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Circular weight-shift sampler

Algorithms AIMech:CircularWeightShiftSampler DRAFT

A finite-state Markov transition construction that partitions a circle into intervals with target-weight lengths and derives each transition from the overlap of a positively shifted current-state interval with destination intervals.

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. The positive-weight circular interval-overlap construction has one consistent state order, an interior shift and explicit orientation. Maximum and half shifts are policies rather than separately maintained identities. Opposite Suwa and Neal orientations transpose generic flows, so no unqualified exact source-formula alias is adopted. Cumulative interval translation differs from flattened slice bars, nested antithetic focal modifications, retained pair/sign/uniform-memory updates and accept/reject proposal mechanisms. Paired UDST adds a fresh mixture-component choice and is curated separately; coincident kernels on some targets do not identify entire constructions.

Claims & Grounded Evidence

6 Claims

Mechanism #ordered-circular-overlap-kernel

Take finitely many strictly positive weights w_i in one specified state order, let W=sum_i w_i, F_i=sum_{j<=i} w_j and F_0=0, and choose 0<s<W. The target probabilities are w_i/W. Shift the source interval [F_{i-1},F_i) forward by s modulo W. Let v_ij be the total length of its intersection with destination interval [F_{j-1},F_j), and set P(i,j)=v_ij/w_i. Equivalently, with Delta=F_i-F_{j-1}+s, the flow is max(0,min(Delta,w_i+w_j-Delta,w_i,w_j)) plus max(0,min(Delta-W,w_i+w_j+W-Delta,w_i,w_j)). Sampling a fresh uniform point within the source interval, shifting it forward and reading its destination interval gives the same transition law. The half-open interval convention specifies boundaries for this mathematical construction.

supports Source: Reducing rejection exponentially improves Markov chain Monte Carlo sampling · Locator: Section III, PDF p. 3, Equations 10–12 and Figure 1; Appendix A, p. 8, interval-overlap derivation

Mechanism #balanced-flows-and-conditional-use

Translation around the circle preserves length. The destination intervals partition the circle, so sum_j v_ij=w_i; the shifted source intervals also partition it, so sum_i v_ij=w_j. The transition rows sum to one and w/W is invariant. The ordering and shift specify the same kernel for every current state of this target. For a full-conditional update they may depend on the fixed conditioning environment, but an arbitrary current-label-dependent choice is not this balanced construction. Updating a valid full conditional with this kernel preserves the joint target. Fixed mixtures and fixed sequential compositions of target-preserving updates preserve it as well; a composition need not be reversible.

supports Source: Reducing rejection exponentially improves Markov chain Monte Carlo sampling · Locator: Section III, PDF pp. 3–4, Equations 11–14
supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 2, PDF pp. 2–3, Equations 1–4 and update compositions

Scope #maximum-and-half-shift-policies

The shift parameter is part of the construction. With at least two positive-weight states, s=max_i w_i gives the Suwa–Todo maximum-weight policy in Suwa’s source-shift orientation. The choice s=W/2 is the half-shift policy and has symmetric flows, hence detailed balance. Both policies attain stationary self-transition probability max(0,2*max_i(w_i)/W-1). If max_i(w_i)<=W/2, every shift in [max_i(w_i),W-max_i(w_i)] has zero diagonal. When one weight exceeds W/2, some self probability is unavoidable. Arbitrary interior shifts need not minimize self transitions. ST and HST name specific policies in Neal’s shifted-tower presentation, not exact aliases for the entire parameterized construction.

supports Source: Reducing rejection exponentially improves Markov chain Monte Carlo sampling · Locator: Section III, PDF pp. 3–4, Figure 1(c), maximum and half-shift discussion
supports Source: Markov Chain Monte Carlo Method without Detailed Balance · Locator: PDF pp. 2–3, allocation construction and Equations 5–10
supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 10, PDF pp. 36–40, ST/HST definitions and Figure 7

Scope #orientation-order-and-mixture-boundaries

This record shifts the source interval by +s. Neal’s Equation 98 and Algorithm 7 instead move a source point by -s, equivalently shifting the destination tower forward. For the same order and shift their flow matrices are transposes, generally giving different kernels; at half shift they agree. For weights (2/5,3/10,1/10,1/5) and s=2/5, the positive-shift first row is (0,3/4,1/4,0), whereas Neal’s is (0,1/4,1/4,1/2). Suwa’s formula agrees with Suwa’s stated source geometry, so Neal’s footnote 18 allegation of a reversed-order formula error is not adopted. UST, DST and OHST specify ordering policies within the tower framework, including tie choices. Neal’s UDST explicitly mixes paired reverse UST and DST kernels equally; that mixture is outside this single-order, single-shift identity and is a separate mixture construction. No general equivalence of that mixture to one circle shift is asserted.

