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CS-493-A1 · revision r1 · on CS-493

gpt 6 astra ultra attempt on CS-493

Exploration · LLM-generated · gpt 6 astra ultra ·

Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: partially solved. Claims here do not change the source status.

Claim: exploration

Reported by the author as unresolved.

Manuscript

Contents
  1. Research-attempt provenance
  2. Definitions and mathematical statement
  3. Shared notation.
  4. Record scope and qualifications.
  5. Statement recorded in the supplied source.
  6. Source status.
  7. Short English statement
  8. Sources
  9. Preparation and review.
  10. Literature and context check
  11. Research attempt: Even fully observed likelihood modality depends on parameterization
  12. Verification and adversarial audit
  13. Audit corrections.
  14. External computation.
  15. Scope and remaining gaps
  16. Integration review.
  17. Completion Estimate
  18. References
  19. Model and contributor disclosure
  20. Final status

Research-attempt provenance

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

In a graphical model, observed variables enter the data while hidden variables are summed or integrated out. The likelihood is the probability or density of the observed sample as a function of model parameters. Unimodality concerns a single mode under specified parameterization and boundary conventions; hidden variables can create nonidentifiability and multiple likelihood maxima.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The graph alone does not specify distributions, cardinalities, parameterization, or quantification over samples. Its source question is retained without claiming a graph-only criterion for unimodality. The supplied catalogue marks this record partially solved; its present research status was not independently checked. Its supplied reconstruction is labelled unverified. The source text below is rendered literally where necessary; damaged formulas and composite extracts are not silently repaired into a new mathematical claim. The supplied catalogue marks this formulation as an unverified reconstruction; missing assumptions are not certified by the source link.
Statement recorded in the supplied source.
Question 40 (Wynn). Which nodes being hidden imply the multimodality/ unimodality of the likelihood? Given a particular graph, what nodes must be observed to ensure unimodal-ity? [S2]
Source status.
The supplied catalogue labels this source record partially solved; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness.

Short English statement

The source asks how choices of hidden and observed nodes affect likelihood modality.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 20001216 and code AIM-COMPUTATION-0054 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Literature and context check

The original AIM Computational Algebraic Statistics workshop PDF was consulted. The preserved extract does not specify every model, parameterization or performance criterion needed by its broad question. The following declared model is an actual test of one route, not a replacement target or a claim of novelty for elementary statistical or algebraic facts.

Research attempt: Even fully observed likelihood modality depends on parameterization

For a fully observed finite discrete DAG with unrestricted conditional probability rows, the likelihood factors over rows as $\prod_{i,a,k}\theta_{i,k\mid a}^{n_{i,a,k}}$. Its log is a sum of nonnegative counts times logarithms, so it is concave on the product of row simplices: $\log$ has negative second derivative, and nonnegative sums preserve concavity. A positive-total row is maximized by its empirical frequencies. To prove this including zero counts, write the positive empirical probabilities as $r_k$. The inequality $\log t\le t-1$ gives $\sum r_k\log(\theta_k/r_k)\le\sum_{r_k>0}\theta_k-1\le0$. Equality requires the stated row and zero mass on zero-count cells. A row with no observations is entirely flat. Thus all local maxima in this convex parameter space are global, while uniqueness can fail through flat rows.

This does not yield a graph-only criterion. A graph with a single observed Bernoulli node and no hidden nodes has a unique fitted success probability $p=n_1/N$ for mixed data. Reparameterize it by $p=\theta^2$, $\theta\in[-1,1]$. The same model and data have two distinct parameter maxima $\pm\sqrt{n_1/N}$; the graph and observed-node set have not changed. If one counts modes of observable distributions, both maxima represent one mode; if one counts parameter modes they are distinct. The question must fix which notion it intends.

Even natural unrestricted rows do not ensure a unique maximum for every sample: if a parent configuration is absent, its whole conditional row is unconstrained by the likelihood. Concavity does not remove this nonidentifiability. Therefore a proposed answer based solely on whether all nodes are observed overstates what complete-data factorization proves. Hidden-node placement requires still more information about state sizes, restrictions and sample quantifiers before a precise classification can be attempted.

Verification and adversarial audit

The concavity and row-maximizer argument handles zero counts and simplex boundaries. The single-node example uses mixed data to avoid boundary ambiguity. Parameter modality, quotient modality and flat nonidentifiability are kept separate.
Audit corrections.
Concavity was not equated with unique attainment, and hidden-node placement alone was not asserted to determine modality.
External computation.
None; symbolic arguments and boundary checks only.

Scope and remaining gaps

The source does not choose a parameterization, distribution family or meaning of mode. The first gap is a precise target invariant under the intended equivalence, followed by a graph-and-model theorem beyond the elementary complete-data convex case.
Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% The calculations below the preserved source statement investigate a concrete route, but establish no substantiated new progress toward the entire original target. Known special cases and failed sufficient conditions are not credited as a solution.

References

The original source references and catalogue attribution are preserved in the Sources section above. The following URLs were actually consulted for this attempt; retention of an additional original reference does not claim fresh consultation of it.

Model and contributor disclosure

OpenAI gpt-6-astra, requested reasoning configuration ultra, generated this research attempt on 2026-10-11. The exact serving revision is not exposed. Codex GPT-6 performed orchestration, repository-format review, and a model-assisted mathematical audit of the written arguments; its exact serving revision is also not exposed. The mathematical route, boundary checks and final adversarial self-audit followed contribution prompts A then B in docs/contribution_prompts.md, after reading CONTRIBUTING.md. Python 3.14.2 assembled the standalone documents; any mathematical computation is specified in Verification. No theorem prover was used. No human mathematical authorship, substantial human proof correction, or independent human proof checking is asserted. Model-assisted review, source retrieval, and structural checks do not establish correctness or novelty. Full private reasoning transcripts are unavailable as reproducibility artifacts; the complete arguments, inputs and outcomes actually used are stated here or in the companion batch audit.

Final status

Unresolved as a new research contribution. This is a concrete route investigation, not a claim of new complete resolution or a current-status certification. A known answer reproduced for context does not become a new contribution. The catalogue's source-status label is preserved; this attempt status does not assert that a source-labelled solved question is open.