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Demystifying Posterior Distributions: A Tutorial on Their Derivation.

Han Du1, Fang Liu2, Zhiyong Zhang3

  • 1Department of Psychology, University of California, Los Angeles, Los Angeles, CA, USA.

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|November 22, 2025
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Summary
This summary is machine-generated.

This tutorial provides a step-by-step guide to deriving posterior distributions in Bayesian statistics. It simplifies complex concepts for researchers lacking advanced mathematical backgrounds, covering univariate and multilevel models.

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Area of Science:

  • Statistics
  • Computational Statistics

Background:

  • Bayesian statistics are increasingly used across disciplines.
  • Textbooks often lack detailed derivations of posterior distributions, assuming advanced mathematical knowledge.
  • This creates a barrier for researchers without extensive linear algebra and probability theory training.

Purpose of the Study:

  • To provide an accessible, step-by-step tutorial on deriving posterior distributions.
  • To bridge the gap between theoretical Bayesian statistics and practical application.
  • To demystify advanced statistical concepts for a broader audience.

Main Methods:

  • Detailed derivation of posterior distributions for two models: univariate normal and multilevel.
  • Illustrative examples with code and analytical forms.
  • Focus on practical, step-by-step explanations.

Main Results:

  • Demonstration of posterior distribution derivation for univariate normal models.
  • Explanation of posterior distribution derivation for multilevel models.
  • Generalizable principles applicable to various statistical models and distributions.

Conclusions:

  • The tutorial successfully simplifies the process of deriving posterior distributions.
  • It enhances accessibility of Bayesian statistical methods for researchers with varied mathematical backgrounds.
  • The presented methods and concepts are broadly applicable beyond the specific models discussed.