Fixed-length Poisson Markov random field

⚠️ Curated Limitations & Negative Results

• The finite support, reference measure and total L are part of the identity. Positive quadratic coefficients need not imply positive marginal covariance under a fixed-sum constraint; for two coordinates covariance equals minus their variance.
• Usual nonedge/conditional-independence conclusions for positive Cartesian-support MRFs cannot be transferred mechanically to fixed-total counts. The name is retained from the source, while the actual support and potential are explicit.
• Parameters are not uniquely identified without conventions: common theta shifts, and some fixed-L changes to diagonal and off-diagonal terms, leave the same law. A reparameterization valid at one L need not preserve a shared-parameter model over varying totals.
• Finite fixed-total support ensures normalization, but does not make partition computation or parameter estimation trivial. Adding a length distribution requires the correctly normalized conditional distribution at each total; one normalizer cannot be reused across lengths.
• The source reuses nodewise Poisson regression from an unbounded model as a surrogate estimator. A fixed-total count coordinate is deterministic given all other count coordinates, so this is not its exact count-coordinate pseudolikelihood. Graph-recovery guarantees for a different Cartesian-support model do not automatically transfer.
• For a general symmetric Phi, a token-label Gibbs update has log weight theta_s+2*(Phi*x_minus)_s+Phi_ss. The printed sampler omits Phi_ss and needs zero/equal diagonals or a specified reparameterization. The printed eigenvalue bound using L^2*lambda_max also needs lambda_max>=0; for arbitrary symmetric matrices a safe loose bound uses its nonnegative part. These are curator qualifications of displayed formulas, not reproduced code defects.
• The length-dependent interaction weighting and curve fitted above AIS estimates are modeling and numerical choices. A curve above finitely many estimated log partitions is not a certified upper bound on every exact partition value. No AIS convergence, likelihood accuracy or author implementation was validated.
• The reported topic-model perplexities use MAP latent assignments, as the source explicitly notes, rather than exact marginalization over all topic allocations. Qualitative word pairs and historical timing curves do not establish causal relations, generic superiority or current performance. No author fit or benchmark was reproduced.