openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /JointExtraction /ConditionalCountEntropy.lean
| import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.PopulationRates | |
| namespace OAI | |
| /-! Joint tensor extraction, compatibility and entropy estimates. -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.JointConditionalCountEntropy | |
| open MatrixMultiplication.Foundation JointPopulationRates | |
| open scoped BigOperators | |
| variable {U A : Type*} [Fintype U] [Fintype A] | |
| attribute [local instance] Classical.propDecidable | |
| def classCounts (n : U → ℕ) (f : U → A) (a : A) (u : U) : ℕ := | |
| if f u = a then n u else 0 | |
| def classSize (n : U → ℕ) (f : U → A) (a : A) : ℕ := | |
| ∑ u, classCounts n f a u | |
| theorem classSize_total (n : U → ℕ) (f : U → A) : | |
| (∑ a, classSize n f a) = ∑ u, n u := by | |
| classical | |
| unfold classSize classCounts | |
| rw [Finset.sum_comm] | |
| simp | |
| theorem sum_masked_entropy (p : U → ℝ) (f : U → A) : | |
| (∑ a, finiteEntropy (fun u => if f u = a then p u else 0)) = | |
| finiteEntropy p := by | |
| classical | |
| have ht (a : A) (u : U) : | |
| entropyTerm (if f u = a then p u else 0) = | |
| if f u = a then entropyTerm (p u) else 0 := by | |
| by_cases h : f u = a <;> simp [h] | |
| simp only [finiteEntropy, ht] | |
| rw [Finset.sum_comm] | |
| simp | |
| theorem sum_classCounts_entropy (n : U → ℕ) (f : U → A) (scale : ℝ) : | |
| (∑ a, finiteEntropy (fun u => (classCounts n f a u : ℝ) / scale)) = | |
| finiteEntropy (fun u => (n u : ℝ) / scale) := by | |
| classical | |
| have hm (a : A) : | |
| (fun u => (classCounts n f a u : ℝ) / scale) = | |
| (fun u => if f u = a then (n u : ℝ) / scale else 0) := by | |
| funext u | |
| by_cases h : f u = a <;> simp [classCounts, h] | |
| simp_rw [hm] | |
| exact sum_masked_entropy _ f | |
| theorem sum_scaled_class_empirical_entropy (n : U → ℕ) (f : U → A) | |
| (scale : ℝ) : | |
| (∑ a, ((classSize n f a : ℝ) / scale) * | |
| finiteEntropy (fun u => | |
| (classCounts n f a u : ℝ) / (classSize n f a : ℝ))) = | |
| finiteEntropy (fun u => (n u : ℝ) / scale) - | |
| finiteEntropy (fun a => (classSize n f a : ℝ) / scale) := by | |
| calc | |
| _ = ∑ a, (finiteEntropy (fun u => (classCounts n f a u : ℝ) / scale) - | |
| entropyTerm ((classSize n f a : ℝ) / scale)) := by | |
| apply Finset.sum_congr rfl | |
| intro a _ | |
| exact weighted_empirical_entropy_eq (classCounts n f a) scale | |
| _ = _ := by | |
| rw [Finset.sum_sub_distrib, sum_classCounts_entropy] | |
| rfl | |
| theorem sum_classSize_mul_empirical_entropy (n : U → ℕ) (f : U → A) : | |
| (∑ a, (classSize n f a : ℝ) * | |
| finiteEntropy (fun u => | |
| (classCounts n f a u : ℝ) / (classSize n f a : ℝ))) = | |
| (∑ u, n u : ℕ) * | |
| (finiteEntropy (fun u => (n u : ℝ) / (∑ v, n v : ℕ)) - | |
| finiteEntropy (fun a => (classSize n f a : ℝ) / (∑ v, n v : ℕ))) := by | |
| have hclasses := sum_scaled_class_empirical_entropy n f 1 | |
| have hjoint := weighted_empirical_entropy_eq n 1 | |
| have hmarginal := weighted_empirical_entropy_eq (classSize n f) 1 | |
| simp only [div_one] at hclasses hjoint hmarginal | |
| rw [classSize_total] at hmarginal | |
| rw [hclasses, mul_sub, hjoint, hmarginal] | |
| ring | |
| theorem sum_classSize_mul_entropy_of_law (n : U → ℕ) (f : U → A) | |
| (p : FiniteLaw U) (scale : ℝ) | |
| (hn : ∀ u, (n u : ℝ) = scale * p.mass u) : | |
| (∑ a, (classSize n f a : ℝ) * | |
| finiteEntropy (fun u => | |
| (classCounts n f a u : ℝ) / (classSize n f a : ℝ))) = | |
| scale * (finiteEntropy p.mass - finiteEntropy (p.map f).mass) := by | |
| classical | |
| have hclass (a : A) : (classSize n f a : ℝ) = scale * (p.map f).mass a := by | |
| rw [classSize, Nat.cast_sum, FiniteLaw.map_mass, Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro u _ | |
| by_cases h : f u = a <;> simp [classCounts, h, hn] | |
| have h := sum_scaled_class_empirical_entropy n f 1 | |
| simp only [div_one] at h | |
| rw [h] | |
| simp_rw [hn, hclass] | |
| rw [finiteEntropy_scalar p.mass scale p.total, | |
| finiteEntropy_scalar (p.map f).mass scale (p.map f).total] | |
| ring | |
| end MatrixMultiplication.JointConditionalCountEntropy | |
| end | |
| end OAI | |