Demystifying Posterior Distributions: A Tutorial on Their Derivation

Bayesian statistics have gained significant traction across various fields over the past few decades. Bayesian statistics textbooks often provide both code and the analytical forms of parameters for simple models. However, they often omit the process of deriving posterior distributions or limit it to basic univariate examples focused on the mean and variance. Additionally, these resources frequently assume a strong background in linear algebra and probability theory, which can present barriers for researchers without extensive mathematical training. This tutorial aims to fill that gap by offering a step-by-step guide to deriving posterior distributions. We aim to make concepts typically reserved for advanced statistics courses more accessible and practical. This tutorial will cover two models: the univariate normal model and the multilevel model. The concepts and properties demonstrated in the two examples can be generalized to other models and distributions.