| import Mathlib |
|
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| namespace OAI |
|
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| open scoped BigOperators Topology NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter Real |
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| noncomputable section |
|
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| namespace SquareRootDegree |
| namespace SmoothGaussian |
|
|
| /-- A continuous probability of passing a rare interval test. It vanishes |
| outside the interval, and is rounded to deterministic tests on each finite |
| input alphabet before constructing any final observation. -/ |
| def tent (a h x : ℝ) : ℝ := max 0 (1 - |x-a| / h) |
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| @[fun_prop] theorem continuous_tent (a h : ℝ) : Continuous (tent a h) := by |
| unfold tent |
| fun_prop |
|
|
| theorem tent_nonneg (a h x : ℝ) : 0 ≤ tent a h x := le_max_left _ _ |
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| theorem tent_le_one {h : ℝ} (hh : 0 < h) (a x : ℝ) : tent a h x ≤ 1 := by |
| apply max_le zero_le_one |
| have := div_nonneg (abs_nonneg (x-a)) hh.le |
| linarith |
|
|
| theorem tent_eq_zero {a h x : ℝ} (hh : 0 < h) (hx : h ≤ |x-a|) : |
| tent a h x = 0 := by |
| unfold tent |
| apply max_eq_left |
| have : 1 ≤ |x-a|/h := (one_le_div hh).mpr hx |
| linarith |
|
|
| theorem tent_ne_zero_iff {a h x : ℝ} (hh : 0 < h) : |
| tent a h x ≠ 0 ↔ |x-a| < h := by |
| constructor |
| · intro hn |
| exact lt_of_not_ge (fun hx => hn (tent_eq_zero hh hx)) |
| · intro hx |
| apply ne_of_gt |
| apply lt_of_lt_of_le _ (le_max_right _ _) |
| have := (div_lt_one hh).mpr hx |
| linarith |
|
|
| theorem tent_support {a h : ℝ} (hh : 0 < h) : |
| Function.support (tent a h) ⊆ Set.Icc (a-h) (a+h) := by |
| intro x hx |
| have he := (tent_ne_zero_iff hh).mp hx |
| rw [abs_lt] at he |
| constructor <;> linarith |
|
|
| theorem tent_compactSupport {a h : ℝ} (hh : 0 < h) : HasCompactSupport (tent a h) := by |
| apply HasCompactSupport.intro (isCompact_Icc (a := a-h) (b := a+h)) |
| intro x hx |
| by_contra hn |
| exact hx (tent_support hh hn) |
|
|
| theorem tent_half {a h x : ℝ} (hh : 0 < h) (hx : |x-a| ≤ h/2) : |
| 1/2 ≤ tent a h x := by |
| apply le_trans _ (le_max_right _ _) |
| have : |x-a|/h ≤ 1/2 := (div_le_iff₀ hh).mpr (by linarith) |
| linarith |
|
|
| theorem integrable_tent_mul {a h : ℝ} (hh : 0 < h) (p : ℝ → ℝ) (hp : Continuous p) |
| (μ : Measure ℝ) [IsFiniteMeasureOnCompacts μ] : |
| Integrable (fun x => tent a h x * p x) μ := by |
| apply ((continuous_tent a h).mul hp).integrable_of_hasCompactSupport |
| exact (tent_compactSupport hh).mul_right |
|
|
| theorem pdf_lower {A x : ℝ} (hx : |x| ≤ A) : |
| (Real.sqrt (2*Real.pi))⁻¹ * Real.exp (-(A^2)/2) ≤ gaussianPDFReal 0 1 x := by |
| simp only [gaussianPDFReal, NNReal.coe_one, mul_one, sub_zero] |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply Real.exp_le_exp.mpr |
| have : x^2 ≤ A^2 := by nlinarith [abs_nonneg x, sq_abs x] |
| linarith |
|
|
| theorem beta_lower {a h : ℝ} (hh : 0 < h) : |
| h/2 * ((Real.sqrt (2*Real.pi))⁻¹ * Real.exp (-((|a|+h)^2)/2)) ≤ |
| ∫ x, tent a h x ∂gaussianReal 0 1 := by |
| let c := (Real.sqrt (2*Real.pi))⁻¹ * Real.exp (-((|a|+h)^2)/2) |
