We define and solve classes of sparse matrix problems that arise in multilevel modelling and data analysis. The classes are indexed by the number of nested units, with two-level problems corresponding to the common situation, in which data on level-1 units are grouped within a two-level structure. We provide full solutions for two-level and three-level problems, and their derivations provide blueprints for the challenging, albeit rarer in applications, higher-level versions of the problem. While our linear system solutions are a concise recasting of existing results, our matrix inverse sub-block results are novel and facilitate streamlined computation of standard errors in frequentist inference as well as allowing streamlined mean field variational Bayesian inference for models containing higher-level random effects.

中文翻译：

---

多级稀疏矩阵问题的简化解决方案

我们定义和解决多级建模和数据分析中出现的稀疏矩阵问题类。这些类以嵌套单元的数量为索引，二级问题对应于常见情况，其中一级单元上的数据被分组在一个二级结构中。我们为两级和三级问题提供完整的解决方案，它们的推导为具有挑战性的问题提供了蓝图，尽管在应用程序中很少见，但更高级别的版本。虽然我们的线性系统解决方案是对现有结果的简明重铸，但我们的矩阵逆子块结果是新颖的，有助于简化频率论推理中标准误差的计算，并允许对包含更高级别随机效应的模型进行简化的平均场变分贝叶斯推理.