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Hamiltonian Monte Carlo

Algorithms AIMech:HamiltonianMonteCarlo DRAFT

Aliases: HMC, Hybrid Monte Carlo

A Markov chain Monte Carlo algorithm for differentiable continuous targets that augments positions with momenta, proposes a state through reversible volume-preserving numerical Hamiltonian dynamics, and applies a Metropolis correction for the Hamiltonian change.

Identity Notes & Concept Scope The record describes conventional Euclidean HMC with Gaussian momenta, leapfrog integration and an end-of-trajectory Metropolis test. The source’s unit-mass baseline is explicit; fixed positive-definite mass matrices are documented in the author technical chapter. HMC is not deterministic optimization, an uncorrected molecular-dynamics simulation, or NUTS’s adaptive tree-selection kernel. The original 1987 publication’s metadata was verified, but its full text was inaccessible in this session. No normalized identity match was found across all 442 pinned AIO labels/scoped synonyms and 153 prior native IDs/names/aliases. Both checkout text searches included ignored and hidden files and returned no matches for the selected exact IDs, full labels, aliases and source titles. Existing HMM and Markov random field records describe probability models, not these sampling or tuning procedures.

Claims & Grounded Evidence

2 Claims

Mechanism #method

For target pi(theta), use potential U=−log pi up to a constant and Gaussian momentum r with kinetic energy K=r·r/2 in the unit-mass baseline. Refresh r independently at each iteration. Apply L leapfrog steps: half a momentum update using the log-density gradient, a full position update, and the remaining half momentum update. Form the reversible proposal with momentum reversal and accept with min(1,exp(H_start−H_proposed)); on rejection retain the current position. The integrator preserves volume and reversibility mathematically, while the Metropolis correction accounts for numerical integration error in the invariant target. The final momentum sign can be omitted when only positions are retained and momenta are fully refreshed before reuse.

supports Source: The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo · Locator: Section 2 and Algorithm 1, printed pp. 1594–1596 / PDF pp. 2–4

Scope #author-method-and-limits

Neal’s author-posted technical chapter defines fixed Gaussian mass matrices, the full-refresh leapfrog/Metropolis algorithm, its invariant-distribution argument, and periodicity failures of ergodicity. It is a methodological review and implementation description, not the original 1987 invention paper.

supports Source: MCMC using Hamiltonian dynamics · Locator: Sections 2.1, 2.3 and 3.1–3.2, Equations (2.5)–(2.8), (2.28)–(2.30), (3.6), Figure 2 and Ergodicity of HMC, PDF/printed pp. 3–4, 8, 10–15

⚠️ Curated Limitations & Negative Results

The baseline needs evaluable log density and gradients on continuous coordinates. Discrete variables or constraints require additional handling; the stated algorithm does not directly solve those representation problems.
Step size, trajectory length and coordinate scaling materially affect behavior. Too-large steps can cause integration failure or rejection, and too-short or resonant trajectories can explore poorly.
Target invariance is not universal ergodicity or finite-sample convergence. Neal explicitly gives periodic trajectories that can cause nonergodicity.
The correction does not remove Monte Carlo error or make floating-point dynamics exactly reversible in every implementation. Rejected positions remain part of the chain.
No universal dimension-scaling law, tuned parameter value, benchmark superiority or current library default is asserted.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:MarkovChain` Markov Chain	related_match	This procedure constructs transitions of a Markov chain to sample a specified target distribution. The exact AIO CLASS describes a stochastic process/model with dependence on the preceding state and is exported beneath Network. The method-to-process relationship is related_match, not equivalence, subclass membership or a claim that the sampler is a trained network.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`method-source`	The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo Matthew D. Hoffman, Andrew Gelman	primary paper full_text	ID: https://www.jmlr.org/papers/v15/hoffman14a.html sha256:bf2a5592015b…
`author-method-and-limits`	MCMC using Hamiltonian dynamics Radford M. Neal	author technical report full_text	ID: arxiv:1206.1901v1 sha256:0582743db691…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-11T06:48:02Z

Updated: 2026-10-11T06:48:02Z

Source File: data/curated/HamiltonianMonteCarlo.yaml