Metropolis–Hastings sampling
Algorithms AIMech:MetropolisHastings DRAFT
Aliases: MH
A Markov chain Monte Carlo algorithm that proposes a new state from a specified transition distribution and accepts it with the smaller of one and the target-density ratio multiplied by the reverse-to-forward proposal ratio, otherwise retaining the current state.
Identity Notes & Concept Scope
This record denotes the usual Metropolis acceptance choice within Hastings’s more general family of reversible updates. It is not the whole MCMC family, a Markov-chain model, the Barker acceptance rule or an independent-draw generator. Asymmetric proposals require the proposal-ratio correction. No normalized identity match was found across all 442 pinned AIO labels/scoped synonyms and 153 prior native IDs/names/aliases. Both checkout text searches included ignored and hidden files and returned no matches for the selected exact IDs, full labels, aliases and source titles. Existing HMM and Markov random field records describe probability models, not these sampling or tuning procedures.
Claims & Grounded Evidence
2 Claims
Mechanism
#method
At a current state x with positive target mass or density pi(x), draw y from q(y|x). For a supported proposal define R=pi(y)q(x|y)/(pi(x)q(y|x)), and accept y with probability min(1,R); on rejection record x again. A common unknown normalizing constant in pi cancels. The accepted-move probabilities satisfy detailed balance with pi. For symmetric q the proposal factors cancel, recovering the Metropolis choice. A valid target-invariant kernel is not by itself a finite-run convergence certificate.
Limitation
#error-and-mixing
Hastings distinguishes invariance from practical error assessment: initial-state effects, correlated samples, incomplete exploration and inaccurate ratio calculations can undermine finite-run estimates.
⚠️ Curated Limitations & Negative Results
- The target must be proper, and the proposal and acceptance calculation must be defined on the relevant support. A proposed move with no reverse support has zero acceptance; zero-over-zero expressions are not a general operational rule.
- Target invariance does not ensure irreducibility, aperiodicity, rapid mixing, independent samples or convergence from every start. Those properties depend on the target and proposal.
- Rejections repeat the current state. Keeping only accepted moves without correcting their weighting changes the sampling process.
- Poor proposals, isolated modes, finite precision and inadequate run length can yield misleading estimates; acceptance rate alone does not establish adequate exploration.
- The record does not reproduce an implementation or assert a universal acceptance target, speed improvement or numerical posterior result.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MarkovChain
Markov Chain
|
related_match | This procedure constructs transitions of a Markov chain to sample a specified target distribution. The exact AIO CLASS describes a stochastic process/model with dependence on the preceding state and is exported beneath Network. The method-to-process relationship is related_match, not equivalence, subclass membership or a claim that the sampler is a trained network. |
Evidence Sources & Primary Literature
2 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
W. K. Hastings
|
primary paper
full_text
|
ID: doi:10.1093/biomet/57.1.97
sha256:dc620d3651d3…
|
error-and-mixing
|
W. K. Hastings
|
primary paper
full_text
|
ID: doi:10.1093/biomet/57.1.97
sha256:dc620d3651d3…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T06:48:02Z
Updated: 2026-10-11T06:48:02Z
Source File:
data/curated/MetropolisHastings.yaml