Sublinear Poisson graphical model
Model Families AIMech:SublinearPoissonGraphicalModel DRAFT
Aliases: SPGM, Sublinear Poisson Markov random field
A finite-graph model for unbounded nonnegative integer counts that retains factorial base measures while replacing counts in unary and pairwise interactions by a specified bounded, saturating sufficient statistic.
Identity Notes & Concept Scope
SPGM denotes the particular sublinear Poisson graphical construction in Yang et al. Its sufficient statistic is linear for small counts, bends quadratically over an interval and is constant for large counts. Counts remain unbounded: the model saturates their interaction statistic rather than truncating their values. The full conditional is a transformed-statistic count distribution, not an ordinary Poisson law, a generic arbitrary sublinear exponential family or a neural activation layer. No normalized identity match was found across all 442 pinned AIO labels and all scoped synonyms and 406 preceding native IDs, names and aliases. Fixed-string full-checkout searches included ignored and hidden files in worktree and main, yielding six and six matched lines; all five unique contents were read. Hits are the prior publication lead and source URLs, not existing records of these identities. All 185 semantic label/synonym hits were read. Forty-five fully annotated exact CLASS nodes and outgoing edges were inspected: 43 byte-equivalent to prior inspected nodes and two freshly read. Fifty-nine contextual YAMLs comprise 45 generated and 14 native records: 54 byte-verified after prior reading and five freshly read. Acronyms were included in normalized identity comparison and the text-search terms. Absence is bounded to the pinned snapshot, maintained corpus and inspected semantic neighborhood.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
Choose finite 0<=R0<R and set B(x)=x for x<=R0; B(x)=-x^2/[2*(R-R0)]+R*x/(R-R0)-R0^2/[2*(R-R0)] for R0<x<=R; and B(x)=(R+R0)/2 for x>R. On a finite graph define p(x)=Z^(-1)*exp[sum_i a_i*B(x_i)+sum_(i<j) b_ij*B(x_i)*B(x_j)-sum_i log(x_i!)], with symmetric edge parameters. The site conditional is proportional to exp[eta_i*B(k)]/k!, where eta_i=a_i+sum_(j!=i) b_ij*B(x_j). Since B is globally bounded, all finite unary and edge parameters give a finite positive Z: bound the parameter-dependent factor by a constant and sum the product of reciprocal factorials. This argument includes zero unary parameters. The conditional log partition D(eta)=log sum_(k>=0) exp[eta*B(k)]/k! satisfies D(eta)<=exp(eta) when eta>=0 because B(k)<=k; this bound is not asserted for negative eta. Once both k and k+1 lie on the plateau, the successive conditional mass ratio is 1/(k+1).
⚠️ Curated Limitations & Negative Results
- Finite distinct thresholds with R0<R are required by the stated formula. B is bounded; a general unbounded function called sublinear need not inherit the same normalization result.
- Neighbor values above the plateau threshold have identical effects through B. This suppresses sensitivity to large-count differences and changes the conditional family; exp(eta_i) is not generally its conditional mean.
- The Poisson log-partition upper bound requires nonnegative eta. For negative eta the underlying termwise inequality reverses, so unconstrained regression updates cannot blindly use the same bound as a certificate.
- The supplement illustrates its normalization proof with two variables and a threshold construction that can fail when a unary coefficient is zero. The direct bounded-statistic domination argument stated here supplies the finite-graph justification without dividing by unary parameters.
- Exact conditional normalization still involves an infinite sum or a justified tail calculation. Existence for all finite parameters does not establish numerical accuracy, sampler mixing or recovery of the graph from finite data.
- The finite graph and the symmetric once-counted edge parameters must be specified. Independently fitted node regressions do not automatically agree on a compatible joint parameterization. Model definition, numerical estimation and graph selection are distinct.
- The source sketches l1-penalized neighborhood fitting, but explicitly leaves extensions of statistical graph-recovery guarantees to future work. A model existence result is not a sparsistency theorem or a guarantee that a chosen approximate optimizer finds the intended fit.
- The reported network experiments are contextual evidence, not a universal performance guarantee. The processed breast-cancer microRNA data and cited biological associations do not establish causal regulation or clinical predictive validity. No author implementation, empirical fit or benchmark was reproduced.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:Model
Model
|
broad_match | This is a specified joint probability family on integer-valued graph sites and falls within the broader mathematical representation described by Model. A particular optimizer or prediction task is not its identity. |
AIO:ProbabilisticGraphicalModel
Probabilistic Graphical Model
|
related_match | The family encodes undirected graphical conditional dependencies. The pinned target is task-scoped, so a related mapping connects this probability model to graphical modeling without asserting a model is-a task relationship. |
Evidence Sources & Primary Literature
2 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Eunho Yang, Pradeep Ravikumar, Genevera I. Allen, Zhandong Liu
|
primary paper
full_text
|
sha256:43358362c5a5…
|
supplement
|
Eunho Yang, Pradeep Ravikumar, Genevera I. Allen, Zhandong Liu
|
primary paper
full_text
|
sha256:38b42f678bb1…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T22:28:17Z
Updated: 2026-10-11T22:28:17Z
Source File:
data/curated/SublinearPoissonGraphicalModel.yaml