Liu modified discrete-state Gibbs sampler
Algorithms AIMech:LiuModifiedGibbsSampler DRAFT
A discrete conditional sampling algorithm that proposes a category from the full conditional distribution with the current category excluded, then applies a Metropolis correction to preserve the original conditional distribution.
Identity Notes & Concept Scope
Whole-corpus normalized checks cover every ID, name, alias and scoped synonym in 442 pinned AIO and 703 native records; four complete checkouts, including ignored and hidden files, are searched for final identities and researched labels. The finite excluded-current conditional proposal with its specific acceptance correction is distinct from exact Gibbs, generic Metropolis-within-Gibbs, multiple-try updates and Neal’s full ordered-pair construction. Bare metropolized Gibbs is a search probe, not an exact alias. Existing source-frontier references are resolved in place. Related AIO:MarkovChain is supported at a different abstraction level; all 18 candidate class contexts and 19 nearest native records are preserved.
Claims & Grounded Evidence
5 Claims
Mechanism
#excluded-current-conditional-proposal
Hold the other components fixed and let p be the normalized finite full-conditional probability mass function for the selected component. At a current category a with p_a>0, if p_a<1, propose a different category c with probability p_c/(1−p_a). Categories with zero conditional mass have zero proposal probability. If p_a=1, retain a without evaluating the undefined excluded-current proposal.
Mechanism
#conditional-metropolis-correction
Accept the proposed category c with probability min(1,(1−p_a)/(1−p_c)); otherwise retain a. For distinct supported categories, the complete transition probability is U(a,c)=min(p_c/(1−p_a),p_c/(1−p_c)), with U(a,a) completing the row to one. The Metropolis correction uses the original conditional masses, not the proposed category probability after excluded-current renormalization. A proposal that differs from the current category can still be rejected.
Mechanism
#conditional-balance-and-joint-invariance
The update satisfies detailed balance with the selected full conditional. Its off-diagonal probability U(a,c) is at least p_c, the corresponding exact-Gibbs transition probability; this is a property of the conditional transition matrix. Holding the remaining components fixed therefore preserves the joint target. This local reversible kernel does not require a persistent direction variable or a stored preceding joint state.
Scope
#scan-schedule-and-comparison-boundary
A fixed-probability mixture of these conditional kernels preserves the target and is reversible. A fixed sequential sweep also preserves the target, but its composition need not be reversible. Neal explicitly distinguishes random-scan and sequential updates when applying Peskun’s comparison: the reversible-chain argument is not automatically a result for a systematic sweep, random-permutation sweep, finite-run error, or computational cost.
Scope
#local-update-versus-pair-state-construction
Neal subsequently applies this conditional kernel inside a separate construction that swaps an ordered pair and updates its second component to reduce immediate backtracking. That augmented pair-state algorithm is not the identity of Liu’s conditional update. In particular, forbidding the present category in a proposal does not forbid all repeated values or eliminate rejection from the resulting chain.
⚠️ Curated Limitations & Negative Results
- The maintained scope is a finite discrete full conditional with compatible joint target and a supported current category. A point-mass conditional requires the explicit stay guard; zero-over-zero ratios are not operational rules.
- Conditional invariance does not establish irreducibility, aperiodicity or convergence of every joint update schedule. Increased off-diagonal transition probabilities are not a universal mixing-time, finite-sample accuracy or runtime guarantee.
- The proposal must be sampled from the correctly normalized excluded-current conditional. Determining its probabilities can be costly for many categories; an arbitrary categorical proposal is not covered by the stated acceptance formula.
- The inspected mathematical method is Neal’s 2004 report, which attributes the modification to Liu’s 1996 paper. The original Liu paper was available only through abstract and metadata; its full text was not obtained. This access gap does not supply unseen original proofs.
- The name does not denote all Metropolis-within-Gibbs methods or every method called metropolized Gibbs. Neal’s pair-state backtracking construction and complete library sampler classes have separate identities.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:MarkovChain
Markov Chain
|
related_match | This procedure constructs conditional Markov transitions for sampling a specified target. AIO MarkovChain denotes a stochastic process/model and is exported beneath Network. The relation is method-to-process association, not equivalence, subclass membership or inheritance of network and hidden-layer assertions. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
neal2004
|
Radford M. Neal
|
author technical report
full_text
|
ID: arxiv:math/0407281v1
sha256:b3f451ce72fb…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-14T21:02:45Z
Updated: 2026-10-14T21:02:45Z
Source File:
data/curated/LiuModifiedGibbsSampler.yaml