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b317718 | 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 | import OAI.Probability.FreeIsing.Trees.CoupledPathProducts
import OAI.Probability.FreeIsing.Trees.PathVariance
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
theorem integral_weighted_path_log_sq_le_twice_unweighted
{ι : Type*} (s : Finset ι) {theta : ℝ} (hθ : theta ^ 2 ≤ 1)
{X Y : ι → Ω → ℝ}
(hX : ∀ i ∈ s, Measurable (X i)) (hY : ∀ i ∈ s, Measurable (Y i))
(hInt : ∀ i ∈ s, Integrable (fun ω => (X i ω - Y i ω) ^ 2) μ) :
(∫ ω, ((∏ i ∈ s, responseAttenuation theta (X i ω)) ^ 2 +
(∏ i ∈ s, responseAttenuation theta (Y i ω)) ^ 2) *
(∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω)))) ^ 2 ∂μ) ≤
2 * ∫ ω, (∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω)))) ^ 2 ∂μ := by
have hQ := (integral_prod_responseAttenuation_sub_sq_le s hθ hX hY hInt).1
have hZ := memLp_sum_log_responseAttenuation_sub s hθ hX hY hInt
calc
_ ≤ ∫ ω, 2 * (∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω)))) ^ 2 ∂μ := by
apply integral_mono hQ (hZ.integrable_sq.const_mul 2)
intro ω
apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)
have hp := prod_responseAttenuation_pos s (fun i => X i ω) hθ
have hp' := prod_responseAttenuation_pos s (fun i => Y i ω) hθ
have hl := prod_attenuation_le_one s (fun i => X i ω) hθ
have hl' := prod_attenuation_le_one s (fun i => Y i ω) hθ
nlinarith
_ = _ := integral_const_mul _ _
theorem integral_weighted_path_log_sq_le_geometric_pair_bound
(s : Finset ℤ) {theta q S : ℝ} (hθ : theta ^ 2 ≤ 1)
(hq0 : 0 ≤ q) (hq1 : q < 1) (hS : 0 ≤ S)
{X Y : ℤ → Ω → ℝ}
(hX : ∀ i ∈ s, Measurable (X i)) (hY : ∀ i ∈ s, Measurable (Y i))
(hInt : ∀ i ∈ s, Integrable (fun ω => (X i ω - Y i ω) ^ 2) μ)
(hpair : ∀ i ∈ s, ∀ j ∈ s,
|∫ ω, (Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω))) *
(Real.log (responseAttenuation theta (X j ω)) -
Real.log (responseAttenuation theta (Y j ω))) ∂μ| ≤
S * pathCorrelationKernel q (j - i)) :
(∫ ω, ((∏ i ∈ s, responseAttenuation theta (X i ω)) ^ 2 +
(∏ i ∈ s, responseAttenuation theta (Y i ω)) ^ 2) *
(∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω)))) ^ 2 ∂μ) ≤
2 * (s.card : ℝ) * S * (2 / (1 - q) ^ 2 - 1) := by
let D := fun i ω => Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω))
have hD : ∀ i ∈ s, MemLp (D i) 2 μ := by
intro i hi
have hm := (lipschitzWith_log_responseAttenuation hθ).continuous.measurable
exact (memLp_two_iff_integrable_sq
(((hm.comp (hX i hi)).sub (hm.comp (hY i hi))).aestronglyMeasurable)).2
(integrable_log_responseAttenuation_sub_sq hθ (hX i hi) (hY i hi) (hInt i hi))
have hsum := integral_sum_sq_le_geometric_pair_bound s D hq0 hq1 hS hD hpair
calc
_ ≤ 2 * ∫ ω, (∑ i ∈ s, D i ω) ^ 2 ∂μ :=
integral_weighted_path_log_sq_le_twice_unweighted s hθ hX hY hInt
_ ≤ 2 * ((s.card : ℝ) * S * (2 / (1 - q) ^ 2 - 1)) :=
mul_le_mul_of_nonneg_left hsum (by norm_num)
_ = _ := by ring
theorem integral_weighted_path_log_sq_le_of_marginals_and_covariance
[IsProbabilityMeasure μ] (s : Finset ℤ)
{theta q S : ℝ} (hθ : theta ^ 2 ≤ 1)
(hq0 : 0 ≤ q) (hq1 : q < 1) (hS : 0 ≤ S)
{X Y : ℤ → Ω → ℝ}
(hXm : ∀ i ∈ s, Measurable (X i)) (hYm : ∀ i ∈ s, Measurable (Y i))
(hX : ∀ i ∈ s, MemLp (X i) 2 μ)
(hlaw : ∀ i ∈ s, IdentDistrib (X i) (Y i) μ μ)
(hcov : ∀ i ∈ s, ∀ j ∈ s,
|covariance
(fun ω => Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω)))
(fun ω => Real.log (responseAttenuation theta (X j ω)) -
Real.log (responseAttenuation theta (Y j ω))) μ| ≤
S * pathCorrelationKernel q (j - i)) :
(∫ ω, ((∏ i ∈ s, responseAttenuation theta (X i ω)) ^ 2 +
(∏ i ∈ s, responseAttenuation theta (Y i ω)) ^ 2) *
(∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω)))) ^ 2 ∂μ) ≤
2 * (s.card : ℝ) * S * (2 / (1 - q) ^ 2 - 1) := by
have hInt : ∀ i ∈ s, Integrable (fun ω => (X i ω - Y i ω) ^ 2) μ := by
intro i hi
exact ((hX i hi).sub ((hlaw i hi).memLp_snd (hX i hi))).integrable_sq
apply integral_weighted_path_log_sq_le_geometric_pair_bound
s hθ hq0 hq1 hS hXm hYm hInt
intro i hi j hj
have hDi := centered_log_response_discrepancy_of_identDistrib hθ (hX i hi) (hlaw i hi)
have hDj := centered_log_response_discrepancy_of_identDistrib hθ (hX j hj) (hlaw j hj)
have h := hcov i hi j hj
rw [covariance_eq_sub hDi.1 hDj.1, hDi.2, hDj.2, zero_mul, sub_zero] at h
exact h
end Problem367
end
end OAI
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