Introduction: Bayesian Inference

At the heart of Bayesian statistics lies Bayes’ theorem, which describes how we update our beliefs about parameters θ given observed data D:

\[P(\theta \mid D) = \frac{P(D \mid \theta) \cdot P(\theta)}{P(D)}\]

Where:

P(θ | D) is the posterior distribution — what we want to learn
P(D | θ) is the likelihood — how probable the data is given the parameters
P(θ) is the prior distribution — our initial beliefs about the parameters
P(D) is the marginal likelihood or evidence — a normalizing constant

The Challenge

The denominator P(D) requires integrating over all possible parameter values:

\[P(D) = \int P(D \mid \theta) \cdot P(\theta) \, d\theta\]

For most real-world problems, this integral is intractable — impossible to compute analytically. This is where sampling algorithms come in.

Why We Need Samplers

Instead of computing the posterior distribution directly, samplers generate representative samples from P(θ | D). With enough samples, we can:

Estimate posterior means, medians, and credible intervals
Visualize the posterior distribution
Make predictions on new data
Perform model comparison

📌 Available Samplers

This page gathers different sampling algorithms I experiment with, mostly in the context of Bayesian inference and high-dimensional problems. Each sampler has its own dedicated page with interactive visualizations, implementation details, and diagnostics.

🔹 Markov Chain Monte Carlo (MCMC)

A foundational Metropolis–Hastings sampler that forms the basis for understanding modern sampling methods. This implementation serves as a reference for exploring fundamental concepts like:

Convergence diagnostics — How to know when your chain has reached the posterior
Autocorrelation — Understanding sample dependencies and effective sample size
Acceptance rates — Tuning proposals for efficient exploration
Scaling behavior — Why vanilla MCMC struggles in high dimensions

While not the most efficient for complex posteriors, MCMC remains invaluable for building intuition and as a diagnostic baseline.

➡️ Explore the MCMC sampler
Tags: Bayesian inference, Metropolis-Hastings, diagnostics, autocorrelation

🔹 Hamiltonian Monte Carlo (HMC)

A gradient-based sampler that treats sampling as a physics simulation problem.

HMC leverages Hamiltonian dynamics to propose distant states with high acceptance probability, making it particularly effective for:

High-dimensional posteriors — Efficient exploration where MCMC fails
Complex geometries — Following curved posterior landscapes
Reduced autocorrelation — Longer jumps mean fewer correlated samples

By simulating the motion of a particle with momentum through the posterior landscape, HMC can traverse the distribution much more efficiently than random-walk methods.

➡️ Explore the HMC sampler Tags: HMC, gradients, high-dimensional inference, Hamiltonian dynamics

🔹 Nested sampling

➡️ Explore Nested sampling Tags: HMC, gradients, high-dimensional inference, Hamiltonian dynamics

🔹 MCMC Parralel tempering

➡️ Explore MCMC parralel tempering Tags: HMC, gradients, high-dimensional inference, Hamiltonian dynamics

🔧 Implementation Notes

All samplers are implemented with:

Interactive visualizations — See the algorithms in action
Step-by-step explanations — Understand what’s happening at each iteration
Diagnostic tools — Assess convergence and sample quality
Comparative benchmarks — Performance across different problem types

📚 Further Reading

Bayesian Data Analysis (Gelman et al.) — Comprehensive treatment of Bayesian methods
MCMC Handbook (Brooks et al.) — Deep dive into sampling algorithms
Stan Documentation — Practical implementation patterns