We study a matrix that arises from a singular form of the Woodbury matrix identity. We present generalized inverse and pseudo-determinant identities for this matrix, which have direct applications for Gaussian process regression, specifically its likelihood representation and precision matrix. We extend the definition of the precision matrix to the Bott-Duffin inverse of the covariance matrix, preserving properties related to conditional independence, conditional precision, and marginal precision. We also provide an efficient algorithm and numerical analysis for the presented determinant identities and demonstrate their advantages under specific conditions relevant to computing log-determinant terms in likelihood functions of Gaussian process regression.

翻译：我们研究了一个由奇异形式的伍德伯里矩阵恒等式导出的矩阵。针对该矩阵，我们提出了广义逆与伪行列式恒等式，这些恒等式可直接应用于高斯过程回归，具体体现在其似然表示与精度矩阵中。我们将精度矩阵的定义推广至协方差矩阵的Bott-Duffin逆，保留了与条件独立性、条件精度及边际精度相关的性质。我们还为所提出的行列式恒等式提供了高效算法与数值分析，并论证了它们在特定条件下计算高斯过程回归似然函数中对数行列式项时的优势。