Preconditioned Crank–Nicolson sampling
Algorithms AIMech:PreconditionedCrankNicolson DRAFT
Aliases: pCN
A Metropolis sampler for a Gaussian-reference target whose proposal combines the current state with an independent draw from that reference using a scalar autoregressive coefficient.
Identity Notes & Concept Scope
This is the pCN proposal and likelihood-ratio test in Cotter et al. Section 4.2. It preserves the Gaussian reference before correction and differs from an additive random walk, the unpreconditioned CN solve and an angular slice search. The displayed nonnegative coefficient form uses the source’s delta≤2 parameterization. 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 0<beta≤1, draw w from Normal(0,C), and set v=sqrt(1−beta squared)u+beta w. Accept with min(1,exp(Phi(u)−Phi(v))); otherwise retain u. For 0<delta≤2 this equals (2+delta)v=(2−delta)u+sqrt(8 delta)w with beta squared=8 delta/(2+delta) squared. Gaussian-reference reversibility cancels the prior and proposal factors in the full Metropolis ratio. At beta=1 the proposal is an independent prior draw, while the retained states can still be correlated through rejection.
⚠️ 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.
- Beta=0 would give no movement and is excluded from this stated sampler. Delta>2 in the original linear form gives a negative autoregressive coefficient and is not represented by the positive square-root parameterization here.
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/PreconditionedCrankNicolson.yaml