The physics of Hamiltonian Monte Carlo, part 3: In the final post in this series, I discuss Hamiltonian Monte Carlo, building off previous discussions of the Euler–Lagrange equation and Hamiltonian dynamics.

1

The physics of Hamiltonian Monte Carlo, part 2: Building off the Euler–Lagrange equation, I discuss Lagrangian mechanics, the principle of stationary action, and Hamilton's equations.

2

The physics of Hamiltonian Monte Carlo, part 1: Lagrangian and Hamiltonian mechanics are based on the principle of stationary action, formalized by the calculus of variations and the Euler–Lagrange equation. I discuss this result.

3

Gibbs sampling is a computationally convenient Bayesian inference algorithm that is a special case of the Metropolis–Hastings algorithm. I discuss Gibbs sampling in the broader context of Markov chain Monte Carlo methods.

4

Many authors introduce Metropolis–Hastings through its acceptance criteria without explaining why such a criteria allows us to sample from our target distribution. I provide a formal justification.

5

A Markov chain is ergodic if and only if it has at most one recurrent class and is aperiodic. A sketch of a proof of this theorem hinges on an intuitive probabilistic idea called "coupling" that is worth understanding.

6

Numerical sampling uses randomized algorithms to sample from and estimate properties of distributions. I explain two basic sampling algorithms, rejection sampling and importance sampling.

7