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namespace OAI
/-!
# Finite uniform sampling
The shared group estimator samples child labels uniformly with replacement.
These lemmas use that finite distribution, including its product law;
they do not assume that different group representatives are independent.
-/
namespace EditApproximation
open Finset MeasureTheory ProbabilityTheory
noncomputable def uniformMean {ι : Type*} [Fintype ι] (value : ι → ℝ) : ℝ :=
(∑ i, value i) / Fintype.card ι
theorem uniform_integral {ι : Type*} [Fintype ι] [Nonempty ι]
[MeasurableSpace ι] [MeasurableSingletonClass ι] (value : ι → ℝ) :
(∫ i, value i ∂(PMF.uniformOfFintype ι).toMeasure) = uniformMean value := by
rw [PMF.integral_eq_sum]
simp only [PMF.uniformOfFintype_apply, ENNReal.toReal_inv, ENNReal.toReal_natCast,
smul_eq_mul, ← mul_sum, uniformMean]
ring
theorem uniformMean_bounds {ι : Type*} [Fintype ι] [Nonempty ι]
(value : ι → ℝ) {bound : ℝ} (hvalue : ∀ i, 0 ≤ value i ∧ value i ≤ bound) :
0 ≤ uniformMean value ∧ uniformMean value ≤ bound := by
have hc : (0 : ℝ) < Fintype.card ι := by exact_mod_cast Fintype.card_pos
constructor
· exact div_nonneg (sum_nonneg fun i _ => (hvalue i).1) hc.le
· apply (div_le_iff₀ hc).2
have hsum := sum_le_sum (s := univ) fun i _ => (hvalue i).2
simpa only [sum_const, card_univ, nsmul_eq_mul, mul_comm] using hsum
theorem uniform_centered_bound {ι : Type*} [Fintype ι] [Nonempty ι]
(value : ι → ℝ) {bound : ℝ} (hvalue : ∀ i, 0 ≤ value i ∧ value i ≤ bound)
(i : ι) : |value i - uniformMean value| ≤ bound := by
have hm := uniformMean_bounds value hvalue
exact abs_le.mpr ⟨by linarith [(hvalue i).1], by linarith [(hvalue i).2]⟩
theorem uniform_centered_integral {ι : Type*} [Fintype ι] [Nonempty ι]
[MeasurableSpace ι] [MeasurableSingletonClass ι] (value : ι → ℝ) :
(∫ i, (value i - uniformMean value)
∂(PMF.uniformOfFintype ι).toMeasure) = 0 := by
rw [integral_sub (.of_finite) (integrable_const _), uniform_integral, integral_const]
simp
theorem uniform_variance_bound {ι : Type*} [Fintype ι] [Nonempty ι]
[MeasurableSpace ι] [MeasurableSingletonClass ι]
(value : ι → ℝ) {bound : ℝ} (hvalue : ∀ i, 0 ≤ value i ∧ value i ≤ bound) :
(∫ i, (value i - uniformMean value) ^ 2
∂(PMF.uniformOfFintype ι).toMeasure) ≤ bound * uniformMean value := by
let μ := (PMF.uniformOfFintype ι).toMeasure
have hmean : (∫ i, value i ∂μ) = uniformMean value := uniform_integral value
have hexpand : (∫ i, (value i - uniformMean value) ^ 2 ∂μ) =
(∫ i, value i ^ 2 ∂μ) - (uniformMean value) ^ 2 := by
calc
_ = ∫ i, (value i ^ 2 - 2 * uniformMean value * value i +
(uniformMean value) ^ 2) ∂μ := by congr 1; funext i; ring
_ = _ := by
rw [integral_add (.of_finite) (integrable_const _),
integral_sub (.of_finite) (.of_finite), integral_const_mul, hmean, integral_const]
simp only [probReal_univ, one_smul]
ring
have hsecond : (∫ i, value i ^ 2 ∂μ) ≤ bound * uniformMean value := by
rw [← hmean, ← integral_const_mul]
apply integral_mono (.of_finite) (.of_finite)
intro i
nlinarith [(hvalue i).1, (hvalue i).2]
rw [hexpand]
nlinarith [sq_nonneg (uniformMean value)]
/-- Actual independent uniform draws, centered and divided by the sample
