Quadratic Poisson graphical model
Model Families AIMech:QuadraticPoissonGraphicalModel DRAFT
Aliases: QPGM, Quadratic Poisson Markov random field
A graphical probability model on nonnegative integer counts whose joint log mass contains unary linear terms, site quadratic terms and symmetric pairwise count products, with parameters restricted to make its infinite-support partition function finite.
Identity Notes & Concept Scope
The QPGM name follows Yang et al., but its conditional laws are quadratic exponential count distributions, not ordinary Poisson laws. The model replaces the Poisson factorial penalty with quadratic site terms. It is not the continuous auto-normal field, nor is assigning quadratic masses at integer points equivalent to rounding a continuous Gaussian. A Gaussian integral approximation is a separate numerical proposal and does not define the discrete model. 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
On a finite graph with X_i in {0,1,...}, define p(x)=Z^(-1)*exp[sum_i a_i*x_i+sum_i d_i*x_i^2+sum_(i<j) b_ij*x_i*x_j], with symmetric edge coefficients and finite Z. The site conditional is proportional to exp(d_i*k^2+eta_i*k), k>=0, where eta_i=a_i+sum_(j!=i) b_ij*x_j. There is no factorial base measure in this construction. Define the symmetric matrix H with H_ii=d_i and H_ij=b_ij/2 for i!=j, so the quadratic contribution is x^T H x without double counting. A sufficient validity condition is x^T H x<=-c*||x||^2 for some c>0 on the nonnegative orthant; finite linear terms are then dominated by quadratic decay. Negative definiteness of H on the whole real space is one sufficient condition and permits either sign of the off-diagonal interactions. It is not asserted to be a necessary characterization of all admissible parameters.
⚠️ Curated Limitations & Negative Results
- Individually summable site quadratics do not alone ensure a finite joint partition function when positive interactions create growing joint directions. The joint validity condition must be checked; flat quadratic boundary cases require separate linear-tail analysis.
- The source claims that a full-real Gaussian integral universally upper-bounds the nonnegative-integer quadratic kernel sum. This inequality fails, for example, with d=-100 and eta=0 or eta=1: the discrete sum includes a unit term at zero, but the Gaussian integral is less than 0.18. This is an original mathematical counterexample to the approximation claim, not a failure of the discrete model.
- Consequently, Gaussian-regression behavior or optimization guarantees cannot be inferred from that claimed bound without an additional justified approximation. Integer support, normalization, moments and fitted parameters must be treated as those of the discrete law.
- The source numerical plots compare TPGM and SPGM with other estimators; they do not supply a general empirical evaluation of QPGM. The record makes no generic accuracy claim.
- 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/QuadraticPoissonGraphicalModel.yaml