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f61e547 | 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 | import OAI.Probability.FreeIsing.Conditioning.CovarianceLimit
namespace OAI
noncomputable section
open MeasureTheory Filter
open scoped Topology
namespace Problem367
variable {Ω : Type*} [mΩ : MeasurableSpace Ω]
def posteriorResponse (μ : Measure Ω) (m : MeasurableSpace Ω)
(f g : Ω → ℝ) (ω : Ω) : ℝ :=
posteriorCovariance (mΩ := mΩ) μ m f g ω / (1 - (μ[f | m] ω) ^ 2)
lemma stronglyMeasurable_posteriorResponse (μ : Measure Ω)
(m : MeasurableSpace Ω) (f g : Ω → ℝ) :
StronglyMeasurable[m] (posteriorResponse (mΩ := mΩ) μ m f g) :=
(stronglyMeasurable_posteriorCovariance (mΩ := mΩ) μ m f g).div
(stronglyMeasurable_const.sub (stronglyMeasurable_condExp.pow 2))
lemma abs_covariance_le_variance_of_binary_moment_bounds
{a b c : ℝ} (ha : |a| ≤ 1)
(h₁ : c ≤ 1 + a - b) (h₂ : c ≤ 1 - a + b)
(h₃ : -1 - a - b ≤ c) (h₄ : -1 + a + b ≤ c) :
|c - a * b| ≤ 1 - a ^ 2 := by
have ham : 0 ≤ 1 - a := by linarith only [(abs_le.mp ha).2]
have hap : 0 ≤ 1 + a := by linarith only [(abs_le.mp ha).1]
apply abs_le.mpr
constructor
· nlinarith [mul_nonneg ham (show 0 ≤ 1 + a + b + c by linarith),
mul_nonneg hap (show 0 ≤ 1 - a - b + c by linarith)]
· nlinarith [mul_nonneg ham (show 0 ≤ 1 + a - b - c by linarith),
mul_nonneg hap (show 0 ≤ 1 - a + b - c by linarith)]
theorem ae_abs_posteriorCovariance_le_binary_variance
(μ : Measure Ω) [IsFiniteMeasure μ]
(m : MeasurableSpace Ω) (hm : m ≤ mΩ)
{f g : Ω → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
(hbinary : ∀ᵐ ω ∂μ, f ω = -1 ∨ f ω = 1)
(hgb : ∀ᵐ ω ∂μ, |g ω| ≤ 1) :
∀ᵐ ω ∂μ, |posteriorCovariance (mΩ := mΩ) μ m f g ω| ≤ 1 - (μ[f | m] ω) ^ 2 := by
have hfb : ∀ᵐ ω ∂μ, |f ω| ≤ 1 := by
filter_upwards [hbinary] with ω hω
rcases hω with hω | hω <;> simp [hω]
have hfg : Integrable (fun ω => f ω * g ω) μ := by
apply Integrable.of_bound (hf.aestronglyMeasurable.mul hg.aestronglyMeasurable) 1
filter_upwards [hfb, hgb] with ω hfw hgw
simpa only [Real.norm_eq_abs, Pi.mul_apply, abs_mul, one_mul] using
mul_le_mul hfw hgw (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 1)
have h₁ : (fun ω => f ω * g ω) ≤ᵐ[μ] (fun ω => 1 + f ω - g ω) := by
filter_upwards [hbinary, hgb] with ω hω hgω
rcases hω with hω | hω <;> rw [hω] <;> nlinarith [(abs_le.mp hgω).1, (abs_le.mp hgω).2]
have h₂ : (fun ω => f ω * g ω) ≤ᵐ[μ] (fun ω => 1 - f ω + g ω) := by
filter_upwards [hbinary, hgb] with ω hω hgω
rcases hω with hω | hω <;> rw [hω] <;> nlinarith [(abs_le.mp hgω).1, (abs_le.mp hgω).2]
have h₃ : (fun ω => -1 - f ω - g ω) ≤ᵐ[μ] (fun ω => f ω * g ω) := by
filter_upwards [hbinary, hgb] with ω hω hgω
rcases hω with hω | hω <;> rw [hω] <;> nlinarith [(abs_le.mp hgω).1, (abs_le.mp hgω).2]
have h₄ : (fun ω => -1 + f ω + g ω) ≤ᵐ[μ] (fun ω => f ω * g ω) := by
filter_upwards [hbinary, hgb] with ω hω hgω
rcases hω with hω | hω <;> rw [hω] <;> nlinarith [(abs_le.mp hgω).1, (abs_le.mp hgω).2]
have hc₁ := condExp_mono (m := m) hfg (((integrable_const (1 : ℝ)).add hf).sub hg) h₁
have hc₂ := condExp_mono (m := m) hfg (((integrable_const (1 : ℝ)).sub hf).add hg) h₂
have hc₃ := condExp_mono (m := m) (((integrable_const (-1 : ℝ)).sub hf).sub hg) hfg h₃
have hc₄ := condExp_mono (m := m) (((integrable_const (-1 : ℝ)).add hf).add hg) hfg h₄
