Adaptive scaling within adaptive Metropolis
Algorithms AIMech:AdaptiveScalingWithinAdaptiveMetropolis DRAFT
Aliases: ASWAM
An adaptive Metropolis algorithm that combines acceptance-rate-driven scalar proposal scaling with recursive mean and covariance-shape adaptation from retained samples.
Identity Notes & Concept Scope
This record gives the bounded mean-and-shape formulation of Vihola Definition 5. Its log scale is unbounded, while the proposed mean/shape pair is accepted only inside a specified compact constraint set. Unconstrained software variants are not automatically the same validity setting. No normalized identity match was found across all 442 pinned AIO labels/scoped synonyms and 181 prior native IDs/names/aliases. Ignored-inclusive searches of both full checkouts found prior discovery leads, neighboring sampler discussions and unrelated author-name/environment substrings. Existing random-walk Metropolis, delayed-rejection, fixed symmetric delayed-rejection and RAM records were inspected; shared components do not make these procedures equivalent.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
Choose alpha_star in (0,1) and initialize x with pi(x)>0, mu=x and feasible positive-definite Sigma. Propose Y=x+exp(S)Sigma^(1/2)W with spherical W, use alpha=min(1,pi(Y)/pi(x)), and retain X_new after the accept/reject test. Update S_new=S+eta_n(alpha−alpha_star). With the old mean mu and shape Sigma, form mu_hat=(1−eta_n)mu+eta_n X_new and Sigma_hat=(1−eta_n)Sigma+eta_n(X_new−mu)(X_new−mu)^T. For fixed zeta≥1, accept the entire proposed mean/shape pair only if ||mu_hat||≤zeta and every eigenvalue of Sigma_hat lies in [zeta^−1,zeta]; otherwise keep both old mu and old Sigma. Start with a feasible mean/shape pair and keep the scalar update separately. Use eta_n in (0,1), with divergent sum and finite squared sum under the source assumptions.
⚠️ Curated Limitations & Negative Results
- The covariance update uses the old mean and the retained new state, including repeated states after rejection. Substituting the new mean or only accepted proposals changes the specified recursion.
- The mean and shape constraints are part of this source’s proved formulation; only the scalar scale lacks a fixed bound. Omitting constraints in practice does not inherit the same theorem.
- The source’s strong-law results restrict alpha_star to (0,1/2) and require the corresponding target/proposal regularity conditions. Algorithm definition alone does not prove ergodicity.
- Shape learning is a recursive stochastic-approximation update, not exactly the original unbiased full-history covariance at every finite iteration.
- No universal efficiency advantage, marginal-distribution convergence or executed benchmark 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 adapts a family of Markov transition kernels that are target-invariant at frozen settings. Its sampled-state history is not assumed to be a homogeneous Markov chain, and adaptive validity requires separate assumptions. AIO MarkovChain is a process/model beneath Network; this is a related method-to-process mapping, not equivalence, an is-a assertion or a convergence guarantee. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Matti Vihola
|
primary paper
full_text
|
ID: arXiv:0903.4061v3
sha256:2de1293c2ed1…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T07:40:20Z
Updated: 2026-10-11T07:41:38Z