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cdec422 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | 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
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