I present all the details in calculating the posterior distribution of the conjugate Normal-Gamma prior in Bayesian Linear Models (BLM), including correlated observations, prediction, model selection and comments on efficient numeric implementations. A Python implementation is also presented. These have been presented and available in many books and texts but, I believe, a general compact and simple presentation is always welcome and not always simple to find. Since correlated observations are also included, these results may also be useful for time series analysis and spacial statistics. Other particular cases presented include regression, Gaussian processes and Bayesian Dynamic Models.

翻译：本文详细阐述了贝叶斯线性模型（BLM）中计算共轭正态-伽马先验后验分布的全部过程，涵盖相关观测、预测、模型选择及高效数值实现方案的评述。同时提供了相应的Python实现。尽管相关理论已在众多著作中有所阐述，但一套通用、简洁且清晰的系统性表述仍具有重要价值且不易获得。由于本工作纳入了相关观测情形，所得结论亦可适用于时间序列分析与空间统计领域。其他具体案例包括回归分析、高斯过程及贝叶斯动态模型。