Bayesian Inference: A Tutorial from First Principles

A ground-up introduction to Bayesian inference, with emphasis on evidence and model comparison in cosmology.

Welcome

This tutorial teaches Bayesian inference from conditional probability through to cutting-edge evidence calculation methods. The core philosophy: Bayesian evidence for model comparison is the pinnacle of statistical inference—more valuable than point estimates or even parameter constraints alone.

Why This Perspective?

In modern cosmology, we often care more about detecting new physics than precisely measuring known parameters. Bayesian evidence provides the rigorous framework to ask:

  • "Is dark energy evolving, or is it a cosmological constant?"
  • "Do we need additional parameters to explain the data?"
  • "Which systematic model best describes our observations?"

Evidence comparison naturally implements Occam's razor and provides direct probability statements about competing theories.

Start Here

If you're completely new to Bayesian inference:

  1. conditional-probability - The foundation
  2. bayes-theorem - The fundamental theorem
  3. bayesian-inference - Applying Bayes to science
  4. evidence - The key to model comparison

If you already understand Bayes' theorem:

If you're here for cosmology applications:

The Bayesian Hierarchy

This tutorial follows a natural progression in inference complexity:

Conditional Probability
        ↓
   Bayes' Theorem
        ↓
Parameter Inference (Posterior distributions)
        ↓
Model Comparison (Evidence)

Each level builds on the previous, increasing in both complexity and scientific value.

  • Pages are interlinked using [[wiki-style]] brackets
  • Click any linked term to explore that concept
  • Mathematical rigor increases gradually through the tutorial
  • Code examples are provided for key methods
  • All claims are cited to primary literature

About This Tutorial

This tutorial draws from PhD research on efficient Bayesian evidence calculation for cosmological model comparison, including:

  • GPU-accelerated nested sampling
  • Non-Gaussian likelihood analysis
  • Simulation-based inference methods
  • Applications to CMB, supernovae, and weak lensing data

The perspective throughout is that of a practitioner: someone who has actually implemented these methods at scale and wrestled with real data.

Core Concepts: - conditional-probability - Foundation of Bayesian reasoning - bayes-theorem - Inverting probabilities - bayesian-inference - Parameter estimation framework - evidence - The marginalized likelihood - model-comparison - Comparing theories with Bayes factors

Computational Methods: - nested-sampling - Direct evidence calculation - mcmc - Markov Chain Monte Carlo sampling - computing-evidence - Practical methods comparison

Applications: - cmb-analysis - Nearly-Gaussian posteriors - supernovae-analysis - Non-linear degeneracies - cosmic-shear - High-dimensional problems

Advanced Topics: - gpu-acceleration - Modern hardware utilization - simulation-based-inference - When likelihoods are intractable

Let's begin: conditional-probability →

Conditional Probability →