Crank–Nicolson Metropolis sampling
Algorithms AIMech:CrankNicolsonMetropolis DRAFT
Aliases: CN sampler
A Metropolis sampler for a Gaussian-reference target using a Crank–Nicolson discretization of the unpreconditioned reference Ornstein–Uhlenbeck dynamics as its proposal.
Identity Notes & Concept Scope
This is the CN proposal of Cotter et al. Section 4.2 paired with its Metropolis test, not the generic Crank–Nicolson numerical integration scheme. Unlike pCN, its proposal requires an operator or matrix solve involving C and the step size. It contains no likelihood-gradient drift. No normalized identity match was found across all 442 pinned AIO labels/scoped synonyms and 167 prior native IDs/names/aliases. Full-checkout searches in both worktrees included ignored and hidden files; the 15 lines in each were shared Tierney source-title references or an adaptive-rejection discovery lead, not matching records. Existing Gibbs, Metropolis-within-Gibbs, random-walk and independence Metropolis mechanisms were inspected to resolve specialization boundaries.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
For finite-dimensional pi(u) proportional to exp(−Phi(u)) Normal(u;0,C), with fixed symmetric positive-definite C, choose delta>0 and draw w from Normal(0,C). Solve (2C+delta I)v=(2C−delta I)u+sqrt(8 delta C)w, where the square root is the symmetric matrix square root. Accept v with min(1,exp(Phi(u)−Phi(v))); otherwise retain u. Equivalently, with L=C inverse, (I+delta L/2)v=(I−delta L/2)u+sqrt(2 delta)xi for standard Gaussian xi. The proposal is reversible with respect to the Gaussian reference, so the Gaussian factors cancel in Metropolis acceptance even though the proposal need not be symmetric in Euclidean coordinates.
⚠️ Curated Limitations & Negative Results
- The target must be proper relative to the Gaussian reference. The explicit density-ratio implementation here is finite-dimensional with symmetric positive-definite C; function-space extensions require the source’s operator, regularity and absolute-continuity assumptions.
- Target invariance does not imply independent draws, good exploration or finite-run accuracy. Mesh-robust behavior under stated assumptions is not a universal dimension-free performance guarantee.
- C and the step size are fixed in the stated kernel. Adaptive covariance or step-size policies need separate validity arguments.
- The reprint’s later Equation (6.2) reverses signs relative to Equation (6.1) and the explicit Section 4 proposals. This record uses Section 4 and does not claim a published erratum or use the inconsistent expression.
- An accurate linear solve and Gaussian draw are required. Replacing the noise by an incorrectly scaled standard Gaussian changes the proposal and invalidates the stated cancellation.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MarkovChain
Markov Chain
|
related_match | This algorithm constructs target-invariant Markov transitions under its stated assumptions. The exact AIO CLASS denotes a stochastic process/model and is exported beneath Network. This is a method-to-process related match, not equivalence, subclass membership, independence of retained draws or a claim that the algorithm is a trained network. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
S. L. Cotter, G. O. Roberts, A. M. Stuart, D. White
|
primary paper
full_text
|
ID: doi:10.1214/13-STS421
sha256:4be2044ef1d8…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T07:13:10Z
Updated: 2026-10-11T07:13:10Z
Source File:
data/curated/CrankNicolsonMetropolis.yaml