The claim
CLAIM-154 says to keep a candidate direction, block or sector when d’² exceeds the Occam term ln(s²/σ²), with a Gaussian prior and a Laplace posterior. “In the conjugate Gaussian case the rule is the Bayes factor exactly”, and that case was re-run here. Its own caveat is about magnitude far from the threshold: “Away from the threshold the Laplace form can understate the evidence badly.”
The limit this claim adds is about the kind of model, not the distance from the threshold:
- The penalty depends on regularity. In a regular model the Occam factor comes from the curvature at the optimum and grows as (d/2) log n. In a singular model, where the Fisher information is degenerate at the optimum, it grows as λ log n, with λ the real log canonical threshold, and λ falls below d/2 when the prior is positive on the optimal set (THEORY-087, from MacKay and Watanabe).
- Laplace-based pruning misprices singular reductions. THEORY-079 says Bayesian model reduction under the Laplace approximation charges the reductions of a singular parent with a Gaussian Occam factor, so its evidence differences can be wrong by a term that grows with log n. The same arithmetic is in CLAIM-154’s rule.
- The candidates are the singular kind. Models “with hierarchical layers or hidden variables are singular” (THEORY-079, citing Watanabe). CLAIM-154 is put to work on near-decomposable blocks, on couplings (CLAIM-174) and on the owner’s dancing couple (CLAIM-175). A model that contains a block or a coupling is singular where the coupling is switched off. There the parameters attached to it are not identified, because any value of them gives the same distribution once the coupling’s strength is zero. When the reduced model is the truth, this is the case at the true parameter. CASE-152’s product model is such a family, and the note of 2026-10-29 checks the rule on it.
The direction of the error follows from THEORY-087. Where λ < d/2, the true penalty is smaller than the Gaussian term, so the rule overcharges a candidate and prunes structure the exact Bayes factor would keep. Because the gap grows with log n, more data does not wash it out, and near the threshold it can flip the decision. That is the kind of failure CLAIM-154’s own defeater counts.
THEORY-031 adds a smaller point. d’ separates sensitivity from criterion only for equal-variance normal signal and noise. The rule’s “effect-to-noise ratio” inherits that assumption.
What it does not say
- It does not say CLAIM-154 is wrong where it was checked. The conjugate case is exact, and nothing here touches it.
- It does not say the error always runs one way. λ < d/2 is the case THEORY-087 names. Other singular structures need their own λ.
- It does not prove Watanabe’s Theorem 2 in general. It rests on THEORY-087, which holds the theorem as cited and checks its prediction exactly in singular models of known λ, and on the threshold computation in the note of 2026-10-29, which covers one family. That a candidate the rule keeps is kept by the Bayes factor too is shown there, not in general.
- d is the dimension of the model with the candidate, in the parametrization the prior is placed on. Singularity belongs to model and prior together. The uniform prior on CASE-152’s a and b induces on c = ab a density that diverges at zero; a Gaussian prior placed on c itself makes the model regular and the rule exact.
- CLAIM-154 is dual-filed from the mathematics-of-meaning record (MOM-CLM-160). The same limit would apply there. Filing it there is that record’s business.
What would answer it
The rule and an exact or WBIC-estimated Bayes factor agree on keep or prune near the threshold, across n, in a family where removing the candidate makes the model singular.
Note of 2026-10-12: an Active ground for the premise
THEORY-082 now grounds this claim. The Fisher of a multilayer perceptron “loses rank wherever an output weight is zero”, and pruning a hidden unit is exactly that case. It supplies, from an Active THEORY, this claim’s premise that the candidates are the singular kind.
Note of 2026-10-14: a second ground for caution
Grünwald & Roos (LIT-238 §6.3, read in NOTE-760) give a second, independent ground for the same caution. When every candidate model is misspecified, as a block or coupling model of a real system is, Bayes and MDL can prefer the wrong model for large n, and the remedy (Safe Bayes, η < 1) changes the effective Occam cost. The review cites the result and does not prove it. THEORY-259 (LIT-225, NOTE-801) adds that a modeller’s prior gives the description-length penalty only for hypotheses that are random under that prior.
Note of 2026-10-29: the check on a concrete family
CASE-152’s model, Y ~ N(ab, 1), with a uniform prior on [−1, 1]², against a = 0, the model with no parameters. The data enter through ĉ ~ N(ab, 1/n). With z = √n ĉ, the exact keep threshold z*², found by quadrature, tracks log n − 2 log log n + 1.2. The regular two-parameter count keeps only when z² > 2 log n − 2 ln(π/2).
| n | exact z*² | log n − 2 log log n | regular count’s threshold |
|---|---|---|---|
| 10² | 2.75 | 1.55 | 8.31 |
| 10³ | 4.34 | 3.04 | 12.91 |
| 10⁴ | 6.08 | 4.77 | 17.52 |
| 10⁶ | 9.84 | 8.56 | 26.73 |
| 10⁸ | 13.83 | 12.59 | 35.94 |
| 10¹⁰ | 17.95 | 16.75 | 45.15 |
| 10¹² | 22.16 | 20.99 | 54.36 |
Between the two thresholds the decisions are opposite while both log Bayes factors are modest. At n = 10⁴ and z² = 12 the exact ln BF is +2.85 and the regular count’s −2.76; at n = 10⁶ and z² = 16, +3.03 and −5.36; at n = 10⁸ and z² = 25, +5.55 and −5.47. Counting the candidate as one direction, with threshold log n plus a constant, still overcharges, by a gap in z² growing as 2 log log n (3.1, 4.6 and 5.5 at n = 10⁴, 10⁸ and 10¹²). That meets the “What would answer it” test above against the rule, in a family where switching the candidate off makes the model singular. The scripts were written by the reviewer, re-run by the reviewing session, and are not in the record.