Home / Curated / Algorithms / Upward-downward nested antithetic mixture

Upward-downward nested antithetic mixture

Algorithms AIMech:UpwardDownwardNestedAntitheticMixture DRAFT

A finite conditional Markov update that chooses with equal probability between complete upward and downward nested antithetic modification kernels for the same target distribution.

Identity Notes & Concept Scope The actual post317 assessment covers all 442 pinned AIO and 710 native IDs, names, aliases and every scoped synonym, plus four complete checkout searches including ignored and hidden files. Historical 706-native source research, 334 selected ID/name rows, 103 selected definitions, 59 additional mixture definitions, 22 exact AIO node/incident/generated contexts and 18 full native neighbors retain their attributed reading scope. The four actual FSS/ZFSS and NAM/ZDNAM records are freshly assessed. AIO:MarkovChain is related_match between a transition construction and a stochastic process; no equivalent or subclass mapping or neural-layer inheritance is asserted. No aliases adopted. UDNAM is a constant equal choice between complete upward and downward NAM kernels, with consistent independently selected valid tie policies. A mixture is not a single supplied-order NAM modification, sequential U-then-D composition or state-dependent component selection. The finite (4,3,2)/9 two-positive-diagonal witness excludes every single NAM ordering. UNAM tie invariance and DNAM tie sensitivity are distinguished. Shared local reversible kernels or special cases with other samplers do not establish identity.

Claims & Grounded Evidence

6 Claims

Mechanism #complete-upward-and-downward-components

Let pi be a strictly positive normalized probability mass function on a finite state space. Construct UNAM from a nondecreasing probability order and DNAM from a nonincreasing order. Each is a complete NAM construction starting with K(i,j)=pi(j). At a stage whose remaining block has K(i,j)=r*pi(j), let q be its focal-state mass and s the nonempty tail mass. Set delta=r*min(q/s,s/q). Increase focal-to-tail entries by delta*pi(j) and tail-to-focal entries by delta*q; decrease the focal diagonal by delta*s and tail-to-tail entries by delta*pi(j)*q/s. Previously determined entries are unchanged. Continue on the tail if q<s; if q>=s its block becomes zero and the construction stops. A one-state target already has the identity kernel. The component is not an incomplete first-stage modification.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 5, PDF p. 10, Equation 19; Section 6, pp. 14–17, Equations 28–34 and Algorithm 2; Sections 7–8, pp. 22–29, upward/downward orders and Algorithms 3–4

Mechanism #constant-half-mixture-selection

Write U and D for the two completed component matrices. UDNAM uses M(i,j)=(U(i,j)+D(i,j))/2. It can sample a fresh fair component choice for each local update and then sample that component row, or sample directly from the averaged row. The half weights do not depend on the current state label. Selecting a component once for a whole chain, applying U followed by D, or using current-state-dependent component weights does not define this specified transition matrix. Both component ordering rules define all rows for the same target.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 8, PDF p. 28, equal random choice and averaged transition probabilities; Section 2, pp. 2–3, distinction between mixtures and compositions

Mechanism #reversible-mixture-and-conditional-use

Both U and D are nonnegative stochastic matrices satisfying detailed balance with pi. Their arithmetic average therefore has row sums one and satisfies pi(i)*M(i,j)=pi(j)*M(j,i), so it preserves pi. Used for a valid full conditional with other coordinates fixed, the update preserves the joint target. The probability ordering may change when that conditioning environment changes. Fixed mixtures and fixed sequential compositions of valid conditional updates preserve the joint target too; the local reversibility statement does not imply that a full sequential sweep is reversible.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Sections 5–6, PDF pp. 10 and 14–20, balance-preserving antithetic construction; Section 8, p. 28, mixture; Section 2, pp. 2–3, Equations 1–4 and composition

Scope #downward-tie-policy

UNAM is unchanged by permutations within equal-probability groups, whereas DNAM can depend on its tie order. UDNAM inherits this dependence and requires a stated consistent downward tie policy. For pi=(1,3,3,5)/12, the original exact calculations use upward orders (1,2,3,4) and (1,3,2,4), with corresponding downward orders (4,3,2,1) and (4,2,3,1), in one-based labels. The UNAM kernel is unchanged, but the UDNAM diagonals are respectively (0,1/42,0,5/44) and (0,0,1/42,5/44). Both mixtures remain reversible. Unlike the paired-tower mixture, UDNAM does not need its two component orders to be exact reversals for reversibility, because each component is already reversible.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 7, PDF p. 24, Equation 75 and tie invariance; Section 8, pp. 28–29, tie discussion and Figure 6(a); original exact tie comparison in research/discovery/mixture-boundaries.py

Scope #distinct-mixture-and-self-mass

The equal mixture is not generally one NAM focal ordering. At pi=(4,3,2)/9, its diagonal is (5/56,1/30,0), with two positive entries, whereas a complete single-order NAM construction has at most one. Original exact calculations confirm the difference for all six focal permutations on this target, retaining duplicate resulting matrices in that enumeration. UDNAM also need not minimize self transitions: this target has no mass exceeding one half, yet its mixture has positive stationary self mass. Agreement with UNAM, DNAM or other samplers on special targets does not identify the full constructions.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 6, PDF pp. 14–20, completed NAM diagonal structure; Section 8, pp. 28–30, mixture and ordering limits; Section 13, pp. 49–51, simple-target comparisons; original six-order calculation in research/discovery/mixture-boundaries.py

Limitation #invariance-is-not-convergence

For a uniform binary target, both components and their mixture flip the state deterministically. For two independent uniform binary coordinates, a fixed full scan maps (a,b) to (1-a,1-b), so it has two closed classes even though the joint uniform target is invariant. Thus a reversible local mixture and reduced self transitions do not establish aperiodicity, joint exploration or convergence from every initial state. The source discusses efficiency comparisons under additional assumptions; this record makes no general variance, spectral-efficiency, burn-in, finite-run accuracy or runtime claim.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 5, PDF pp. 13–14 and footnotes 7–8, scan boundaries; Section 8, p. 28, scoped efficiency discussion; Section 13, p. 49, uniform-target behavior; original scan checks in research/discovery/mixture-boundaries.py

⚠️ Curated Limitations & Negative Results

The maintained scope is a finite strictly positive normalized target and consistent complete component kernels. Zero-current conventions, approximate normalization and countably infinite upward ordering are outside this scope.
The source calls this equal mixture UDNAM. Fixed/upward/downward NAM policies, ZDNAM completion, UDST and arbitrary mixtures of invariant kernels are not exact aliases for it.
The original FHY1992 full text was not obtained. Historical attribution of the UNAM component does not identify every earlier estimator or truncated modification with the full UDNAM mixture.
Original exact curator calculations support bounded identities and counterexamples. No author code, numerical implementation, empirical benchmark or full 84-page performance review is claimed.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:MarkovChain` Markov Chain	related_match	The algorithm constructs target-preserving finite Markov transitions. AIO MarkovChain denotes the stochastic process/model; this method-to-process association does not assert 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: 2026-10-14T23:04:24Z

Updated: 2026-10-14T23:04:24Z

Source File: data/curated/UpwardDownwardNestedAntitheticMixture.yaml