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Zero-self downward nested antithetic modification

Algorithms AIMech:ZeroSelfDownwardNestedAntitheticModification DRAFT

A finite conditional sampling construction that follows downward nested antithetic modification and replaces stages retaining avoidable self probability with a reversible two-focal-state completion that minimizes self transitions.

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. ZDNAM is a distinct two-focal/tail completion following the downward prefix, with an explicit tie policy and dominant-mass boundary. It differs from every single NAM ordering in the supported (4,3,2)/9 example. Zero-self denotes removal of avoidable self mass, not unconditional zero diagonal or an alias of every minimum-self sampler. No aliases adopted.

Claims & Grounded Evidence

6 Claims

Mechanism #downward-prefix-and-look-ahead

Let pi be a normalized strictly positive probability mass function on a finite state space. Choose a consistent nonincreasing order sigma, including a specified tie rule. Handle a maximum mass of at least one half separately. Otherwise follow the downward NAM stages, but at each stage i examine the next focal mass q2=pi(sigma(i+1)) and the mass s2 of states after it. When q2>=s2, complete the remaining block jointly instead of taking separate DNAM steps for the two focal states. Previously determined transitions to and from earlier focal states remain unchanged. Equality belongs to the completion branch of Algorithm 5.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 9, PDF pp. 31–34, Equations 91–94 and Algorithm 5; Section 8, pp. 28–29, downward ordering

Mechanism #two-focal-state-completion

At completion let a=sigma(i), b=sigma(i+1), q=pi(a), q2=pi(b), and let f be the transition mass still assigned to the block containing a, b and the remaining tail. When s2>0, set A=(q+q2-s2)/2, B=(q-q2+s2)/(2*s2), and C=(s2+q2-q)/(2*s2). The block entries are P(a,b)=f*A/q, P(b,a)=f*A/q2, P(a,j)=f*B*pi(j)/q, P(b,j)=f*C*pi(j)/q2, P(j,a)=f*B and P(j,b)=f*C for tail states j. Both focal diagonals and all tail-to-tail entries are zero. The relations A+B*s2=q, A+C*s2=q2 and B+C=1 give the required residual row sums; the paired flows satisfy detailed balance. Algorithm 5 has a separate zero-tail guard; formulas dividing by s2 are not used at s2=0.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 9, PDF p. 32, Equations 92–94; Algorithm 5 Part 2, p. 34

Mechanism #dominant-mass-boundary

If a maximum state a has mass h>=1/2, the positive-support kernel moves every other state to a with probability one, while P(a,j)=pi(j)/h for j!=a and P(a,a)=(2*h-1)/h. At h=1/2 the maximum has no self transition. For a one-state point mass, it stays with probability one. Thus zero-self in the name does not mean that positive self probability is eliminated when a state has more than half the target mass.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Algorithm 5 Part 1, PDF p. 33, both half-mass branches; Section 2, p. 3, footnote 1

Mechanism #invariance-and-minimum-self-mass

The NAM prefix and final balanced block preserve pi. If no state has mass greater than one half, every diagonal is zero. Otherwise only the dominant state retains the unavoidable self probability. The stationary probability of a self transition is therefore max(0,2*max(pi)-1), attaining the lower bound from invariance. This is a statement about the specified finite transition kernel, not a guarantee of convergence for every multi-variable update schedule.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 9, PDF pp. 31–34, reversible completion; Section 2, p. 3, footnote 1; Section 13, p. 51, Equations 112–114

Scope #tie-policy-and-distinct-kernel-boundaries

The choice among tied downward focal states can change ZDNAM even on positive support. For pi=(1/7,2/7,2/7,2/7), orders (2,3,4,1) and (3,2,4,1), using one-based labels, give different supported rows in the original exact calculations. ZDNAM is also not generally a different single NAM order: for pi=(4/9,3/9,2/9), its diagonal is zero while every one of the six NAM orders retains a positive diagonal entry. The source’s DNAM/ZDNAM comparison does not assert that reducing self transitions always improves efficiency relative to DNAM.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 9, PDF pp. 31–35, Algorithm 5 and Equation 95; Section 8, p. 30, limits of single-order NAM; explicit tie and six-order comparisons in research/discovery/nam-boundaries.py

Limitation #conditional-and-composition-scope

Using the reversible kernel for a valid full conditional while keeping other coordinates unchanged preserves the joint target. Fixed mixtures and fixed sweeps also preserve it, but a fixed sweep need not be reversible or explore all states. Uniform binary conditionals give deterministic flips, so eliminating self transitions alone does not establish aperiodicity or joint exploration. This record does not adopt a universal variance, finite-run accuracy, burn-in or wall-clock advantage.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 2, PDF pp. 2–3, conditional invariance and composition; Section 5, p. 14, footnotes 7–8; Algorithm 5, p. 33

⚠️ Curated Limitations & Negative Results

The maintained scope is a finite positive-support distribution, exact normalization and a consistent downward ordering with explicit tie handling. Zero-probability current states and countably infinite extensions are outside this initial operational scope.
Algorithm 5 contains printed missing brackets in its loop and index notation. The mathematical interpretation used here follows the surrounding equations and includes the equality completion condition. It is not a claim that literal author implementation code was executed or floating-point guards fully verified.
Curator calculations cover a bounded set of exact rational kernels. They do not certify an arbitrary floating-point implementation, execute author code or reproduce empirical benchmarks.
NAM, UNAM, DNAM and the UDNAM mixture have related but different scopes. No broad exact aliases or equivalence with all zero-self sampling methods are asserted. The source family and empirical literature remain incompletely assessed.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:MarkovChain` Markov Chain	related_match	The procedure constructs reversible conditional Markov transitions with a specified target. AIO MarkovChain denotes a stochastic process/model. The relation is method-to-process association without equivalence, subclass membership 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/ZeroSelfDownwardNestedAntitheticModification.yaml