| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal NNReal Real |
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| namespace Problem367 |
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| theorem gaussianPDFReal_le_peak (m : ℝ) (v : ℝ≥0) (x : ℝ) : |
| gaussianPDFReal m v x ≤ (Real.sqrt (2 * Real.pi * v))⁻¹ := by |
| unfold gaussianPDFReal |
| have hexp : Real.exp (-(x - m) ^ 2 / (2 * v)) ≤ 1 := by |
| apply Real.exp_le_one_iff.mpr |
| exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (sq_nonneg _)) (by positivity) |
| calc |
| _ ≤ (Real.sqrt (2 * Real.pi * v))⁻¹ * 1 := |
| mul_le_mul_of_nonneg_left hexp (by positivity) |
| _ = _ := mul_one _ |
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| theorem gaussianReal_le_peak_mul_volume |
| (m : ℝ) {v : ℝ≥0} (hv : v ≠ 0) (s : Set ℝ) : |
| gaussianReal m v s ≤ |
| ENNReal.ofReal ((Real.sqrt (2 * Real.pi * v))⁻¹) * volume s := by |
| rw [gaussianReal_apply m hv s] |
| calc |
| _ ≤ ∫⁻ _ in s, ENNReal.ofReal ((Real.sqrt (2 * Real.pi * v))⁻¹) := by |
| apply lintegral_mono |
| intro x |
| exact ENNReal.ofReal_le_ofReal (gaussianPDFReal_le_peak m v x) |
| _ = _ := by simp |
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| theorem gaussianReal_Icc_le |
| (m a b : ℝ) {v : ℝ≥0} (hv : v ≠ 0) : |
| gaussianReal m v (Set.Icc a b) ≤ |
| ENNReal.ofReal ((b - a) / Real.sqrt (2 * Real.pi * v)) := by |
| have h := gaussianReal_le_peak_mul_volume m hv (Set.Icc a b) |
| rw [Real.volume_Icc, ← ENNReal.ofReal_mul (by positivity)] at h |
| simpa only [div_eq_mul_inv, mul_comm] using h |
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| theorem gaussianReal_abs_le |
| (m ε : ℝ) {v : ℝ≥0} (hv : v ≠ 0) : |
| gaussianReal m v {x | |x| ≤ ε} ≤ |
| ENNReal.ofReal (2 * ε / Real.sqrt (2 * Real.pi * v)) := by |
| have hset : {x : ℝ | |x| ≤ ε} = Set.Icc (-ε) ε := by |
| ext x |
| simp only [Set.mem_ofPred_eq, Set.mem_Icc, abs_le] |
| rw [hset] |
| convert gaussianReal_Icc_le m (-ε) ε hv using 2 |
| ring |
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| theorem gaussianReal_abs_le_of_variance_ge |
| (m : ℝ) {ε : ℝ} (hε : 0 ≤ ε) {δ v : ℝ≥0} |
| (hδ : 0 < δ) (hδv : δ ≤ v) : |
| gaussianReal m v {x | |x| ≤ ε} ≤ |
| ENNReal.ofReal (2 * ε / Real.sqrt (2 * Real.pi * δ)) := by |
| have hv : v ≠ 0 := ne_of_gt (lt_of_lt_of_le hδ hδv) |
| apply (gaussianReal_abs_le m ε hv).trans |
| apply ENNReal.ofReal_le_ofReal |
| have hsqrt : Real.sqrt (2 * Real.pi * (δ : ℝ)) ≤ |
| Real.sqrt (2 * Real.pi * (v : ℝ)) := by |
| apply Real.sqrt_le_sqrt |
| exact mul_le_mul_of_nonneg_left (by exact_mod_cast hδv) (by positivity) |
| apply div_le_div_of_nonneg_left (by positivity) _ hsqrt |
| apply Real.sqrt_pos.mpr |
| exact mul_pos (by positivity) (by exact_mod_cast hδ) |
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| def gaussianSmallRadius (δ : ℝ≥0) : ℝ := |
| Real.sqrt (2 * Real.pi * δ) / 4 |
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| lemma gaussianSmallRadius_pos {δ : ℝ≥0} (hδ : 0 < δ) : |
| 0 < gaussianSmallRadius δ := by |
| unfold gaussianSmallRadius |
| positivity |
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| lemma gaussianReal_smallBall_le_half |
| (m : ℝ) {δ v : ℝ≥0} (hδ : 0 < δ) (hδv : δ ≤ v) : |
| gaussianReal m v {x | |x| ≤ gaussianSmallRadius δ} ≤ (1 / 2 : ℝ≥0∞) := by |
