Pairwise Markov tree
Model Families AIMech:PairwiseMarkovTree DRAFT
Aliases: Arbre de Markov couple
A latent-variable tree-model family whose paired hidden and observed states factor into a root distribution and parent-to-child conditional kernels on a fixed rooted tree, without requiring the hidden marginal to be Markov on that tree.
Identity Notes & Concept Scope
Pairwise denotes Z_s=(X_s,Y_s), not generic pairwise edge potentials. The tree is the stochastic dependence topology across sites, rather than a decision-tree classifier, hierarchical mixture-of-experts gate structure, or hidden Markov decision tree’s within-time state hierarchy. An ordinary memoryless-emission hidden Markov tree is a constrained special case. A path is a degenerate tree, but the family permits branching dependencies absent from a sequential chain. No normalized identity match was found across all 442 pinned AIO labels and all scoped synonyms and 391 preceding native IDs, names and aliases for the three selected concepts. Full-checkout fixed-string searches included ignored and hidden files in both worktree and main, yielding 9 and 9 matched lines; all 7 unique contents were read. Hits are prior leads and references, not records of the proposed identities. All 89 semantic label/synonym hits and 21 fully annotated exact AIO CLASS nodes with outgoing edges were read. Twenty-two contextual YAMLs were byte-verified after prior full reading and five freshly read, totaling 21 generated and six native contextual records. Short acronym identities were compared across the complete normalized inventory. Absence is bounded to the pinned source, maintained collection and inspected semantic neighborhood.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
On a finite fixed rooted tree with one parent for each nonroot site, specify a normalized root law p(z_r) and normalized kernels K_s(z_s|z_parent(s)); their product defines the joint paired distribution. At fixed complete y, the hidden posterior factors on the same rooted tree. Set beta_s(x_s)=1 at leaves and compute each internal beta_s(x_s) as the product over children t of sum_(x_t) K_t(x_t,y_t|x_s,y_s)*beta_t(x_t). The root posterior is proportional to p(x_r,y_r)*beta_r(x_r). Each child posterior transition at fixed parent is proportional to K_t*beta_t and normalized over x_t. Propagate the root marginal downward through these conditional kernels to obtain all hidden-site marginals. Classical independent-emission hidden Markov trees result from root pi(x_r)*b(y_r|x_r) and child kernels A(x_t|x_s)*b(y_t|x_t). General joint kernels can leave hidden siblings dependent given only their hidden parent, so the hidden marginal need not itself be a Markov tree.
⚠️ Curated Limitations & Negative Results
- The definition uses a fixed finite rooted tree; it is not a graph-structure learning algorithm, a loopy-field model or a decision-making policy. Exact tree recursions require finite hidden states or separately tractable integrals.
- Posterior formulas require nonzero observation evidence and valid conditional support. Numerical scaling can be needed on large trees; the factorization alone does not address parameter estimation, missing observations or arbitrary topology changes.
- Independence of the child hidden-state transition from the parent observation is sufficient for a hidden Markov-tree marginal. Proposition 3.2 asserts necessity under the additional hypotheses that each internal node has at least two children with identical conditional laws given its paired parent; that converse is not adopted without those hypotheses.
- Correlated observations can coexist with a Markov hidden marginal. Non-Markov projection is permitted, not required, and memoryless site emissions are not necessary for every hidden-Markov special case.
- The four-page source proves structural and inference results; it does not report a comparative empirical evaluation or establish universal segmentation superiority. Later parameter-estimation and image-processing papers remain separate evidence leads.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:BayesianNetwork
Bayesian Network
|
broad_match | The joint paired-state law has a normalized directed acyclic factorization on a rooted tree, a restricted Bayesian-network topology. The mapping concerns paired stochastic states and imposes no causal interpretation or requirement that the hidden marginal preserve that graph. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Wojciech Pieczynski
|
primary paper
full_text
|
ID: doi:10.1016/S1631-073X(02)02430-5
sha256:7834f53dcb3b…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T21:03:30Z
Updated: 2026-10-11T21:03:30Z
Source File:
data/curated/PairwiseMarkovTree.yaml