Bayesian Linear Models: A compact general set of results

TL;DR

This paper provides a comprehensive, simplified presentation of Bayesian Linear Models, including conjugate priors, correlated observations, and practical implementation, useful for various statistical applications.

Contribution

It offers a compact, accessible derivation of Bayesian Linear Model results with detailed calculations and a Python implementation, covering correlated data and multiple model cases.

Findings

Explicit formulas for posterior distributions with conjugate priors

Implementation details for efficient computation

Applicability to time series and spatial data

Abstract

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.