Spectral Uncertainty Decomposition for Sparse Bayesian Learning

Estimating distributed neural sources from M/EEG measurements is an ill-posed inverse problem where principled uncertainty quantification remains challenging. The usual conditional Gaussian posterior covariance does not explicitly distinguish uncertainty along likelihood-visible source directions from uncertainty that is invisible to the measurements.

从脑磁图/脑电图（M/EEG）测量中估计分布式神经源是一个不适定的反问题，其中原则性的不确定性量化仍然具有挑战性。通常的条件高斯后验协方差并不能明确区分沿似然可见源方向的不确定性与对测量不可见的不确定性。

We propose a spectral framework based on the singular value structure of the prior-preconditioned forward operator, which provides an exact, interpretable decomposition of the posterior covariance into data-informed and indeterminacy epistemic components. This formulation identifies uncertainty contributions that are reducible by better measurements from those that are fundamentally irreducible by sensing geometry.

我们提出了一种基于先验预处理前向算子奇异值结构的谱框架，该框架提供了后验协方差的精确且可解释的分解，将其分为数据驱动成分和不确定性认知成分。这种公式化方法识别出了可以通过更好的测量来减少的不确定性贡献，以及那些因传感几何结构而从根本上无法减少的不确定性。

We further derive an equivalent subtractive decomposition that expresses the prior covariance as the sum of posterior and data-driven reduction terms, enabling precise accounting of uncertainty reduction through Bayesian learning. Together, these complementary representations offer both theoretical insight and practical tools for interpretable uncertainty quantification in M/EEG source imaging for linear-Gaussian Bayesian inverse solvers where the posterior covariance is available in closed form.

我们进一步推导了一种等效的减法分解，将先验协方差表示为后验项与数据驱动缩减项之和，从而能够精确计算通过贝叶斯学习实现的不确定性缩减。总之，这些互补的表示方法为线性高斯贝叶斯反演求解器（在后验协方差可闭式求解的情况下）中的 M/EEG 源成像提供了理论见解和可解释不确定性量化的实用工具。