Auto-Poisson Markov random field
Model Families AIMech:AutoPoissonMarkovRandomField DRAFT
Aliases: Auto-Poisson model, Auto-Poisson scheme
A finite pairwise Markov-field model for unbounded nonnegative integer counts whose full conditionals are Poisson with log means linear in neighboring counts and whose symmetric interaction coefficients are nonpositive.
Identity Notes & Concept Scope
This record denotes the classical unbounded auto-Poisson joint model. A regression using neighboring counts can resemble its site formula while failing joint compatibility or normalizability. Truncated, Winsorized, transformed-statistic and other modified count graphical models require separate identities; the broad phrase Poisson graphical model is therefore not used as an exact alias. Poisson refers to each full conditional, not necessarily each marginal distribution. A finite count field is also distinct from a Poisson point process with random locations. No normalized identity match was found across all 442 pinned AIO labels and all scoped synonyms and 403 preceding native IDs, names and aliases. Fixed-string full-checkout searches included ignored and hidden files in worktree and main, yielding 37 and 37 matched lines; all 20 unique contents were read. Hits are citations, contextual statements and discovery leads rather than records of these identities. All 185 semantic label/synonym hits were read. Forty-three fully annotated exact CLASS nodes and outgoing edges were inspected: 38 byte-equivalent to prior inspected nodes and five freshly read. Fifty-four contextual YAMLs comprise 43 generated and 11 native records: 46 byte-verified after prior reading and eight freshly read. Absence is bounded to the pinned snapshot, maintained corpus and inspected semantic neighborhood.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
For finite real alpha_i and symmetric beta_ij=beta_ji, the full conditional is X_i|X_(-i)~Poisson(mu_i), where mu_i=exp[alpha_i+sum_(j!=i) beta_ij*x_j]. The compatible pairwise mass is p(x)=Z^(-1)*exp[sum_i(alpha_i*x_i-log(x_i!))+sum_(i<j) beta_ij*x_i*x_j] on all nonnegative integer configurations, with zero coefficients on nonedges. For this unrestricted Cartesian support and finite site set, Z is finite if and only if every off-diagonal beta_ij is nonpositive. Sufficiency follows by bounding the interaction factor by one and summing the independent Poisson kernels. A positive coefficient fails summability along a two-site growing-count slice even though every proposed site Poisson conditional is individually normalized. With valid beta_ij, increasing a neighbor count cannot increase the site conditional mean when other neighbors are held fixed; zero interactions yield independent Poisson variables.
⚠️ Curated Limitations & Negative Results
- The nonpositive-interaction restriction concerns this unbounded pairwise model and its conditional dependence. It is not a theorem that every marginal pairwise correlation in every count graphical model must be negative.
- A finite cutoff makes the state space summable but changes the support and site conditionals to truncated laws. A successful truncated calculation or finite Gibbs run cannot certify the original unbounded positive-interaction model.
- The factorial base measure is part of the model. Omitting it changes conditional count laws and their existence conditions.
- The source presents a formal validity restriction, not a generic accuracy or efficiency result for ecological or genomic count prediction. Later modified Poisson families are research leads, not silently absorbed into this record.
- The graph, site supports and any conditioning boundary values must be specified. The construction assumes compatible positive full conditionals on the stated Cartesian support; arbitrary separately fitted regressions or asymmetric neighbor coefficients do not automatically define a joint model.
- The finite fixed-site model does not by itself specify a random-location point process, temporal causal mechanism, hidden observation layer or infinite-lattice existence theorem. Conditional association does not establish causation.
- A closed-form site conditional does not make the joint partition function, maximum-likelihood fit or uncertainty calculation trivial. Estimation, sampling and model diagnostics require their own procedures. 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 | The finite joint probability family is a mathematical representation within the broader Model class. Its defining support and compatible full conditionals do not require a particular regression task or neural architecture. |
AIO:ProbabilisticGraphicalModel
Probabilistic Graphical Model
|
related_match | The field represents undirected graphical conditional dependence. The pinned AIO target is task-scoped, so related_match connects the model family to that graphical-modeling task without asserting model is-a task. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Julian Besag
|
primary paper
full_text
|
ID: doi:10.1111/j.2517-6161.1974.tb00999.x
sha256:796199ef5d87…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T22:15:55Z
Updated: 2026-10-11T22:15:55Z
Source File:
data/curated/AutoPoissonMarkovRandomField.yaml