[Mag25]: Constraint Interaction in Probabilistic Phonology #
Replication of [Mag25] "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
([ZH17]; [Hay22]) 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 [Mag25]'s theory with the Tagalog nasal substitution case study from the paper, verifying:
- The six constraints satisfy
ConstraintIndependence - The violation differences inherit independence (
ViolDiffIndependence) - The per-cell symbolic logit rates and empirical odds ratios
- The separable forward direction at the probability level
The 2×2 square data, constraint inventory, and the constant-difference
identity itself (maxent_predicts_hz_tagalog,
hz_constant_value_tagalog) come from Studies/ZurawHayes2017.lean
([Mag25] inherits the sub-square setup from [ZH17]).
§ 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 [ZH17] ex. (3) (NasSub is the markedness driver against nasal+obstruent sequences).
C₂ = *NC is insensitive to the prefix. Per [Zur10] 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: Concrete Logit-Rate Computations #
The constant-difference identity itself is
ZurawHayes2017.maxent_predicts_hz_tagalog (with closed form
ZurawHayes2017.hz_constant_value_tagalog), stated with that paper's
data. This section verifies [Mag25]'s per-cell symbolic logit rates
LR(x) = Σₖ wₖ · Δₖ(x).
LR(maŋb) = w₁ − w₅
LR(/maŋk/) = w₁ + w₂ − w₃ − w₄ − w₅
LR(/paŋb/) = w₁ − w₆
LR(/paŋk/) = w₁ + w₂ − w₃ − w₄ − w₆
Per-cell rates recover ZurawHayes2017.hz_constant_value_tagalog's
constant difference −w₂ + w₃ + w₄; w 2 and w 3 are not separately
identifiable from the b-vs-k square — only their sum matters, since
*[stemŋ] and *[stemŋ]/n coincide on the b/k restriction.
§ 4: 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.
§ 5: 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.