| import OAI.Probability.FreeIsing.Energy.CoupledAttenuation |
| import OAI.Probability.FreeIsing.Energy.MomentTail |
| import OAI.Probability.FreeIsing.Estimates.LogInterpolation |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
|
|
| namespace Problem367 |
|
|
| def cavityLogVarianceCoefficient (theta : ℝ) : ℝ := |
| 4 / (1 - theta) ^ 2 |
|
|
| def cavityLogTailCoefficient (theta M : ℝ) : ℝ := |
| 8 * (M / (1 - theta)) ^ 2 |
|
|
| theorem cavityLogVarianceCoefficient_nonneg (theta : ℝ) : |
| 0 ≤ cavityLogVarianceCoefficient theta := by |
| unfold cavityLogVarianceCoefficient |
| positivity |
|
|
| theorem cavityLogTailCoefficient_nonneg (theta M : ℝ) : |
| 0 ≤ cavityLogTailCoefficient theta M := by |
| unfold cavityLogTailCoefficient |
| positivity |
|
|
| section Coupling |
|
|
| variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
| variable {theta : ℝ} {X Y X' Y' H K H' K' : Ω → ℝ} |
|
|
| theorem integral_cavity_log_discrepancy_sq_le_of_marginal_tails |
| (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) |
| (hX : Measurable X) (hX' : Measurable X') |
| (hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ) |
| (hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ) |
| (heq : ∀ᵐ ω ∂μ, |
| X ω + cavityMessage theta (Y ω) = H ω ∧ |
| Y ω + cavityMessage theta (X ω) = K ω ∧ |
| X' ω + cavityMessage theta (Y' ω) = H' ω ∧ |
| Y' ω + cavityMessage theta (X' ω) = K' ω) |
| {V M R B₁ B₂ : ℝ} (hM : 0 ≤ M) |
| (hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M) |
| (hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V) |
| (hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V) |
| (hTail : μ.real {ω | R < |X ω|} ≤ B₁) |
| (hTail' : μ.real {ω | R < |X' ω|} ≤ B₂) : |
| (∫ ω, (Real.log (responseAttenuation theta (X ω)) - |
| Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ |
| cavityLogVarianceCoefficient theta * R ^ 2 * V + |
| min (cavityLogVarianceCoefficient theta * V) |
| (4 * (M / (1 - theta)) ^ 2 * (B₁ + B₂)) := by |
| have hθsq : theta ^ 2 ≤ 1 := by nlinarith |
| have hInt := integrable_cavity_pair_sub_sq hθ0 hθ1 hX hX' hH hK heq |
| have hVar := integral_cavity_pair_sub_sq_le hθ0 hθ1 hX hX' hH hK heq hHV hKV |
| have hBound : ∀ᵐ ω ∂μ, |X ω - X' ω| ≤ M / (1 - theta) := by |
| filter_upwards [heq, hbound] with ω he hb |
| exact cavity_pair_sub_abs_le hθ0 hθ1 he.1 he.2.1 he.2.2.1 he.2.2.2 hb.1 hb.2 |
| have h := integral_log_responseAttenuation_sub_sq_le_of_marginal_tails |
| hθsq hX hX' hInt (div_nonneg hM (by linarith)) hBound hVar hTail hTail' |
| have hfirst : 4 * R ^ 2 * (V / (1 - theta) ^ 2) = |
| cavityLogVarianceCoefficient theta * R ^ 2 * V := by |
| unfold cavityLogVarianceCoefficient |
| ring |
| have hsecond : 4 * (V / (1 - theta) ^ 2) = |
| cavityLogVarianceCoefficient theta * V := by |
| unfold cavityLogVarianceCoefficient |
| ring |
| simpa only [hfirst, hsecond] using h |
|
|
| theorem integral_cavity_log_discrepancy_small_time |
| (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) |
| (hX : Measurable X) (hX' : Measurable X') |
| (hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ) |
| (hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ) |
| (heq : ∀ᵐ ω ∂μ, |
