Computing posterior distributions, often represented as $p(\theta | D)$, is central to Bayesian analysis. However, these distributions frequently involve high-dimensional integrals that lack analytical solutions, making direct calculation infeasible. Markov Chain Monte Carlo (MCMC) methods provide a powerful computational toolkit to address this challenge by generating samples from the target posterior distribution, allowing us to approximate it and compute relevant summaries.

This chapter covers the theory and practice of MCMC techniques for Bayesian inference. We will begin with the foundational ideas of Monte Carlo integration and the necessary Markov chain theory that guarantees these methods work. You will learn to implement and analyze several key algorithms, including:

• The Metropolis-Hastings algorithm and its variants.
• Gibbs sampling, particularly useful for models with specific conditional structures.
• Hamiltonian Monte Carlo (HMC), a gradient-based method enabling efficient exploration of complex probability spaces.
• The No-U-Turn Sampler (NUTS), an adaptive extension of HMC.

Furthermore, we will address the practical necessity of evaluating whether MCMC simulations have converged correctly and are performing efficiently. This includes applying standard convergence diagnostics. A hands-on section will guide you through implementing advanced MCMC samplers using contemporary probabilistic programming libraries.