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Upward-downward shifted tower mixture

Algorithms AIMech:UpwardDownwardShiftedTowerMixture DRAFT

A finite conditional Markov update that equally mixes maximum-shift tower kernels built from paired upward and downward cyclic state orders, making their stationary flows symmetric.

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. UDST is a constant equal choice between maximum-shift components using the same maximum anchor and exactly reversed tails, including tie ordering. Its paired flows transpose, while independently weak-sorted ties need not. It adds a component choice absent from a single circular shift. The uniform-four-state witness excludes every single cyclic order and shift by exact certificates and a separate analytic argument. It differs from UDNAM and HST as constructions despite possible special-case coincidences; no claim that all resulting kernel sets are disjoint.

Claims & Grounded Evidence

6 Claims

Mechanism #paired-orders-and-maximum-shift

Let pi be a strictly positive normalized probability mass function on a finite state space. Choose one maximum-probability anchor a and an order of all other states with nondecreasing probability, including an explicit tie policy. Use the cyclic order (a,tail) for UST and (a,reverse(tail)) for DST, preserving the same anchor. Both components use shift h=max(pi). For at least two positive states, 0<h<1; a one-state target is handled separately by staying. The component orders must define every row for the same target, not be chosen independently from the current state label.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 10, PDF p. 37, upward/downward ordering definitions and Equation 101; p. 40, Figure 9 and maximum-shift policy

Mechanism #component-overlap-and-half-mixture

For either component order, let C_i be the total pi mass before state i. Following Neal’s orientation, move a point sampled uniformly in [C_i,C_i+pi(i)) by -h modulo one and select its destination interval. Equivalently, let d=pi(i)-h+C_i-C_j and g(z)=max(0,min(z,pi(i)+pi(j)-z,pi(i),pi(j))). The flow from i to j is v_ij=g(d)+g(d+1), and its transition probability is v_ij/pi(i). If U and D denote the resulting matrices, UDST uses M=(U+D)/2. Draw a fresh fair component choice for each local update or sample the averaged row directly. A sequential U-then-D product, a once-per-chain choice or current-label-dependent component weights is not this constant half mixture.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 10, PDF pp. 37–41, Equations 98–104 and Algorithm 7; Figure 9 and the UDST definition on p. 40

Mechanism #symmetric-flow-and-minimum-self-mass

The paired cyclic reversal gives pi(i)*U(i,j)=pi(j)*D(j,i). Averaging the two component flows therefore gives pi(i)*M(i,j)=pi(j)*M(j,i): the mixture is stochastic, reversible and preserves pi. Both maximum-shift components attain the minimum stationary self-transition mass max(0,2*h-1), so their mixture does too. If h>1/2, only the dominant state has positive self probability, (2*h-1)/h; every other state moves to it with probability one and its outgoing probabilities to them are pi(j)/h. At h=1/2 the self probability is zero. These statements concern the positive-support local transition kernel.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 10, PDF pp. 37 and 40, Equations 101–104 and minimum-self discussion; Section 13, p. 51, Equations 112–114

Scope #tie-pairing-and-orientation-qualification

Independent weak sorting is insufficient for the reversal relation when probabilities tie. For a uniform three-state target, choosing order (1,2,3) for both UST and DST satisfies both weak probability orders but makes both components the same directed cycle. Their average is invariant but not reversible. Pairing (1,2,3) with (1,3,2) gives the intended reversible mixture. Tie pairing is therefore an explicit qualification of the source’s reversal argument. Changing the common shift orientation transposes both component flows; for the correctly paired mixture they exchange, leaving its averaged kernel unchanged. This does not identify the unaveraged fixed-order component kernels.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 10, PDF p. 37, tie discussion and Equation 101; p. 40, Figure 9 and Equations 102–104; original tie and orientation checks in research/discovery/mixture-boundaries.py

Scope #distinct-mixture-construction

For four equally probable states in cyclic order (1,2,3,4), UDST moves to the two neighbors with probability one half each. Its first row is (0,1/2,0,1/2), while same-order HST moves deterministically to the opposite state and UDNAM has first row (0,1/3,1/3,1/3). This UDST kernel cannot be one circular weight shift in any order: for equal quarter-length intervals, a single shift gives either one cyclic offset or two consecutive offsets. Two half-probability outputs require a consecutive pair that is not closed under negation modulo four, contrary to reversibility. Original exact calculations also exclude all shifts through 96 closed affine segments across all 24 orders. This one-target distinction does not deny agreement on other targets, including dominant-mass cases.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 10, PDF pp. 37–40, component geometry and mixture; Section 13, pp. 49–51, uniform and dominant-mass comparisons; original all-order certificate in research/discovery/mixture-boundaries.py

Limitation #conditional-scan-and-convergence-limits

Using the kernel for a valid full conditional with other coordinates fixed preserves the joint target. Fixed mixtures and fixed sweeps of such updates preserve it too, although a full sweep need not be reversible. Uniform even-state UDST is a period-two random walk on a cycle. For two independent uniform binary coordinates, a fixed full scan maps (a,b) to (1-a,1-b) and has two closed classes. Thus minimum self mass and local reversibility do not establish aperiodicity, joint exploration or convergence from every initial state. No universal variance, spectral-efficiency, burn-in, finite-run accuracy or runtime advantage is claimed.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 2, PDF pp. 2–3, conditional invariance and composition; Section 5, pp. 13–14 and footnotes 7–8; Section 13, pp. 49–51, uniform-cycle comparisons; original scan checks in research/discovery/mixture-boundaries.py

⚠️ Curated Limitations & Negative Results

The maintained scope uses finite strictly positive normalized probabilities and exact paired orders, including ties. Zero-current completions, approximate normalization and adaptive ordering are outside this scope.
The source calls this equal mixture UDST. Generic circular shifting, the individual ST/UST/DST/HST/OHST policies and arbitrary mixtures of invariant kernels are not exact aliases for this construction.
Figure 9’s middle caption uses a DST subscript while its reversed construction is identified as UST. The paired geometry and Equations 101–104 support the maintained relation. No author-issued correction is claimed.
The exact overlap specification does not certify the printed direct sampler at closed-uniform endpoints or floating-point boundaries. Original curator checks and source lineage are retained; author software and empirical benchmarks were not executed.

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 a stochastic process/model; this 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/UpwardDownwardShiftedTowerMixture.yaml