| X ω + cavityMessage theta (Y ω) = H ω ∧ |
| Y ω + cavityMessage theta (X ω) = K ω ∧ |
| X' ω + cavityMessage theta (Y' ω) = H' ω ∧ |
| Y' ω + cavityMessage theta (X' ω) = K' ω) |
| {V M t c : ℝ} (hM : 0 ≤ M) (ht : 0 < t) |
| (hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M) |
| (hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V) |
| (hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V) |
| (hTail : μ.real {ω | t ^ (1 / 8 : ℝ) < |X ω|} ≤ |
| Real.exp (-c * (Real.log (1 / t)) ^ 2)) |
| (hTail' : μ.real {ω | t ^ (1 / 8 : ℝ) < |X' ω|} ≤ |
| Real.exp (-c * (Real.log (1 / t)) ^ 2)) : |
| (∫ ω, (Real.log (responseAttenuation theta (X ω)) - |
| Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ |
| cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V + |
| min (cavityLogVarianceCoefficient theta * V) |
| (cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2)) := by |
| have h := integral_cavity_log_discrepancy_sq_le_of_marginal_tails |
| hθ0 hθ1 hX hX' hH hK heq hM hbound hHV hKV hTail hTail' |
| have hr : (t ^ (1 / 8 : ℝ)) ^ 2 = t ^ (1 / 4 : ℝ) := by |
| rw [← Real.rpow_natCast, ← Real.rpow_mul ht.le] |
| norm_num |
| have htail : 4 * (M / (1 - theta)) ^ 2 * |
| (Real.exp (-c * (Real.log (1 / t)) ^ 2) + Real.exp (-c * (Real.log (1 / t)) ^ 2)) = |
| cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2) := by |
| unfold cavityLogTailCoefficient |
| ring |
| simpa only [hr, htail] using h |
|
|
| theorem integral_cavity_log_discrepancy_small_time_of_moments |
| (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) |
| (hX : Measurable X) (hX' : Measurable X') |
| (hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ) |
| (hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ) |
| (heq : ∀ᵐ ω ∂μ, |
| X ω + cavityMessage theta (Y ω) = H ω ∧ |
| Y ω + cavityMessage theta (X ω) = K ω ∧ |
| X' ω + cavityMessage theta (Y' ω) = H' ω ∧ |
| Y' ω + cavityMessage theta (X' ω) = K' ω) |
| {V M t A : ℝ} (hM : 0 ≤ M) (ht : 0 < t) (hA : 0 < A) |
| (hsmall : 32 * A ≤ Real.log (1 / t)) |
| (hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M) |
| (hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V) |
| (hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V) |
| (hInt : ∀ p : ℕ, 1 ≤ p → Integrable (fun ω => |X ω| ^ p) μ) |
| (hInt' : ∀ p : ℕ, 1 ≤ p → Integrable (fun ω => |X' ω| ^ p) μ) |
| (hmom : ∀ p : ℕ, 1 ≤ p → (∫ ω, |X ω| ^ p ∂μ) ≤ |
| Real.exp (A * (p : ℝ) ^ 2) * t ^ ((p : ℝ) / 4)) |
| (hmom' : ∀ p : ℕ, 1 ≤ p → (∫ ω, |X' ω| ^ p ∂μ) ≤ |
| Real.exp (A * (p : ℝ) ^ 2) * t ^ ((p : ℝ) / 4)) : |
| (∫ ω, (Real.log (responseAttenuation theta (X ω)) - |
| Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ |
| cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V + |
| min (cavityLogVarianceCoefficient theta * V) |
| (cavityLogTailCoefficient theta M * |
| Real.exp (-(1 / (512 * A)) * (Real.log (1 / t)) ^ 2)) := by |
| have he : -(1 / (512 * A)) * (Real.log (1 / t)) ^ 2 = |
| -((Real.log (1 / t)) ^ 2) / (512 * A) := by ring |
| apply integral_cavity_log_discrepancy_small_time hθ0 hθ1 hX hX' hH hK heq |
| hM ht hbound hHV hKV |
| · rw [he] |
| exact measureReal_abs_gt_rpow_le_of_moments hA ht hsmall hInt hmom |
