The sole aim of this book is to give a self-contained introduction to concepts and mathematical tools in Bayesian matrix decomposition in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning Bayesian matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of variational inference for conducting the optimization. We refer the reader to literature in the field of Bayesian analysis for a more detailed introduction to the related fields. This book is primarily a summary of purpose, significance of important Bayesian matrix decomposition methods, e.g., real-valued decomposition, nonnegative matrix factorization, Bayesian interpolative decomposition, and the origin and complexity of the methods which shed light on their applications. The mathematical prerequisite is a first course in statistics and linear algebra. Other than this modest background, the development is self-contained, with rigorous proof provided throughout.

翻译：本书的唯一目的是对贝叶斯矩阵分解中的概念和数学工具进行自成体系的介绍，以便在后继章节中无缝引入矩阵分解技术及其应用。然而，我们清楚地意识到自身无法涵盖贝叶斯矩阵分解所有有用且有趣的成果，且受讨论范围所限（例如，未能对执行优化的变分推断进行单独分析）。我们建议读者参阅贝叶斯分析领域的文献以获取相关领域的更详细介绍。本书主要总结了重要贝叶斯矩阵分解方法（如实值分解、非负矩阵分解、贝叶斯插值分解）的目的、意义，以及这些方法的起源与复杂度，从而阐明其应用场景。所需的数学基础为统计学和线性代数的初级课程。除这一基本背景要求外，全书内容自成体系，并提供严谨的证明过程。