| have hc : 0 ≤ c := by dsimp [c]; positivity |
| have ha := integrable_tent_mul (a := a) hh (gaussianPDFReal 0 1) (by |
| unfold gaussianPDFReal; fun_prop) volume |
| rw [integral_gaussianReal_eq_integral_smul (by norm_num : (1:ℝ≥0) ≠ 0)] |
| change h/2*c ≤ ∫ x, gaussianPDFReal 0 1 x * tent a h x |
| have hi : Integrable (fun x => gaussianPDFReal 0 1 x * tent a h x) volume := by |
| simpa only [mul_comm] using ha |
| have hset : ∫ x in Set.Icc (a-h/2) (a+h/2), c/2 ≤ |
| ∫ x in Set.Icc (a-h/2) (a+h/2), gaussianPDFReal 0 1 x * tent a h x := by |
| apply integral_mono_ae (integrable_const _) hi.integrableOn |
| filter_upwards [ae_restrict_mem measurableSet_Icc] with x hx |
| have hab : |x-a| ≤ h/2 := abs_le.mpr (by constructor <;> linarith [hx.1,hx.2]) |
| have hp : c ≤ gaussianPDFReal 0 1 x := by |
| apply pdf_lower |
| calc |x| = |(x-a)+a| := by congr 1; ring |
| _ ≤ |x-a|+|a| := abs_add_le _ _ |
| _ ≤ |a|+h := by linarith |
| have hw := tent_half hh hab |
| calc c/2 = c * (1/2) := by ring |
| _ ≤ gaussianPDFReal 0 1 x * tent a h x := |
| mul_le_mul hp hw (by norm_num) (gaussianPDFReal_nonneg ..) |
| have hs : ∫ x in Set.Icc (a-h/2) (a+h/2), gaussianPDFReal 0 1 x * tent a h x ≤ |
| ∫ x, gaussianPDFReal 0 1 x * tent a h x := by |
| exact integral_mono_measure Measure.restrict_le_self |
| (ae_of_all _ (fun x => mul_nonneg (gaussianPDFReal_nonneg ..) (tent_nonneg ..))) hi |
| apply le_trans _ (hset.trans hs) |
| simp only [integral_const, Measure.real, Measure.restrict_apply_univ, Real.volume_Icc, smul_eq_mul] |
| rw [ENNReal.toReal_ofReal (by linarith : 0 ≤ a+h/2-(a-h/2))] |
| ring_nf |
| exact le_rfl |
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| def mass (μ : Measure ℝ) (w : ℝ → ℝ) : ℝ := ∫ x, w x ∂μ |
|
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| def mean (μ : Measure ℝ) (w : ℝ → ℝ) : ℝ := (∫ x, w x * x ∂μ) / mass μ w |
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| def variance (μ : Measure ℝ) (w : ℝ → ℝ) : ℝ := |
| (∫ x, w x * x^2 ∂μ) / mass μ w - mean μ w ^ 2 |
|
|
| theorem integral_weight_sub_sq (μ : Measure ℝ) (w : ℝ → ℝ) |
| (h0 : Integrable w μ) (h1 : Integrable (fun x => w x * x) μ) |
| (h2 : Integrable (fun x => w x * x^2) μ) (c : ℝ) : |
| (∫ x, w x * (x-c)^2 ∂μ) = (∫ x, w x * x^2 ∂μ) - |
| 2*c*(∫ x, w x * x ∂μ) + c^2*mass μ w := by |
| have he : (fun x => w x * (x-c)^2) = fun x => |
| w x * x^2 - (2*c)*(w x*x) + c^2*w x := by funext x; ring |
| rw [he, integral_add (f := fun x => w x*x^2-(2*c)*(w x*x)) |
| (g := fun x => c^2*w x) (h2.sub (h1.const_mul (2*c))) (h0.const_mul (c^2)), |
| integral_sub h2 (h1.const_mul (2*c)), integral_const_mul, integral_const_mul] |
| rfl |
|
|
| theorem mean_mul_mass (μ : Measure ℝ) (w : ℝ → ℝ) (hb : mass μ w ≠ 0) : |
| mean μ w * mass μ w = ∫ x, w x * x ∂μ := by |
| simp [mean, hb] |
|
|
| theorem variance_mul_mass (μ : Measure ℝ) (w : ℝ → ℝ) |
| (h0 : Integrable w μ) (h1 : Integrable (fun x => w x * x) μ) |
| (h2 : Integrable (fun x => w x * x^2) μ) (hb : mass μ w ≠ 0) : |
| variance μ w * mass μ w = ∫ x, w x * (x-mean μ w)^2 ∂μ := by |
| rw [integral_weight_sub_sq μ w h0 h1 h2] |
| have hm := mean_mul_mass μ w hb |
| rw [← hm] |
| unfold variance |
| field_simp [hb] |
| ring |
|
|
