This letter considers an $M$-ary hypothesis testing problem on an $n$-dimensional random vector perturbed by the addition of Gaussian noise. A novel expression for the gradient of the error probability, with respect to the covariance matrix of the noise, is derived and shown to be a function of the cross-covariance matrix between the noise matrix (i.e., the matrix obtained by multiplying the noise vector by its transpose) and Bernoulli random variables associated with the correctness event.

中文翻译：

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多元高斯噪声下多元假设检验问题的误差概率梯度

这封信认为 百万美元上的假设检验问题 $n$被高斯噪声扰动的 维随机向量。推导出了一种新的关于噪声协方差矩阵的误差概率梯度表达式，并显示为噪声矩阵之间的互协方差矩阵的函数（即，通过乘以噪声向量获得的矩阵）通过其转置）和与正确性事件相关的伯努利随机变量。