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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 | 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
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