Interactive demo · Bayesian and Probabilistic Machine Learning · Chapter 11

Slice sampling and Hamiltonian Monte Carlo

Two ways to take large steps without a hand-tuned proposal. Slice sampling draws a height u under the curve and then a point uniformly from the slice {z : p̃(z) > u}, finding the slice by stepping out and shrinking. Hybrid Monte Carlo gives the state a random momentum and follows Hamiltonian dynamics with the leapfrog integrator, which preserves volume and nearly preserves the energy H, so long moves are still accepted.

Method

Initial width w

Run

Step size ε

Number of steps L

Target

Step size ε

Leapfrog steps L

Run

Try these

Slice sampling target: p̃(z) = 0.55 exp(−(z+2)²/(2·0.75²)) + 0.45 exp(−(z−2)²/(2·0.95²)), the two-bump shape of Figure 11.13. Leapfrog: E(z) = z²/2 from (z, r) = (1, 1). Hybrid Monte Carlo: a Gaussian with standard deviations 1 and 0.1 along the diagonals, or the mixture of N(±(1, 1), 0.5²I); each iteration picks the direction of integration at random and jitters ε by up to ±10% (Section 11.5.2). The comparison chain is random-walk Metropolis with ρ = σ_min, run for L steps per HMC iteration so both use the same number of density or gradient evaluations.

In the lecture notes

Chapter 11 notes: Section 11.4 (slice sampling, stepping out and shrinking, Figure 11.13), Section 11.5 (the hybrid Monte Carlo algorithm), Section 11.5.1 (Hamiltonian dynamics, Liouville's theorem, the leapfrog scheme, Figure 11.14), Section 11.5.2 (the accept step (11.67), detailed balance, the (σ_max/σ_min) scaling).
Random-walk behaviour of Metropolis and Gibbs: the companion demo Markov chain Monte Carlo: Metropolis and Gibbs.
Python code: code/ch11_sampling.py (functions slice_sample, leapfrog, hmc; Figures 11.13 and 11.14) and the checks in code/checks_ch11.py (volume preservation, reversibility, O(ε²) energy error, detailed balance). JSRandom(seed) reproduces this page's chains number for number.