| import OAI.Combinatorics.EditApproximation.Probability.BernsteinEnvelope |
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| namespace OAI |
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| namespace EditApproximation |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators |
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| section MGF |
| variable {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] |
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| 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 |
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| 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)) |
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| 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 |
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| end MGF |
| end EditApproximation |
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| end OAI |
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