Adaptive Metropolis
Algorithms AIMech:AdaptiveMetropolis DRAFT
Aliases: Adaptive Metropolis algorithm
A history-adaptive random-walk Metropolis sampler whose Gaussian proposal covariance is estimated from the accumulated retained states, including repetitions after rejected proposals.
Identity Notes & Concept Scope
This record identifies covariance adaptation from the full retained history, distinguished from a fixed random-walk kernel, finite sliding-window adaptation, scale-only tuning and RAM direction-based adaptation. The original empirical-covariance form and later recursive modifications share this core but differ in update timing and regularization. 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
Start at a state with positive target density. In the original form, after an initial period with a chosen positive-definite C0, propose Y from N(X_(t−1), C_t), where C_t=s_d[Cov(X_0,...,X_(t−1))+epsilon I], epsilon>0 and s_d>0. The empirical covariance uses every retained state, with denominator t−1 for t available states; choose the adaptation start after at least two states. Accept with min(1,pi(Y)/pi(X_(t−1))) and otherwise repeat the old state. The fixed-covariance Metropolis rule is used at each iteration, but adapting C_t from the sampled history requires a separate validity argument.
⚠️ Curated Limitations & Negative Results
- The 2010 source explicitly analyzes a recursive modification, not an exact restatement of the original unbiased sample covariance. Its displayed outline also shifts the covariance index; the original timing is stated explicitly here.
- The original full-method text was read through a public indexed PDF. Direct download returned 403 and no visible page images were obtained; no original-PDF hash or visual inspection is claimed. The 2010 method pages were read and visually checked.
- The original bounded-support result and the later regular superexponential-tail results have different hypotheses. Neither licenses arbitrary targets or unrestricted adaptive variants.
- The usual dimension-based scale is a tuning heuristic under particular target assumptions, not a universal optimum. Empirical covariance estimation can be unsuitable for distributions without finite second moments.
- The sampled-state sequence alone is history dependent; invariant kernels at frozen settings do not by themselves prove adaptive convergence.
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
2 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Eero Saksman, Matti Vihola
|
primary paper
full_text
|
ID: doi:10.1214/10-AAP682
sha256:87ccd3ec1113…
|
additional-1
|
Heikki Haario, Eero Saksman, Johanna Tamminen
|
primary paper
full_text
|
ID: doi:10.2307/3318737
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T07:40:20Z
Updated: 2026-10-11T07:41:38Z
Source File:
data/curated/AdaptiveMetropolis.yaml