@cite{magri-2025}: Constraint Interaction in Probabilistic Phonology #
@cite{magri-2025}
Replication of @cite{magri-2025} "Constraint Interaction in Probabilistic Phonology: Deducing Maximum Entropy Grammars from Hayes and Zuraw's Shifted Sigmoids Generalization" (Linguistic Inquiry, Early Access).
Main result #
Within harmony-based probabilistic phonology, an n-ary harmony function
predicts the shifted-sigmoids generalization of Hayes and Zuraw
(@cite{zuraw-hayes-2017}; @cite{hayes-2022}) if and only
if the harmony is separable — it decomposes as ∏ₖ hₖ(Cₖ)^{wₖ}.
Since MaxEnt harmony is separable (each hₖ = exp(−·)), ME predicts HZ
as a corollary. And since any separable harmony can be construed as ME
through constraint rescaling Ĉₖ = −log hₖ(Cₖ), the characterization
is complete.
Formalization #
This study file instantiates @cite{magri-2025}'s theory with the Tagalog nasal substitution case study from the paper, verifying:
- The six constraints satisfy
ConstraintIndependence - The violation differences inherit independence (
ViolDiffIndependence) - ME predicts HZ's constant logit-rate difference identity
- The identity holds for any weight assignment (not just specific values)
The 2×2 square data and constraint inventory come from
Phenomena/Phonology/Studies/ZurawHayes2017.lean (Magri 2025 inherits
the sub-square setup from Z&H 2017).
§ 1: Constraint Independence #
The constraint violation profiles viewed as functions on underlying forms (ignoring the candidate dimension, since we work with violation differences Δₖ). For the independence check, we verify that each raw constraint is insensitive to at least one dimension.
C₁ = NasSub is insensitive to the prefix (row dimension): the violation is 1 for NO and 0 for YES regardless of prefix. Per @cite{zuraw-hayes-2017} ex. (3) (NasSub is the markedness driver against nasal+obstruent sequences).
C₂ = *NC is insensitive to the prefix. Per @cite{zuraw-2010} ex. (17): "*NC: A [+nasal] segment must not be immediately followed by a [-voice, -sonorant] segment".
C₃ = *[stem] is insensitive to the prefix.
C₄ = *[stem]/n is insensitive to the prefix.
C₅ = UNIF(maŋ) is insensitive to the stem-initial obstruent (column).
C₆ = UNIF(paŋ) is insensitive to the stem-initial obstruent.
Constraint independence: for each fixed output, the six
constraints satisfy ConstraintIndependence on the nasal substitution
square.
C₁–C₄ (markedness) are insensitive to row (prefix); C₅–C₆ (faithfulness) are insensitive to column (stem obstruent).
§ 2: Violation Difference Consistency #
The violation differences are consistent with the raw constraint
profiles: Δₖ(x) = Cₖ(x, NO) − Cₖ(x, YES).
§ 3: ME Predicts HZ #
ME predicts HZ for Tagalog nasal substitution:
for any weight assignment w : Fin 6 → ℝ, the MaxEnt logit rates
of nasal substitution satisfy the constant-difference identity.
LR(/maŋb/) − LR(/maŋk/) = LR(/paŋb/) − LR(/paŋk/)
This is a direct instantiation of me_predicts_hz with the
Tagalog violation differences and their verified independence.
§ 4: Concrete Logit-Rate Computations #
The logit rate is LR(x) = Σₖ wₖ · Δₖ(x). We verify the
symbolic expressions for each cell.
LR(maŋb) = w₁ − w₅
LR(/maŋk/) = w₁ + w₂ − w₃ − w₄ − w₅
LR(/paŋb/) = w₁ − w₆
LR(/paŋk/) = w₁ + w₂ − w₃ − w₄ − w₆
The constant logit-rate difference equals −w₂ + w₃ + w₄
for both rows, regardless of weights. This follows from the
insensitivity structure of the six constraints (§ 1).
Note that w 2 and w 3 are not separately identifiable from the
b-vs-k square data — only the sum w 2 + w 3 matters here, since
*[stemŋ] and *[stemŋ]/n coincide on the b/k restriction.
The HZ identity verified concretely: both row-differences are equal.
§ 5: Empirical Rate Verification #
The empirical rates satisfy HZ's identity to good approximation.
The exact identity is logit(R(tl)) − logit(R(tr)) = logit(R(bl)) − logit(R(br)).
We verify the approximate version on the rational rates.
Rates are in (0, 1).
Logit-odds ratio for top row: (916/1000)·(7/1000) / ((993/1000)·(84/1000)) = 916·7 / (993·84) = 6412 / 83412.
Logit-odds ratio for bottom row: (434/1000)·(91/1000) / ((909/1000)·(566/1000)) = 434·91 / (909·566) = 39494 / 514494.
The two odds ratios are close: 6412/83412 ≈ 0.0769 and
39494/514494 ≈ 0.0768 — a remarkable match confirming HZ's
empirical observation. Equality of these ratios would mean
logit(R(tl)) − logit(R(tr)) = logit(R(bl)) − logit(R(br))
exactly.
§ 6: Separable Forward Direction #
ME predicts HZ at the probability level: the log-probability-ratio
log(P(YES|x)/P(NO|x)) under ME satisfies HZ's constant-difference
identity for Tagalog nasal substitution, for any weight assignment.
This instantiates separable_predicts_hz with meSeparable and the
Tagalog constraints. Since ME rescaling is the identity
(meSeparable_rescale), the rescaled violation differences reduce to
the raw violation differences, and violDiff_independence provides
the independence hypothesis.