DLM ERM-invariance under positive rescaling

[BCSBB19] [GB24]

Sibling to Training.lean. Two theorems making explicit that ERM solutions for the DLM weighted loss are invariant under positive rescaling of the frequency vector. The substrate operations FrequencyVector.totalMass and FrequencyVector.normalize live in Training.lean next to FrequencyVector itself; this file consumes them to prove the invariance theorems.

Why a separate file

These two theorems form the bridge that lets cross-tradition comparisons go through: when comparing DLM predictions to Bayesian word-learning posteriors over the same event inventory, the "frequency-vector-as-counts" view (DLM-paper-faithful) and the "frequency-vector-as-empirical-distribution" view (Bayesian-tradition) make identical predictions about which G : MeaningVec → FormVec is ERM-optimal.

The DLM literature (paper appendix §A1.3 of [GB24]) uses Q as a diagonal matrix of raw counts, not a normalised distribution. Paper §6.4 explicitly cautions against reifying any single mathematical representation. The rescaling-invariance theorems make the equivalence formal.

PMF view

To cast a FrequencyVector m into a mathlib PMF (Fin m) (e.g., for KL-divergence or total-variation comparisons against probabilistic models), call PMF.ofRealWeightFn from Core.Probability.Constructions directly:

PMF.ofRealWeightFn q hq htot

where htot : 0 < ∑ i, q i is the positive-total hypothesis (q.totalMass unfolded).

There is no project-side toEmpiricalPMF wrapper — the inline PMF.ofRealWeightFn call is the canonical entry point. (An earlier toEmpiricalPMF reinvented PMF.ofRealWeightFn; removed 0.231.X.)

theorem Processing.Lexical.Discriminative.isERMSolution_normalize_iff {m n d : ℕ} (data : TrainingExperience m n d) (q : FrequencyVector m) (G : MeaningVec d →ₗ[ℝ] FormVec n) (h : 0 < q.totalMass) : IsERMSolution data q.normalize G ↔ IsERMSolution data q G

Normalisation preserves ERM solutions. A FrequencyVector and its empirical distribution agree on which production maps are ERM solutions. This is the formal statement that the cognitive content is in the relative weights (= the normalised distribution) and not in the absolute scale.

Direct application of ermSolution_iff_rescaled with c = 1/totalMass.

theorem Processing.Lexical.Discriminative.isERMSolution_of_same_normalize {m n d : ℕ} (data : TrainingExperience m n d) (q₁ q₂ : FrequencyVector m) (G : MeaningVec d →ₗ[ℝ] FormVec n) (h₁ : 0 < q₁.totalMass) (h₂ : 0 < q₂.totalMass) (hnorm : q₁.normalize = q₂.normalize) : IsERMSolution data q₁ G ↔ IsERMSolution data q₂ G

Two FrequencyVectors with identical empirical distributions are ERM-equivalent. The cognitive content lives at the level of the normalised distribution; theories that assign different absolute scales but the same relative frequencies make the same predictions.