisher Information Lower Bounds
摘要：
确定的数字化表现出最小的复杂性可实现的灵敏度的问题得到解决。在这种情况下，测量仅以硬限制形式提供。通过Cramér-Rao下界评估可实现的灵敏度需要表征似然函数，这对于多变量二进制分布是难以处理的。在这方面，指数族的Fisher矩阵和任意概率模型的下界进行了讨论。Fisher信息矩阵的保守近似依赖于由两个紧凑等价连接到实际数据生成系统的代理指数族分布。在不描述可能性及其支持的情况下，这种概率概念使得能够设计一致地实现由保守信息矩阵的逆所定义的灵敏度的估计量。对于多变量二进制样本的参数估计，二次指数代理分布驯服了统计复杂性，使得可实现的灵敏度水平的定量评估变得易于处理。这一事实是利用一个低复杂度的二进制传感器阵列的参数估计的性能分析相比，具有无限幅度分辨率的理想系统。此外，数据驱动的评估，通过估计递归二进制采样下的Fisher矩阵的保守近似值，在$ΣΔ$调制模数转换器中实现的证明。
英文摘要：
The problem of determining the achievable sensitivity with digitization exhibiting minimal complexity is addressed. In this case, measurements are exclusively available in hard-limited form. Assessing the achievable sensitivity via the Cramér-Rao lower bound requires characterization of the likelihood function, which is intractable for multivariate binary distributions. In this context, the Fisher matrix of the exponential family and a lower bound for arbitrary probabilistic models are discussed. The conservative approximation for Fisher's information matrix rests on a surrogate exponential family distribution connected to the actual data-generating system by two compact equivalences. Without characterizing the likelihood and its support, this probabilistic notion enables designing estimators that consistently achieve the sensitivity as defined by the inverse of the conservative information matrix. For parameter estimation with multivariate binary samples, a quadratic exponential surrogate distribution tames statistical complexity such that a quantitative assessment of an achievable sensitivity level becomes tractable. This fact is exploited for the performance analysis concerning parameter estimation with an array of low-complexity binary sensors in comparison to an ideal system featuring infinite amplitude resolution. Additionally, data-driven assessment by estimating a conservative approximation for the Fisher matrix under recursive binary sampling as implemented in $ΣΔ$-modulating analog-to-digital converters is demonstrated.
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