Metropolis-within-Gibbs sampling
Algorithms AIMech:MetropolisWithinGibbs DRAFT
Aliases: MWG
A componentwise Monte Carlo scheme that uses Metropolis–Hastings transitions targeting full conditional distributions for one or more coordinate blocks, optionally combined with exact Gibbs updates for other blocks.
Identity Notes & Concept Scope
The distinction is between an exact draw from a full conditional and a Markov transition that preserves it. The latter can be used within a cycle or fixed mixture of block updates. This record does not define every hybrid MCMC composition as Metropolis-within-Gibbs and does not require a nested chain to be run to equilibrium at each update. No normalized identity match was found across all 442 pinned AIO labels/scoped synonyms and 159 prior native IDs/names/aliases. Ignored-inclusive full-checkout searches found prior discovery leads for Gibbs and slice sampling plus unrelated environment substring hits, not existing concept records. Existing HMM/MRF models and general Metropolis–Hastings were inspected to distinguish model, general transition and specialized algorithm scope.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
Hold all coordinates outside the selected block fixed at their current values. Propose a block value from q(new|old,rest), and accept with min(1, pi(new|rest)q(old|new,rest)/(pi(old|rest)q(new|old,rest))) where defined; otherwise retain the old block. This is the substitution of the full conditional into Tierney’s Metropolis rule. The conditional normalization cancels, so an evaluable joint target ratio at the fixed rest can be used. Combine these conditional-invariant transitions, and any exact Gibbs steps, by a fixed cycle or fixed-probability mixture. A finite valid transition per selected block preserves the joint target without independently resampling that block.
⚠️ Curated Limitations & Negative Results
- The proposal-density ratio is required unless the block proposal is symmetric at the fixed values of the other coordinates.
- A rejected proposal leaves the current block unchanged. Repeating until acceptance changes the transition unless its effect is separately accounted for.
- A valid conditional kernel need not explore its support well. Joint convergence and mixing require additional conditions on the target, proposals and update schedule.
- Fixed mixtures and cycles preserve the target, but Tierney explicitly warns that convergence properties of an individual component do not transfer indiscriminately to a cycle.
- History-dependent tuning, stale parallel updates and approximate target evaluations are outside this fixed-kernel definition. No optimal number of inner steps is claimed.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MarkovChain
Markov Chain
|
related_match | The procedure constructs Markov transitions for sampling a specified target, with the invariance and convergence qualifications stated in this record. AIO MarkovChain denotes the stochastic process/model and is exported beneath Network. This is a method-to-process related match, not equivalence, subclass membership 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
|
Luke Tierney
|
author technical report
full_text
|
ID: https://conservancy.umn.edu/server/api/core/bitstreams/f87bc468-91ac-45bc-a1cf-5d4b0f8fbcd6/content
sha256:cfde8fc2fa3c…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T07:00:31Z
Updated: 2026-10-11T07:00:31Z
Source File:
data/curated/MetropolisWithinGibbs.yaml