| import OAI.NumberTheory.PrimePackets.EntropyMixtures |
|
|
| namespace OAI |
|
|
| /-! |
| # Finite conditional entropy, including zero-mass cells |
|
|
| These algebraic chain rules are used to compare finite observations with |
| the expanding leaf partitions in the metric entropy calculation. |
| -/ |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace PrimeDegreePackets |
|
|
| lemma finiteShannon_mul {α : Type*} [Fintype α] (r : ℝ) (p : α → ℝ) : |
| finiteShannon (fun a => r * p a) = |
| Real.negMulLog r * (∑ a, p a) + r * finiteShannon p := by |
| simp only [finiteShannon, Real.negMulLog_mul, Finset.sum_add_distrib, |
| ← Finset.mul_sum, ← Finset.sum_mul] |
| ring |
|
|
| lemma finiteShannon_mass_decomposition {α : Type*} [Fintype α] |
| (p : α → ℝ) (hp : ∀ a, 0 ≤ p a) : |
| finiteShannon p = Real.negMulLog (∑ a, p a) + |
| (∑ a, p a) * finiteShannon (fun a => p a / (∑ b, p b)) := by |
| classical |
| let r := ∑ a, p a |
| by_cases hr : r = 0 |
| · have hp0 : ∀ a, p a = 0 := by |
| intro a |
| have h := Finset.single_le_sum (fun b _ => hp b) (Finset.mem_univ a) |
| change p a ≤ r at h |
| exact le_antisymm (hr ▸ h) (hp a) |
| simp only [hp0, finiteShannon, Finset.sum_const_zero, Real.negMulLog_zero, |
| zero_div, zero_mul, zero_add] |
| · have hs : ∑ a, p a / r = 1 := by rw [← Finset.sum_div]; exact div_self hr |
| have heq : (fun a => r * (p a / r)) = p := by |
| funext a |
| exact mul_div_cancel₀ (p a) hr |
| have h := finiteShannon_mul r (fun a => p a / r) |
| rw [heq, hs, mul_one] at h |
| exact h |
|
|
| /-- Subadditivity for a nonnegative finite measure; its total mass need not be one. -/ |
| theorem finiteShannon_joint_mass_le {α β : Type*} [Fintype α] [Fintype β] |
| (p : α × β → ℝ) (hp : ∀ z, 0 ≤ p z) : |
| finiteShannon p + Real.negMulLog (∑ z, p z) ≤ |
| finiteShannon (fun a => ∑ b, p (a,b)) + |
| finiteShannon (fun b => ∑ a, p (a,b)) := by |
| classical |
| let r := ∑ z, p z |
| have hr0 : 0 ≤ r := Finset.sum_nonneg (fun z _ => hp z) |
| by_cases hr : r = 0 |
| · have hp0 : ∀ z, p z = 0 := by |
| intro z |
| have h := Finset.single_le_sum (fun w _ => hp w) (Finset.mem_univ z) |
| change p z ≤ r at h |
| exact le_antisymm (hr ▸ h) (hp z) |
| simp [hp0, finiteShannon] |
| have hnorm : ∑ z, p z / r = 1 := by rw [← Finset.sum_div]; exact div_self hr |
| have H := mul_le_mul_of_nonneg_left |
| (finiteShannon_joint_le (p := fun z => p z / r) |
| (fun z => div_nonneg (hp z) hr0) hnorm) hr0 |
| have hrow : (∑ a, ∑ b, p (a,b)) = r := by simp only [r, Fintype.sum_prod_type] |
| have hcol : (∑ b, ∑ a, p (a,b)) = r := by rw [Finset.sum_comm]; exact hrow |
| have h1 := finiteShannon_mass_decomposition p hp |
| have h2 := finiteShannon_mass_decomposition (fun a => ∑ b, p (a,b)) |
| (fun a => Finset.sum_nonneg (fun b _ => hp (a,b))) |
| have h3 := finiteShannon_mass_decomposition (fun b => ∑ a, p (a,b)) |
| (fun b => Finset.sum_nonneg (fun a _ => hp (a,b))) |
| simp only [hrow] at h2 |
| simp only [hcol] at h3 |
| simp only [← Finset.sum_div] at H |
| change finiteShannon p = Real.negMulLog r + r * _ at h1 |
| linarith |
|
|
| /-- The finite entropy submodularity inequality `H(ABC)+H(A) ≤ H(AB)+H(AC)`. -/ |
| theorem finiteShannon_submodular {α β γ : Type*} |
| [Fintype α] [Fintype β] [Fintype γ] |
| (p : α × (β × γ) → ℝ) (hp : ∀ z, 0 ≤ p z) : |
| finiteShannon p + finiteShannon (fun a => ∑ z : β × γ, p (a,z)) ≤ |
| finiteShannon (fun z : α × β => ∑ c, p (z.1,(z.2,c))) + |
| finiteShannon (fun z : α × γ => ∑ b, p (z.1,(b,z.2))) := by |
| have H := Finset.sum_le_sum (s := Finset.univ) (fun a _ => |
| finiteShannon_joint_mass_le (fun z : β × γ => p (a,z)) (fun z => hp (a,z))) |
| simpa only [finiteShannon, Fintype.sum_prod_type, Finset.sum_add_distrib] using H |
|
|
| lemma finiteShannon_fst_le {α β : Type*} [Fintype α] [Fintype β] |
| (p : α × β → ℝ) (hp : ∀ z, 0 ≤ p z) : |
| finiteShannon (fun a => ∑ b, p (a,b)) ≤ finiteShannon p := by |
| simpa only [finiteShannon, Fintype.sum_prod_type] using |
| Finset.sum_le_sum (s := Finset.univ) (fun a _ => |
| negMulLog_sum_le (fun b => p (a,b)) (fun b => hp (a,b))) |
|
|
| lemma finiteShannon_joint_decomposition {α β : Type*} [Fintype α] [Fintype β] |
| (p : α × β → ℝ) (hp : ∀ z, 0 ≤ p z) : |
| finiteShannon p = finiteShannon (fun a => ∑ b, p (a,b)) + |
| ∑ a, (∑ b, p (a,b)) * |
| finiteShannon (fun b => p (a,b) / (∑ c, p (a,c))) := by |
| have H := Finset.sum_congr (s₁ := Finset.univ) rfl (fun a _ => |
| finiteShannon_mass_decomposition (fun b => p (a,b)) (fun b => hp (a,b))) |
| simpa only [finiteShannon, Fintype.sum_prod_type, Finset.sum_add_distrib] using H |
|
|
| end PrimeDegreePackets |
|
|
| end |
|
|
| end OAI |
|
|