filter_upwards [hc₁, hc₂, hc₃, hc₄,
condExp_sub ((integrable_const (1 : ℝ)).add hf) hg m,
condExp_add (integrable_const (1 : ℝ)) hf m,
condExp_add ((integrable_const (1 : ℝ)).sub hf) hg m,
condExp_sub (integrable_const (1 : ℝ)) hf m,
condExp_sub ((integrable_const (-1 : ℝ)).sub hf) hg m,
condExp_sub (integrable_const (-1 : ℝ)) hf m,
condExp_add ((integrable_const (-1 : ℝ)).add hf) hg m,
condExp_add (integrable_const (-1 : ℝ)) hf m,
ae_bdd_abs_condExp_of_ae_bdd_abs (m := m) hfb]
with ω hc₁ hc₂ hc₃ hc₄ he₁ he₂ he₃ he₄ he₅ he₆ he₇ he₈ hb
simp only [Pi.add_apply, Pi.sub_apply, condExp_const hm] at *
exact abs_covariance_le_variance_of_binary_moment_bounds
(a := μ[f | m] ω) (b := μ[g | m] ω)
(c := μ[fun x => f x * g x | m] ω) hb
(by linarith) (by linarith) (by linarith) (by linarith)
theorem ae_abs_posteriorResponse_le_one
(μ : Measure Ω) [IsFiniteMeasure μ]
(m : MeasurableSpace Ω) (hm : m ≤ mΩ)
{f g : Ω → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
(hbinary : ∀ᵐ ω ∂μ, f ω = -1 ∨ f ω = 1)
(hgb : ∀ᵐ ω ∂μ, |g ω| ≤ 1) :
∀ᵐ ω ∂μ, |posteriorResponse (mΩ := mΩ) μ m f g ω| ≤ 1 := by
filter_upwards [ae_abs_posteriorCovariance_le_binary_variance (mΩ := mΩ) μ m hm hf hg hbinary hgb]
with ω hω
have hnonneg : 0 ≤ 1 - (μ[f | m] ω) ^ 2 := (abs_nonneg _).trans hω
change |posteriorCovariance (mΩ := mΩ) μ m f g ω / (1 - (μ[f | m] ω) ^ 2)| ≤ 1
by_cases hz : 1 - (μ[f | m] ω) ^ 2 = 0
· simp [hz]
· rw [abs_div, abs_of_nonneg hnonneg]
exact (div_le_one (lt_of_le_of_ne hnonneg (Ne.symm hz))).mpr hω
theorem tendsto_ae_posteriorResponse
(μ : Measure Ω) [IsFiniteMeasure μ] (F : Filtration ℕ mΩ)
(f g : Ω → ℝ)
(hinterior : ∀ᵐ ω ∂μ, μ[f | ⨆ n, F n] ω ∈ Set.Ioo (-1 : ℝ) 1) :
∀ᵐ ω ∂μ, Tendsto (fun n => posteriorResponse μ (F n) f g ω) atTop
(𝓝 (posteriorResponse μ (⨆ n, F n) f g ω)) := by
filter_upwards [tendsto_ae_posteriorCovariance μ F f g,
tendsto_ae_condExp (μ := μ) (ℱ := F) f, hinterior] with ω hc hm hi
have hden : 1 - (μ[f | ⨆ n, F n] ω) ^ 2 ≠ 0 := by
have hsq : (μ[f | ⨆ n, F n] ω) ^ 2 < 1 := by
nlinarith [mul_pos (show 0 < 1 - μ[f | ⨆ n, F n] ω by linarith [hi.2])
(show 0 < 1 + μ[f | ⨆ n, F n] ω by linarith [hi.1])]
exact ne_of_gt (sub_pos.mpr hsq)
exact hc.div (tendsto_const_nhds.sub (hm.pow 2)) hden
theorem tendsto_integral_sq_sub_posteriorResponse
(μ : Measure Ω) [IsFiniteMeasure μ] (F : Filtration ℕ mΩ)
{f g : Ω → ℝ} (hf : Integrable f μ) (hg : Integrable g μ)
(hbinary : ∀ᵐ ω ∂μ, f ω = -1 ∨ f ω = 1)
(hgb : ∀ᵐ ω ∂μ, |g ω| ≤ 1)
(hinterior : ∀ᵐ ω ∂μ, μ[f | ⨆ n, F n] ω ∈ Set.Ioo (-1 : ℝ) 1) :
Tendsto (fun n => ∫ ω,
(posteriorResponse μ (F n) f g ω -
posteriorResponse μ (⨆ n, F n) f g ω) ^ 2 ∂μ) atTop (𝓝 0) := by
apply tendsto_integral_sq_sub_of_bounded_ae μ
(fun n => ((stronglyMeasurable_posteriorResponse μ (F n) f g).mono
(F.le n)).aestronglyMeasurable)
(((stronglyMeasurable_posteriorResponse μ (⨆ n, F n) f g).mono
(iSup_le fun n => F.le n)).aestronglyMeasurable) (by norm_num : (0 : ℝ) ≤ 1)
· intro n
simpa only [Real.norm_eq_abs] using
ae_abs_posteriorResponse_le_one μ (F n) (F.le n) hf hg hbinary hgb
· simpa only [Real.norm_eq_abs] using
ae_abs_posteriorResponse_le_one μ (⨆ n, F n) (iSup_le fun n => F.le n)
hf hg hbinary hgb
· exact tendsto_ae_posteriorResponse μ F f g hinterior
end Problem367
end
end OAI
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