Auto-normal Markov random field
Model Families AIMech:AutoNormalMarkovRandomField DRAFT
Aliases: Auto-normal model, Auto-normal scheme, Proper Gaussian conditional autoregressive model
A proper finite Gaussian Markov-field model specified by linear full-conditional means and positive, value-independent site conditional variances, with compatibility enforced by a symmetric positive-definite joint precision matrix.
Identity Notes & Concept Scope
The record is scoped to the proper finite auto-normal construction. It is a Gaussian conditional-autoregressive representation of a multivariate normal law with a specified conditional-dependence graph. Intrinsic improper Gaussian fields and infinite-lattice extensions require separate existence or constraint conditions and are not exact aliases here. Spatial conditional autoregression is not temporal autoregression and is not the simultaneous autoregressive model obtained by treating the same local residual equations as independent noise equations. Gaussian-process regression conditions a process prior on training observations for prediction; this record specifies the finite graphical joint law itself. It can be used as a prior but does not inherently introduce hidden labels or independent observation emissions. No normalized identity match was found across all 442 pinned AIO labels and all scoped synonyms and 400 preceding native IDs, names and aliases. Full-checkout fixed-string searches included ignored and hidden files in worktree and main, yielding 35 and 35 matched lines; all 23 unique contents were read. Hits are citations, contextual statements and discovery leads rather than records of these identities. All 182 semantic label/synonym hits were read. Thirty-one fully annotated exact CLASS nodes and outgoing edges were inspected: 23 byte-equivalent to prior inspected nodes and eight freshly read. Thirty-nine contextual YAMLs comprise 31 generated and eight native records: 29 byte-verified after prior reading and 10 freshly read. Short acronyms were included in the normalized identity comparison. Absence is bounded to the pinned snapshot, maintained corpus and inspected semantic neighborhood.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
For site i, specify X_i given all other sites as normal with mean mu_i+sum_(j!=i) beta_ij*(x_j-mu_j) and variance v_i>0 independent of the conditioning values. Let B have diagonal 1 and off-diagonal -beta_ij, and let Lambda=diag(v_i). These conditionals define the proper finite joint model when Q=Lambda^(-1)*B is symmetric positive definite. Symmetry requires beta_ij*v_j=beta_ji*v_i. The joint law is N(mu,Q^(-1)), with density proportional to |Q|^(1/2)*exp[-(x-mu)^T Q(x-mu)/2]. A zero off-diagonal precision entry denotes conditional independence given the other sites; marginal covariance need not vanish. With common v_i=sigma^2, compatibility reduces to symmetric positive-definite B and covariance sigma^2*B^(-1). In contrast, the simultaneous equations B*(X-mu)=epsilon with independent epsilon~N(0,Lambda) imply covariance B^(-1)*Lambda*B^(-T) and precision B^T*Lambda^(-1)*B. For the proper conditional model, a factorization Q=C^T*C allows exact Gaussian sampling as mu+C^(-1)*z with independent standard-normal z.
⚠️ Curated Limitations & Negative Results
- Every site conditional can be individually normalized while the proposed collection fails to define a proper joint law. Weighted symmetry and positive definiteness must both hold. With unequal conditional variances, raw beta coefficients need not be symmetric.
- A singular precision can describe an intrinsic improper construction only with additional interpretation or constraints; it is not the ordinary normalized Gaussian density stated here. An indefinite precision gives a nonintegrable quadratic model. Positive diagonal entries alone are insufficient.
- Finite-graph eigenvalue bounds depend on the actual graph and boundary conditions. Infinite-lattice stationarity criteria are not automatically necessary finite-graph conditions. The source’s stationary or homogeneous examples do not require every finite auto-normal model to share those assumptions.
- Missing precision edges encode conditional independence, not marginal independence or causal effects. A sparse simultaneous-equation matrix can induce additional precision edges.
- Unconstrained conditional least-squares, coding-likelihood or pseudolikelihood estimates may leave the proper joint parameter domain. The Gaussian fitting objective and parameter identifiability depend on the specified mean, graph and interaction parameterization; existence of a covariance representation does not ensure identifiable scientific parameters.
- The source’s wheat and plant examples expose aggregation, nonstationarity and model-fit concerns; a fitted Gaussian field does not establish a causal spatial mechanism or universal predictive adequacy. The record does not certify source implementations, empirical fits or infinite-system convergence.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:Model
Model
|
broad_match | A proper finite Gaussian field is a mathematical probability representation, within the broader AIO Model class. The inspected model is defined by compatible full conditionals and joint precision, not by a required supervised regression task. |
AIO:ProbabilisticGraphicalModel
Probabilistic Graphical Model
|
related_match | Precision zeros encode the field’s graphical conditional-independence structure. The pinned ProbabilisticGraphicalModel class is task-scoped, so the relationship from this model family is related rather than an is-a assertion. |
Evidence Sources & Primary Literature
2 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Julian Besag
|
primary paper
full_text
|
ID: doi:10.2307/2987782
sha256:0db3c34554e1…
|
lattice-model
|
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: 2026-10-11T22:02:22Z
Updated: 2026-10-11T22:02:22Z
Source File:
data/curated/AutoNormalMarkovRandomField.yaml