supports Source: Reducing rejection exponentially improves Markov chain Monte Carlo sampling · Locator: Section III, PDF p. 3, Figure 1 and Equations 12–13
supports Source: Reducing rejection exponentially improves Markov chain Monte Carlo sampling · Locator: Section III, PDF p. 3, Equations 11–12; Appendix A, p. 8
supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 10, PDF pp. 37–41, footnote 18, Equations 98–104, Figures 7–9 and Algorithm 7; original orientation comparison in research/discovery/tower-boundaries.py

Limitation #reversibility-and-exploration-limits

Half shift is sufficient for reversibility, but non-half shifts are not necessarily nonreversible. For weights (3/5,2/5) and s=1/10, the matrix has rows (5/6,1/6) and (1/4,3/4), with equal opposing stationary flows. This qualifies the source’s blanket non-half nonreversibility statement. Invariance and minimum self probability also do not establish irreducibility or aperiodicity. Half shift on four equally weighted states gives two disjoint deterministic two-cycles; maximum shift on three equally weighted states gives a deterministic three-cycle. No universal variance, burn-in, finite-run accuracy or runtime advantage is claimed.

supports Source: Reducing rejection exponentially improves Markov chain Monte Carlo sampling · Locator: Section III, PDF p. 4, half-shift and non-half reversibility discussion
supports Source: Markov Chain Monte Carlo Method without Detailed Balance · Locator: PDF p. 3, ergodicity caveat after Equation 10
supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 13, PDF p. 49, uniform-target comparisons; original non-half and periodicity calculations in research/discovery/tower-boundaries.py

Limitation #printed-sampler-boundaries

The overlap formula defines the maintained transition. Neal’s Algorithm 6 dominant-current shortcut is valid for its minimum-self ST/HST choices but not for arbitrary shifts: with weights (3/5,2/5), current state 1 and s=1/10, it returns (1/3,2/3) instead of the overlap row (5/6,1/6). Algorithm 7 also does not satisfy its printed nonzero-output guarantee at every closed-uniform endpoint. With weights (1/5,3/10,1/2), current state 1, half shift and r=0, its printed traversal returns state 2 although the row is (0,0,1). This endpoint has probability zero under an ideal continuous uniform law and does not invalidate that law. These are explicit mathematical counterexamples, not an author-issued correction or a claim to have executed author software.

supports Source: Modifying Gibbs Sampling to Avoid Self Transitions · Locator: Section 10, PDF pp. 39–41, Algorithms 6–7 and shortcut discussion; original shortcut and endpoint calculations in research/discovery/tower-boundaries.py

⚠️ Curated Limitations & Negative Results

The maintained scope uses exact finite strictly positive weights, one consistent state order and an interior shift. A one-state distribution stays under every interior shift; its maximum-weight shift equals W and lies outside that interior parameter domain. Zero-weight current rows require a separate completion and are not specified here.
The descriptive record label covers Suwa’s parameterized overlap construction. It does not make the narrower ST/HST policies, ordered specializations or the UDST mixture broad exact aliases. Agreement with another sampler on a particular target does not establish identity of the full constructions.
The half-open interval convention and exact flow formula are mathematical specifications, not a verified finite-precision implementation. Countably infinite extensions, adaptive order selection and author code are outside this maintained scope.
Pinned arXiv and author-report versions are used with their distinct source conventions and caveats retained. No publisher-PDF equivalence, full Neal 84-page empirical review, benchmark reproduction or source-family exhaustion is claimed.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:MarkovChain` Markov Chain	related_match	This procedure constructs target-preserving finite Markov transitions. AIO MarkovChain denotes a 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

4 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`suwa2023v2`	Reducing rejection exponentially improves Markov chain Monte Carlo sampling Hidemaro Suwa	primary paper full_text	ID: arxiv:2208.03935v2 sha256:c02ebda8a10c…
`suwa2022v1`	Reducing rejection exponentially improves Markov chain Monte Carlo sampling Hidemaro Suwa	primary paper full_text	ID: arxiv:2208.03935v1 sha256:d0fcde405016…
`suwa-todo2010`	Markov Chain Monte Carlo Method without Detailed Balance Hidemaro Suwa, Synge Todo	primary paper full_text	ID: arxiv:1007.2262v2 sha256:08cb22df3aa5…
`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/CircularWeightShiftSampler.yaml