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Nested antithetic modification

Algorithms AIMech:NestedAntitheticModification DRAFT

A finite conditional transition construction that starts with independent draws from the conditional distribution and successively applies maximal antithetic changes between a focal state and its remaining tail in a supplied ordering.

Identity Notes & Concept Scope The post315 assessment covers all 442 pinned AIO and 708 native IDs, labels, aliases and scoped synonyms, plus four complete checkout searches including ignored and hidden files. Historical 706-native research, 334 selected ID/name rows, 103 definitions and 15 fully read native neighbors retain their attributed scope. The two actual FSS/ZFSS records were freshly assessed; shared dominant-mass boundaries do not equate full kernels. The finite local reversible construction differs from exact Gibbs resampling, one-stage Liu MHGS, pair-state/direction/uniform-memory samplers, continuous slice procedures and multiple-proposal frameworks. AIO:MarkovChain is related_match at the process-versus-construction level; neural Network/ProbabilisticHiddenLayer inheritance is not asserted. A single supplied-order NAM construction includes upward and downward policy specializations without exact aliases or duplicate records. UDNAM equal mixture and ZDNAM completion are not exact single-order aliases. Tjelmeland T2 equality concerns only its finite index kernel; its entire proposal/weighted-estimator framework is distinct. Historical FHY attribution remains through inspected Neal2024 because original full text was not obtained.

Claims & Grounded Evidence

6 Claims

Mechanism #finite-target-and-consistent-focal-order

Let pi be a normalized, strictly positive probability mass function on a finite state space and let sigma be a permutation of its states. Start with P(u,v)=pi(v). At stage i, the focal set is the singleton {sigma(i)} and the tail contains the states later in sigma. The same ordering defines every row for this target. When used as a conditional update, a probability-based ordering may change when the other variables change the full conditional, but independently choosing a new ordering from the current label is not the specified construction.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 6, PDF pp. 14–15 and 21; Algorithm 2, p. 17; current-dependent-order negative control in research/discovery/nam-boundaries.py

Mechanism #maximal-singleton-tail-modification

Before a stage, write the remaining block as P(u,v)=r*pi(v), let q be the focal mass and s the total tail mass. For a nonempty tail, use delta=r*min(q/s,s/q). Increase focal-to-tail entries by delta*pi(v) and tail-to-focal entries by delta*q; decrease the focal diagonal by delta*s and each tail-to-tail entry by delta*pi(v)*q/s. Transitions involving previous focal states remain unchanged. If q<s, the focal diagonal becomes zero and the next stage operates on the tail. If q>=s, the entire tail block becomes zero and the construction stops. With only one state, the initial one-by-one identity matrix already defines the update.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 5, PDF p. 10, Equation 19; Section 6, pp. 14–15, Equations 28–34

Mechanism #reversible-kernel-and-row-computation

Every antithetic stage preserves nonnegative row sums of one and detailed balance with pi. The final NAM kernel therefore preserves pi and has at most one positive diagonal entry. Algorithm 2 computes only the current row by tracking remaining target mass s and remaining transition mass f. Before the current state becomes focal, an earlier focal mass q receives f*q/s after q has been removed from s, unless q>=s, in which case it receives all f. A zero residual f assigns zero to all remaining destinations. Once the current state is focal, its remaining probabilities follow the same maximal stage.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 5, PDF p. 10, Equation 19; Section 6, pp. 16–20, Algorithm 2 and Equations 35–63

Scope #fixed-upward-and-downward-policies

This parameterized construction includes a supplied fixed order, the upward policy UNAM using nondecreasing pi, and the downward policy DNAM using nonincreasing pi. These policies can produce different kernels. UNAM is unchanged by permutations within equal-probability groups and does not decrease any off-diagonal probability relative to the independent draw. DNAM can depend on its tie order, can zero an entire tail block and can decrease off-diagonal probabilities. Neither direction is an exact alias for the entire parameterized construction. Neal also uses unprefixed NAM specifically for a fixed-order policy; this record states the broader construction-level scope explicitly.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 6, PDF p. 21, naming and ordering discussion; Section 7, pp. 22–24, Algorithm 3 and Equations 66–75; Section 8, pp. 28–30, Algorithm 4 and Figure 6

Scope #single-order-versus-completed-mixtures

NAM is distinct from one Liu excluded-current Metropolis correction. The upward full construction has historical counterparts discussed by Neal, but an incomplete first-stage variant is not automatically the same kernel. The separately named UDNAM chooses UNAM or DNAM with equal probability and averages their matrices. Such a mixture is not generally one NAM ordering: for pi=(4/9,3/9,2/9), it has two positive diagonal entries, whereas each of the six supplied-order NAM matrices has one. The zero-self ZDNAM completion also changes the construction rather than merely choosing a different focal permutation.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 7, PDF pp. 25–27, full and truncated historical constructions and iterated Metropolis modifications; Section 8, p. 28, UDNAM definition; Section 9, pp. 31–35; exact six-order comparison in research/discovery/nam-boundaries.py

Limitation #conditional-invariance-and-scan-limits

Using this kernel for a valid full conditional while holding other variables fixed preserves the joint target. Fixed mixtures and fixed sequential compositions preserve that target, although a sequential sweep need not be reversible. Reduced self transitions do not establish aperiodicity or exploration of every joint state. For a uniform binary conditional the modified update flips deterministically; systematic compositions can therefore remain within a proper subset of joint states. This record makes no general variance, burn-in, finite-run accuracy or runtime claim.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 2, PDF pp. 2–3, Equations 1–4 and composition; Section 5, pp. 13–14 and footnotes 7–8

⚠️ Curated Limitations & Negative Results

The maintained mathematical scope uses a finite positive-support distribution with exact normalization and a consistent focal ordering. Zero-current conventions, floating arithmetic and countably infinite extensions are not specified here.
The source Figure 3 caption on PDF p. 16 says off-diagonal probabilities become smaller, contrary to its displayed matrix and the Section 7 inequalities. This caption direction error is not adopted. No author implementation was executed.
Historical attribution to the full Frigessi–Hwang–Younes construction is mediated through Neal’s report; that original full text was not obtained. The local Tjelmeland index-selection kernel does not identify its entire multiple-proposal and weighted-estimator framework with NAM.
All fixed/upward/downward policies are covered as parameter choices, without broad exact aliases. UDNAM remains a distinct mixture identity and ZDNAM a separate completion; this construction does not exhaust the source family.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:MarkovChain` Markov Chain	related_match	The construction defines conditional Markov transitions preserving a target. AIO MarkovChain denotes a stochastic process/model, so this is a method-to-process association without equivalence, subclass status or inheritance of Network and ProbabilisticHiddenLayer assertions.

Evidence Sources & Primary Literature

1 Source

Source ID	Publication / Document	Type / Access	Reference & Checksum
`neal2024`	Modifying Gibbs Sampling to Avoid Self Transitions Radford M. Neal	author technical report full_text	ID: arxiv:2403.18054 sha256:3d637d3eec04…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-14T22:25:01Z

Updated: 2026-10-14T22:25:01Z

Source File: data/curated/NestedAntitheticModification.yaml