| · rw [he] |
| exact measureReal_abs_gt_rpow_le_of_moments hA ht hsmall hInt' hmom' |
|
|
| theorem integral_cavity_log_discrepancy_small_time_of_common_law |
| {Ξ : Type*} [MeasurableSpace Ξ] {ν : Measure Ξ} {Z : Ξ → ℝ} |
| (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) |
| (hX : Measurable X) (hX' : Measurable X') |
| (hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ) |
| (hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ) |
| (heq : ∀ᵐ ω ∂μ, |
| X ω + cavityMessage theta (Y ω) = H ω ∧ |
| Y ω + cavityMessage theta (X ω) = K ω ∧ |
| X' ω + cavityMessage theta (Y' ω) = H' ω ∧ |
| Y' ω + cavityMessage theta (X' ω) = K' ω) |
| {V M t c : ℝ} (hM : 0 ≤ M) (ht : 0 < t) |
| (hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M) |
| (hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V) |
| (hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V) |
| (hLaw : IdentDistrib X Z μ ν) (hLaw' : IdentDistrib X' Z μ ν) |
| (hTail : ν {z | t ^ (1 / 8 : ℝ) < |Z z|} ≤ |
| ENNReal.ofReal (Real.exp (-c * (Real.log (1 / t)) ^ 2))) : |
| (∫ ω, (Real.log (responseAttenuation theta (X ω)) - |
| Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ |
| cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V + |
| min (cavityLogVarianceCoefficient theta * V) |
| (cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2)) := by |
| have transfer {W : Ω → ℝ} (hW : IdentDistrib W Z μ ν) : |
| μ.real {ω | t ^ (1 / 8 : ℝ) < |W ω|} ≤ |
| Real.exp (-c * (Real.log (1 / t)) ^ 2) := by |
| apply ENNReal.toReal_le_of_le_ofReal (Real.exp_nonneg _) |
| have hm := (hW.comp measurable_abs).measure_mem_eq |
| (measurableSet_Ioi : MeasurableSet (Set.Ioi (t ^ (1 / 8 : ℝ)))) |
| change μ {ω | t ^ (1 / 8 : ℝ) < |W ω|} = |
| ν {z | t ^ (1 / 8 : ℝ) < |Z z|} at hm |
| rw [hm] |
| exact hTail |
| exact integral_cavity_log_discrepancy_small_time hθ0 hθ1 hX hX' hH hK heq |
| hM ht hbound hHV hKV (transfer hLaw) (transfer hLaw') |
|
|
| omit [IsProbabilityMeasure μ] in |
|
|
| theorem integral_cavity_log_discrepancy_osgood_of_small_time |
| {V M t c L : ℝ} (hc : 0 < c) (hL : 1 ≤ L) |
| (hV : 0 < V) (hVL : V ≤ L) (ht : 0 < t) (ht1 : t < 1) |
| (hsmall : (∫ ω, (Real.log (responseAttenuation theta (X ω)) - |
| Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ |
| cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V + |
| min (cavityLogVarianceCoefficient theta * V) |
| (cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2))) : |
| (∫ ω, (Real.log (responseAttenuation theta (X ω)) - |
| Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ |
| (cavityLogVarianceCoefficient theta + |
| 2 * (cavityLogVarianceCoefficient theta + cavityLogTailCoefficient theta M) / c) * |
| V * (t ^ (1 / 4 : ℝ) + (1 + Real.log (L / V)) / (Real.log (1 / t)) ^ 2) := by |
| exact small_time_log_loss_of_minimum (cavityLogVarianceCoefficient_nonneg theta) |
| (cavityLogTailCoefficient_nonneg theta M) hc hL hV hVL ht ht1 hsmall |
|
|
| end Coupling |
|
|
| end Problem367 |
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| end |
|
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| end OAI |
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|
