Unlock Faster Bayesian Sampling: How Hamiltonian Monte Carlo Works

Hamiltonian Monte Carlo (HMC) is a faster sampling method compared with traditional Markov Chain Monte Carlo (MCMC). It combines concepts from Hamiltonian dynamics and Monte Carlo sampling.

MCMC Overview

MCMC is a common Bayesian learning technique that can model physical systems. It often uses the Gibbs canonical distribution, where the probability of a system state depends on its energy and temperature, and can define model parameters such as multivariate normal distributions.

HMC Principle

Hamiltonian dynamics describes an object's motion using its position (potential energy) and velocity (kinetic energy). The total energy (Hamiltonian) is the sum of potential and kinetic energy. Position and velocity each follow independent canonical (normal) distributions. Assuming velocity follows a standard normal distribution, the kinetic energy function becomes a quadratic form.

HMC Procedure

Initialize a state for the variable of interest.

Sample momentum from a normal distribution.

Using the Hamiltonian equations, perform L leapfrog steps to propose a new state.

Compute the Metropolis acceptance probability.

Draw a uniform random number in [0,1]; if it is less than the acceptance probability, accept the new state, otherwise retain the current state and repeat.

HMC Example

Assume the position follows a multivariate normal distribution with mean \(\mu\) and covariance \(\Sigma\). The potential energy function is defined accordingly, and its gradient is computed. Together with the kinetic energy, these expressions enable the sampling process described above.