Stabilization of Hidden Alphabet Distribution Models

Definitions and mathematical statement

Interpret the hidden-alphabet size as an integer $d\in{\mathbb{Z}}_{\geq 1}$. The record describes classes $C_d$ of observable probability distributions in a common, unspecified ambient space of probability laws, satisfying \[ C_d\subseteq C_{d'}\qquad\text{whenever }1\leq d\leq d'. \] The hidden-variable model, its parameter constraints, the observable sample space, and the map from hidden parameters to observable distributions are not supplied. Thus the notation $C_d$ specifies only an abstract nested family, rather than a defined statistical model.

The stated stabilization question is whether, for the intended family, \[ \exists d_0\in{\mathbb{Z}}_{\geq 1}\;\forall d'\in{\mathbb{Z}}_{\geq 1},\qquad d'\geq d_0\Longrightarrow C_{d'}=C_{d_0}. \] Equivalently, the question asks whether $\bigcup_{d\geq 1}C_d=C_{d_0}$ for some finite $d_0$. Any dependence of $d_0$ on observable alphabets or other fixed model data cannot be specified from the record.

The record also asks whether the question simplifies, or stabilization holds, for distributions of three observed variables invariant under cyclic permutations or all permutations, particularly without increasing the hidden alphabet. If coordinate permutations act on the intended observable sample space, write $g_*\mu$ for the law obtained by permuting coordinates by $g$. For \[ G=\langle(123)\rangle\quad\text{or}\quad G=S_3, \qquad C_d^G=\{\mu\in C_d:g_*\mu=\mu\text{ for every }g\in G\}. \] Under the interpretation that symmetry restricts only the observable distribution, a precise possible stabilization question is, separately for each of these groups, \[ \exists d_G\in{\mathbb{Z}}_{\geq 1}\;\forall d'\geq d_G, \qquad C_{d'}^G=C_{d_G}^G. \] This is a conditional interpretation, not a uniquely specified target in the source record. The observable domains needed for coordinate permutations are unspecified. Moreover, the phrase "without increasing the hidden alphabet" does not specify whether a hidden representation must itself respect the symmetry at its original alphabet size, or whether it refers only to stabilization of the invariant observable classes. No symmetric hidden-parameter model is defined. These missing specifications prevent a complete model-specific formulation. Closure is mentioned in the supplied literature assessment, but no closure question or topology is supplied.

Short English statement

The problem asks whether increasing the size of a hidden alphabet eventually stops producing new observable distributions, and whether cyclic or full permutation symmetry of three observed variables helps achieve this without enlarging the hidden alphabet. The underlying model and the intended symmetry requirement are missing, so only an abstract stabilization question and a conditional symmetry formulation can be stated precisely.

Sources

• Local UnsolvedMath registry, permanent ID $30003482$, title Stabilization of Hidden Alphabet Distribution Models, source problem code catalogue entry; supplied complete source record. The project identifies data/unsolvedmath/problems.json and catalog/problems.json as registry files; membership in either file was not independently checked.
• Supplied report citation: Algebraic Statistics (2018), Oberwolfach Reports, pp. 1207--1279, DOI 10.4171/owr/2017/20. Source URL and supplied literature source labeled "original OWR report": https://doi.org/10.4171/owr/2017/20. The background attributes the collection to mathematical workshop open-problem discussions documented in Oberwolfach Reports. No individual proposer is supplied.
• The registry gives proposed year $2017$ and tag owr-year:2017, while the supplied report citation labels the report $2018$; both are retained without resolving their relationship.
• Literature checked at: 2026-08-21. This is supplied dated curation, not an independently verified assessment. The supplied assessment labels the problem open and states that stabilization, closure, and symmetry questions remain unresolved in the stated generality, that the report presents them as open problems, and that no later general theorem was identified. The background expressly calls this dated literature triage and requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified; its accompanying explanation says that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This records the supplied editorial assessment and does not establish independent mathematical verification of this draft.