An Introduction to Hamiltonian Monte Carlo Method for Sampling

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" summary

Algorithm samples by drawing a Gaussian momentum, simulating Hamiltonian flow for time T, then refreshing momentum; it removes the Metropolis accept/reject in the idealized continuous case because the flow preserves the Hamiltonian and phase-space volume (time-reversible, energy-conserving ODEs). For smooth strongly log‑concave targets contraction and Wasserstein mixing are controlled by the condition number (κ) not dimension, e.g. harmonic‑oscillator gives explicit T and contraction ≈1−m/M. Practical use requires symplectic (leapfrog) integrators and discretization reintroduces ε and implementation costs.

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" chapters (30)

1. 0:00Introduction
2. 0:22Problem overview
3. 1:18Markov Chain Monte Carlo
4. 3:05Sampling challenges
5. 4:14Hamiltonian Monte Carlo
6. 4:58Hamiltonian dynamics basics
7. 7:26Idealized HMC algorithm
8. 8:46Density preservation property
9. 10:22HMC simulation overview
10. 11:49Visualizing HMC
11. 13:16Crash course on dynamics
12. 15:05Hamilton's equations
13. 16:56Time reversibility
14. 18:12Conservation of Hamiltonian
15. 20:15Volume preservation
16. 24:23HMC convergence theorems
17. 25:47Dimension independence
18. 28:43Preserving target density
19. 32:29Velocity refresh step
20. 34:23Trajectory mapping
21. 38:41Volume preservation mechanics
22. 40:49Running time bounds
23. 41:58Coupling for HMC
24. 44:06Harmonic oscillator analysis
25. 46:11Contraction proof
26. 52:13Coordinate-wise analysis
27. 54:50Final convergence bound
28. 55:49Q&A and intuition
29. 1:00:24Leapfrog integrator
30. 1:03:24Conclusion and workshop

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