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e63eb7f | 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 | import OAI.Combinatorics.EditApproximation.Probability.BernsteinEnvelope
namespace OAI
namespace EditApproximation
open MeasureTheory ProbabilityTheory
open scoped BigOperators
section MGF
variable {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
theorem bounded_square_integrable (X : Ω → ℝ) (b : ℝ) (hb : 0 ≤ b)
(hm : Measurable X) (hbound : ∀ᵐ ω ∂μ, |X ω| ≤ b) :
Integrable (fun ω => (X ω) ^ 2) μ := by
apply Integrable.of_mem_Icc 0 (b ^ 2) (hm.pow_const 2).aemeasurable
filter_upwards [hbound] with ω hω
refine ⟨sq_nonneg _, ?_⟩
have h := mul_le_mul hω hω (abs_nonneg _) hb
simpa only [← pow_two, sq_abs] using h
theorem bounded_centered_mgf (X : Ω → ℝ) (b t : ℝ) (hb : 0 < b) (ht : 0 ≤ t)
(htb : t * b < 3) (hm : Measurable X) (hi : Integrable X μ)
(hmean : (∫ ω, X ω ∂μ) = 0) (hbound : ∀ᵐ ω ∂μ, |X ω| ≤ b) :
mgf X μ t ≤ Real.exp ((t ^ 2 / (2 * (1 - t * b / 3))) * ∫ ω, (X ω) ^ 2 ∂μ) := by
let c := t ^ 2 / (2 * (1 - t * b / 3))
have hd : 0 < 2 * (1 - t * b / 3) := by linarith
have hi2 := bounded_square_integrable μ X b hb.le hm hbound
have hexp := integrable_exp_mul_of_le t b ht hm.aemeasurable
(hbound.mono fun ω hω => (le_abs_self _).trans hω)
have hpoint : ∀ᵐ ω ∂μ, Real.exp (t * X ω) ≤ 1 + t * X ω + c * (X ω) ^ 2 := by
filter_upwards [hbound] with ω hω
have hu : t * X ω ≤ t * b := mul_le_mul_of_nonneg_left ((le_abs_self _).trans hω) ht
have hx := exp_le_bernstein_envelope (t * X ω) (hu.trans_lt htb)
have hdiv : (t * X ω) ^ 2 / (2 * (1 - t * X ω / 3)) ≤
(t * X ω) ^ 2 / (2 * (1 - t * b / 3)) := by
apply div_le_div_of_nonneg_left (sq_nonneg _) hd
linarith
calc
_ ≤ 1 + t * X ω + (t * X ω) ^ 2 / (2 * (1 - t * X ω / 3)) := hx
_ ≤ 1 + t * X ω + (t * X ω) ^ 2 / (2 * (1 - t * b / 3)) := add_le_add le_rfl hdiv
_ = _ := by dsimp only [c]; ring
have h1 : Integrable (fun _ : Ω => (1 : ℝ)) μ := integrable_const 1
have hit := hi.const_mul t
have hic := hi2.const_mul c
have hineq := integral_mono_ae hexp ((h1.add hit).add hic) hpoint
simp only [Pi.add_apply] at hineq
have hadd := integral_add (h1.add hit) hic
have hadd' := integral_add h1 hit
simp only [Pi.add_apply] at hadd hadd'
rw [hadd, hadd', integral_const_mul, integral_const_mul, hmean] at hineq
simp only [integral_const, probReal_univ, smul_eq_mul, one_mul, mul_zero, add_zero] at hineq
exact hineq.trans (by simpa only [add_comm] using Real.add_one_le_exp (c * ∫ ω, (X ω) ^ 2 ∂μ))
theorem bounded_centered_sum_mgf {n : ℕ} (X : Fin n → Ω → ℝ)
(b v t : ℝ) (hb : 0 < b) (ht : 0 ≤ t) (htb : t * b < 3)
(hind : iIndepFun X μ) (hm : ∀ i, Measurable (X i))
(hi : ∀ i, Integrable (X i) μ) (hmean : ∀ i, (∫ ω, X i ω ∂μ) = 0)
(hbound : ∀ i, ∀ᵐ ω ∂μ, |X i ω| ≤ b)
(hvar : (∑ i, ∫ ω, (X i ω) ^ 2 ∂μ) ≤ v) :
mgf (fun ω => ∑ i, X i ω) μ t ≤ Real.exp (t ^ 2 * v / (2 * (1 - t * b / 3))) := by
have hmgf := hind.mgf_sum hm Finset.univ (t := t)
have hsum : (∑ i, X i) = fun ω => ∑ i, X i ω := by
ext ω
simp only [Finset.sum_apply]
rw [hsum] at hmgf
rw [hmgf]
let c := t ^ 2 / (2 * (1 - t * b / 3))
have hc : 0 ≤ c := div_nonneg (sq_nonneg _) (by linarith)
calc
_ ≤ ∏ i, Real.exp (c * ∫ ω, (X i ω) ^ 2 ∂μ) :=
Finset.prod_le_prod₀ (fun i _ => mgf_nonneg) (fun i _ =>
bounded_centered_mgf μ (X i) b t hb ht htb (hm i) (hi i) (hmean i) (hbound i))
_ = Real.exp (c * ∑ i, ∫ ω, (X i ω) ^ 2 ∂μ) := by
rw [Finset.mul_sum, Real.exp_sum]
_ ≤ Real.exp (c * v) := Real.exp_le_exp.mpr (mul_le_mul_of_nonneg_left hvar hc)
_ = _ := by congr 1; dsimp only [c]; ring
end MGF
end EditApproximation
end OAI
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