count. All measurability, centering and variance obligations for Bernstein
are derived from this product distribution. -/
theorem uniform_sample_average_concentration
{ι : Type*} [Fintype ι] [Nonempty ι]
[MeasurableSpace ι] [MeasurableSingletonClass ι]
(n : ℕ) (hn : 0 < n) (value : ι → ℝ) (bound meanBound deviation : ℝ)
(hbound : 0 < bound) (hmeanBound : 0 < meanBound) (hdev : 0 < deviation)
(hvalue : ∀ i, 0 ≤ value i ∧ value i ≤ bound)
(hmean : uniformMean value ≤ meanBound)
(bernstein : FiniteBernsteinInput
(Measure.pi fun _ : Fin n => (PMF.uniformOfFintype ι).toMeasure)) :
(Measure.pi fun _ : Fin n => (PMF.uniformOfFintype ι).toMeasure).real
{ω | deviation ≤ |(∑ j : Fin n, value (ω j)) / n - uniformMean value|} ≤
2 * Real.exp (-(n : ℝ) * deviation ^ 2 /
(2 * bound * meanBound + (2 / 3 : ℝ) * bound * deviation)) := by
let μ := (PMF.uniformOfFintype ι).toMeasure
let X : Fin n → (Fin n → ι) → ℝ :=
fun j ω => (value (ω j) - uniformMean value) / n
have hnr : (0 : ℝ) < n := by exact_mod_cast hn
have hindep : iIndepFun X (Measure.pi fun _ : Fin n => μ) := by
exact iIndepFun_pi (μ := fun _ : Fin n => μ)
(X := fun _ i => (value i - uniformMean value) / n)
(fun _ => (measurable_of_finite _).aemeasurable)
have hcenter : ∀ j, (∫ ω, X j ω ∂Measure.pi (fun _ : Fin n => μ)) = 0 := by
intro j
dsimp [X]
rw [integral_comp_eval (μ := fun _ : Fin n => μ) (i := j)
(f := fun i => (value i - uniformMean value) / n)
(measurable_of_finite _).aestronglyMeasurable]
rw [integral_div, uniform_centered_integral, zero_div]
have hbounded : ∀ j, ∀ᵐ ω ∂Measure.pi (fun _ : Fin n => μ),
|X j ω| ≤ bound / n := by
intro j
filter_upwards [] with ω
dsimp [X]
rw [abs_div, abs_of_pos hnr]
exact div_le_div_of_nonneg_right (uniform_centered_bound value hvalue _) hnr.le
have hvariance : (∑ j : Fin n, ∫ ω, (X j ω) ^ 2
∂Measure.pi (fun _ : Fin n => μ)) ≤ bound * meanBound / n := by
have hpoint (j : Fin n) : (∫ ω, (X j ω) ^ 2
∂Measure.pi (fun _ : Fin n => μ)) ≤ bound * meanBound / n ^ 2 := by
dsimp [X]
rw [integral_comp_eval (μ := fun _ : Fin n => μ) (i := j)
(f := fun i => ((value i - uniformMean value) / n) ^ 2)
(measurable_of_finite _).aestronglyMeasurable]
simp_rw [div_pow]
rw [integral_div]
apply div_le_div_of_nonneg_right _ (sq_nonneg _)
exact (uniform_variance_bound value hvalue).trans
(mul_le_mul_of_nonneg_left hmean hbound.le)
have hsum := sum_le_sum (s := univ) fun j _ => hpoint j
simp only [sum_const, card_univ, Fintype.card_fin, nsmul_eq_mul] at hsum
apply hsum.trans_eq
field_simp
have htail := bernstein n X (bound / n) (bound * meanBound / n) deviation
(div_pos hbound hnr) (div_pos (mul_pos hbound hmeanBound) hnr) hdev hindep
(fun _ => measurable_of_finite _) (fun _ => .of_finite) hcenter hbounded hvariance
have hsum (ω : Fin n → ι) : (∑ j : Fin n, X j ω) =
(∑ j : Fin n, value (ω j)) / n - uniformMean value := by
dsimp [X]
rw [← sum_div, sum_sub_distrib]
simp only [sum_const, card_univ, Fintype.card_fin, nsmul_eq_mul]
field_simp
simp_rw [hsum] at htail
convert htail using 2
congr 1
field_simp
end EditApproximation
end OAI
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