| theorem weighted_interval_bounds (μ : Measure ℝ) (w : ℝ → ℝ) {a h : ℝ} |
| (hh : 0 ≤ h) (hw : ∀ x, 0 ≤ w x) |
| (hs : ∀ x, w x ≠ 0 → |x-a| ≤ h) |
| (h0 : Integrable w μ) (h1 : Integrable (fun x => w x*x) μ) |
| (h2 : Integrable (fun x => w x*x^2) μ) (hb : 0 < mass μ w) : |
| |mean μ w-a| ≤ h ∧ 0 ≤ variance μ w ∧ variance μ w ≤ h^2 := by |
| have hlo : (a-h)*mass μ w ≤ ∫ x, w x*x ∂μ := by |
| rw [mass, ← integral_const_mul] |
| apply integral_mono (h0.const_mul _) h1 |
| intro x |
| by_cases hx : w x = 0 |
| · simp [hx] |
| · have ht := (abs_le.mp (hs x hx)).1 |
| nlinarith [mul_nonneg (hw x) (show 0 ≤ x-(a-h) by linarith)] |
| have hhi : (∫ x, w x*x ∂μ) ≤ (a+h)*mass μ w := by |
| rw [mass, ← integral_const_mul] |
| apply integral_mono h1 (h0.const_mul _) |
| intro x |
| by_cases hx : w x = 0 |
| · simp [hx] |
| · have ht := (abs_le.mp (hs x hx)).2 |
| nlinarith [mul_nonneg (hw x) (show 0 ≤ (a+h)-x by linarith)] |
| have hm := mean_mul_mass μ w hb.ne' |
| have hmean : |mean μ w-a| ≤ h := by |
| rw [abs_le] |
| constructor <;> nlinarith |
| have hv := variance_mul_mass μ w h0 h1 h2 hb.ne' |
| have hv0 : 0 ≤ variance μ w := by |
| have hn : 0 ≤ ∫ x, w x*(x-mean μ w)^2 ∂μ := |
| integral_nonneg (fun x => mul_nonneg (hw x) (sq_nonneg (x-mean μ w))) |
| rw [← hv] at hn |
| exact nonneg_of_mul_nonneg_left hn hb |
| have hsq : ∫ x, w x*(x-a)^2 ∂μ ≤ h^2*mass μ w := by |
| rw [mass, ← integral_const_mul] |
| have he : Integrable (fun x => w x*(x-a)^2) μ := by |
| convert! (h2.sub (h1.const_mul (2*a))).add (h0.const_mul (a^2)) using 1 |
| funext x; simp only [Pi.sub_apply, Pi.add_apply]; ring |
| apply integral_mono he (h0.const_mul _) |
| intro x |
| by_cases hx : w x = 0 |
| · simp [hx] |
| · have ht : (x-a)^2 ≤ h^2 := by nlinarith [sq_abs (x-a),hs x hx, abs_nonneg (x-a)] |
| nlinarith [mul_nonneg (hw x) (sub_nonneg.mpr ht)] |
| rw [integral_weight_sub_sq μ w h0 h1 h2] at hsq |
| have hmoment : (∫ x, w x*x^2 ∂μ) = (variance μ w + mean μ w^2)*mass μ w := by |
| dsimp [variance] |
| field_simp [hb.ne'] |
| ring |
| have hvar : variance μ w ≤ h^2 := by |
| have he := mul_nonneg hb.le (sq_nonneg (mean μ w-a)) |
| rw [hmoment, ← hm] at hsq |
| nlinarith |
| exact ⟨hmean,hv0,hvar⟩ |
|
|
| theorem tent_mass_pos {a h : ℝ} (hh : 0 < h) : |
| 0 < mass (gaussianReal 0 1) (tent a h) := by |
| apply lt_of_lt_of_le _ (beta_lower hh) |
| positivity |
|
|
| theorem tent_moments {a h : ℝ} (hh : 0 < h) : |
| |mean (gaussianReal 0 1) (tent a h)-a| ≤ h ∧ |
| 0 ≤ variance (gaussianReal 0 1) (tent a h) ∧ |
| variance (gaussianReal 0 1) (tent a h) ≤ h^2 := by |
| apply weighted_interval_bounds (gaussianReal 0 1) (tent a h) hh.le (tent_nonneg a h) |
| · intro x hx |
| exact le_of_lt ((tent_ne_zero_iff hh).mp hx) |
| · simpa using integrable_tent_mul (a := a) hh (fun _ => 1) continuous_const (gaussianReal 0 1) |
| · exact integrable_tent_mul hh id continuous_id (gaussianReal 0 1) |
| · exact integrable_tent_mul hh (fun x => x^2) (by fun_prop) (gaussianReal 0 1) |
| · exact tent_mass_pos hh |
|
|
| end SmoothGaussian |
| end SquareRootDegree |
|
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| end |
|
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| end OAI |
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