| have h := gaussianReal_abs_le_of_variance_ge m |
| (gaussianSmallRadius_pos hδ).le hδ hδv |
| have hs : Real.sqrt (2 * Real.pi * (δ : ℝ)) ≠ 0 := by |
| apply ne_of_gt |
| apply Real.sqrt_pos.mpr |
| exact mul_pos (by positivity) (by exact_mod_cast hδ) |
| have hc : 2 * gaussianSmallRadius δ / Real.sqrt (2 * Real.pi * δ) = (1 / 2 : ℝ) := by |
| unfold gaussianSmallRadius |
| field_simp |
| ring |
| rw [hc] at h |
| have hhalf : ENNReal.ofReal (1 / 2 : ℝ) = (1 / 2 : ℝ≥0∞) := by |
| norm_num [ENNReal.ofReal_div_of_pos] |
| simpa only [hhalf] using h |
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| theorem smallBall_le_half_of_hasGaussianLaw |
| {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {X : Ω → ℝ} |
| {m : ℝ} {δ v : ℝ≥0} (hX : HasLaw X (gaussianReal m v) P) |
| (hδ : 0 < δ) (hδv : δ ≤ v) : |
| P {ω | |X ω| ≤ gaussianSmallRadius δ} ≤ (1 / 2 : ℝ≥0∞) := by |
| rw [hX.measure_eq (p := fun x : ℝ => |x| ≤ gaussianSmallRadius δ) |
| (measurableSet_le measurable_id.abs measurable_const)] |
| exact gaussianReal_smallBall_le_half m hδ hδv |
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| theorem integral_le_one_sub_half_gap |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| {f : Ω → ℝ} {s : Set Ω} {c : ℝ} |
| (hf : Integrable f P) (hs : MeasurableSet s) (hc : 0 ≤ c) |
| (hupper : ∀ᵐ x ∂P, f x ≤ 1) |
| (houtside : ∀ᵐ x ∂P, x ∉ s → f x ≤ 1 - c) |
| (hmass : P s ≤ (1 / 2 : ℝ≥0∞)) : |
| ∫ x, f x ∂P ≤ 1 - c / 2 := by |
| have hsreal : P.real s ≤ 1 / 2 := by |
| change (P s).toReal ≤ 1 / 2 |
| have h := ENNReal.toReal_mono (by norm_num : (1 / 2 : ℝ≥0∞) ≠ ∞) hmass |
| simpa only [ENNReal.toReal_div, ENNReal.toReal_one, ENNReal.toReal_ofNat] using h |
| have hi : Integrable (fun x => (1 - c) + s.indicator (fun _ => c) x) P := |
| (integrable_const _).add ((integrable_const _).indicator hs) |
| have hvalue : (∫ x, (1 - c) + s.indicator (fun _ => c) x ∂P) = |
| 1 - c + P.real s * c := by |
| rw [integral_add (integrable_const _) ((integrable_const _).indicator hs), |
| integral_const, integral_indicator_const _ hs] |
| simp |
| have hle : ∫ x, f x ∂P ≤ ∫ x, (1 - c) + s.indicator (fun _ => c) x ∂P := by |
| apply integral_mono_ae hf hi |
| filter_upwards [hupper, houtside] with x hx ho |
| by_cases hxs : x ∈ s |
| · simp only [Set.indicator_of_mem hxs] |
| linarith |
| · simpa only [Set.indicator_of_notMem hxs, add_zero] using ho hxs |
| rw [hvalue] at hle |
| nlinarith |
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| theorem integral_gaussian_le_one_sub_half_gap |
| (m : ℝ) {δ v : ℝ≥0} (hδ : 0 < δ) (hδv : δ ≤ v) |
| {f : ℝ → ℝ} {c : ℝ} (hf : Integrable f (gaussianReal m v)) (hc : 0 ≤ c) |
| (hupper : ∀ x, f x ≤ 1) |
| (houtside : ∀ x, gaussianSmallRadius δ < |x| → f x ≤ 1 - c) : |
| ∫ x, f x ∂gaussianReal m v ≤ 1 - c / 2 := by |
| apply integral_le_one_sub_half_gap (gaussianReal m v) hf |
| (measurableSet_le measurable_id.abs measurable_const) hc |
| (Filter.Eventually.of_forall hupper) |
| · exact Filter.Eventually.of_forall fun x hx => houtside x (lt_of_not_ge hx) |
| · exact gaussianReal_smallBall_le_half m hδ hδv |
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| end Problem367 |
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| end |
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| end OAI |
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