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：本书的唯一目标是提供关于贝叶斯矩阵分解中概念与数学工具的自包含介绍，以便在后续章节中无缝引入矩阵分解技术及其应用。然而，我们清楚地认识到，受限于讨论范围（例如对变分推断优化方法的分离分析），无法涵盖贝叶斯矩阵分解所有有用且有趣的结果。我们建议读者参考贝叶斯分析领域的文献以获取相关领域的更详细引介。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性，从而揭示其应用价值。所需的数学基础仅为统计学与线性代数的入门课程。除这一适度背景外，全书自包含且提供严谨证明。