| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ExactStatisticWords |
| import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexTypeEntropy |
|
|
| namespace OAI |
|
|
| /-! Finite entropy, rate estimates and ordered asymptotic limits. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.LeafRates |
|
|
| open MatrixMultiplication.Foundation Filter |
| open scoped BigOperators Topology |
|
|
| variable {A U : Type*} [Fintype A] [DecidableEq A] [Fintype U] |
|
|
| theorem log_card_supported (statistic : U → A) (counts b : A → ℕ) |
| (hb : ∀ a, 0 < counts a → Fintype.card {u : U // statistic u = a} = b a) |
| (hbpos : ∀ a, 0 < b a) : |
| Real.log (Fintype.card (ExactStatisticWords.Words statistic counts) : ℝ) = |
| Real.log (Nat.multinomial Finset.univ counts : ℝ) + |
| ∑ a, (counts a : ℝ) * Real.log (b a : ℝ) := by |
| rw [ExactStatisticWords.card_of_supported statistic counts b hb, Nat.cast_mul, Nat.cast_prod] |
| have hm : (Nat.multinomial Finset.univ counts : ℝ) ≠ 0 := by |
| exact_mod_cast (Nat.ne_of_gt (Nat.multinomial_pos Finset.univ counts)) |
| have hp : (∏ a, (b a ^ counts a : ℕ) : ℝ) ≠ 0 := by |
| apply Finset.prod_ne_zero_iff.mpr |
| intro a _ |
| exact_mod_cast (Nat.ne_of_gt (pow_pos (hbpos a) _)) |
| rw [Real.log_mul hm hp, Real.log_prod (s := Finset.univ) |
| (f := fun a => ((b a ^ counts a : ℕ) : ℝ)) |
| (fun a _ => by exact_mod_cast (Nat.ne_of_gt (pow_pos (hbpos a) (counts a))))] |
| simp only [Nat.cast_pow, Real.log_pow] |
|
|
| theorem log_card (statistic : U → A) (counts b : A → ℕ) |
| (hb : ∀ a, Fintype.card {u : U // statistic u = a} = b a) |
| (hbpos : ∀ a, 0 < b a) : |
| Real.log (Fintype.card (ExactStatisticWords.Words statistic counts) : ℝ) = |
| Real.log (Nat.multinomial Finset.univ counts : ℝ) + |
| ∑ a, (counts a : ℝ) * Real.log (b a : ℝ) := |
| log_card_supported statistic counts b (fun a _ => hb a) hbpos |
|
|
| theorem tendsto_log_card_supported (statistic : U → A) (counts b : A → ℕ) |
| (hb : ∀ a, 0 < counts a → Fintype.card {u : U // statistic u = a} = b a) |
| (hbpos : ∀ a, 0 < b a) (hD : 0 < ∑ a, counts a) : |
| Tendsto (fun t : ℕ => |
| Real.log (Fintype.card (ExactStatisticWords.Words statistic (fun a => t * counts a)) : ℝ) / |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop |
| (𝓝 (finiteEntropy (fun a => (counts a : ℝ) / (∑ a, counts a : ℕ)) + |
| ∑ a, ((counts a : ℝ) / (∑ a, counts a : ℕ)) * Real.log (b a : ℝ))) := by |
| have hl := (tendsto_log_multinomial_mul counts hD).add_const |
| (∑ a, ((counts a : ℝ) / (∑ a, counts a : ℕ)) * Real.log (b a : ℝ)) |
| apply hl.congr' |
| filter_upwards [eventually_ne_atTop (0 : ℕ)] with t ht |
| have hbt : ∀ a, 0 < t * counts a → Fintype.card {u : U // statistic u = a} = b a := |
| fun a ha => hb a (by nlinarith) |
| rw [log_card_supported statistic (fun a => t * counts a) b hbt hbpos, add_div] |
| congr 1 |
| rw [Finset.sum_div] |
| apply Finset.sum_congr rfl |
| intro a _ |
| have ht' : (t : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr ht |
| have hD' : ((∑ a, counts a : ℕ) : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hD) |
| have hDsum : (∑ a, (counts a : ℝ)) ≠ 0 := by |
| simpa only [Nat.cast_sum] using hD' |
| simp only [Nat.cast_mul] |
| field_simp [ht', hD', hDsum] |
|
|
| theorem tendsto_log_card (statistic : U → A) (counts b : A → ℕ) |
| (hb : ∀ a, Fintype.card {u : U // statistic u = a} = b a) |
| (hbpos : ∀ a, 0 < b a) (hD : 0 < ∑ a, counts a) : |
| Tendsto (fun t : ℕ => |
| Real.log (Fintype.card (ExactStatisticWords.Words statistic (fun a => t * counts a)) : ℝ) / |
| ((t : ℝ) * (∑ a, counts a : ℕ))) atTop |
| (𝓝 (finiteEntropy (fun a => (counts a : ℝ) / (∑ a, counts a : ℕ)) + |
| ∑ a, ((counts a : ℝ) / (∑ a, counts a : ℕ)) * Real.log (b a : ℝ))) := |
| tendsto_log_card_supported statistic counts b (fun a _ => hb a) hbpos hD |
|
|
| end MatrixMultiplication.LeafRates |
|
|
| end |
|
|
| end OAI |
|
|
