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import OAI.Probability.FreeIsing.Estimates.RadialSummability
import OAI.Probability.FreeIsing.Trees.CoupledPathDrift
import OAI.Probability.FreeIsing.Trees.SubcriticalPathSums
namespace OAI
noncomputable section
open MeasureTheory
open scoped ENNReal
namespace Problem367
theorem abs_tanh_weight_drift_le (h h' p p' : ℝ) :
|(h - h') * (Real.tanh h * p ^ 2 - Real.tanh h' * p' ^ 2)| ≤
(h - h') ^ 2 * p ^ 2 + |h - h'| * |p ^ 2 - p' ^ 2| := by
have hdiff : |Real.tanh h - Real.tanh h'| ≤ |h - h'| := by
simpa only [Real.dist_eq, NNReal.coe_one, one_mul] using
lipschitzWith_tanh_response.dist_le_mul h h'
have hm : |Real.tanh h'| ≤ 1 := by
have hsq := Real.tanh_sq_lt_one h'
nlinarith [sq_abs (Real.tanh h'), abs_nonneg (Real.tanh h')]
have hsplit : Real.tanh h * p ^ 2 - Real.tanh h' * p' ^ 2 =
(Real.tanh h - Real.tanh h') * p ^ 2 +
Real.tanh h' * (p ^ 2 - p' ^ 2) := by ring
rw [abs_mul, hsplit]
calc
|h - h'| * |(Real.tanh h - Real.tanh h') * p ^ 2 +
Real.tanh h' * (p ^ 2 - p' ^ 2)| ≤
|h - h'| * (|(Real.tanh h - Real.tanh h') * p ^ 2| +
|Real.tanh h' * (p ^ 2 - p' ^ 2)|) :=
mul_le_mul_of_nonneg_left (abs_add_le _ _) (abs_nonneg _)
_ = |h - h'| * (|Real.tanh h - Real.tanh h'| * p ^ 2 +
|Real.tanh h'| * |p ^ 2 - p' ^ 2|) := by
rw [abs_mul, abs_mul, abs_of_nonneg (sq_nonneg p)]
_ ≤ |h - h'| * (|h - h'| * p ^ 2 + |p ^ 2 - p' ^ 2|) := by
apply mul_le_mul_of_nonneg_left _ (abs_nonneg _)
apply add_le_add (mul_le_mul_of_nonneg_right hdiff (sq_nonneg p))
exact mul_le_of_le_one_left (abs_nonneg _) hm
_ = _ := by rw [mul_add, ← mul_assoc, ← sq, sq_abs]
section Probability
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
theorem integral_abs_tanh_path_drift_le
{ι : Type*} (s : Finset ι) {theta : ℝ} (hθ : theta ^ 2 ≤ 1)
{X Y : ι → Ω → ℝ} {H H' : 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)
(hH : Measurable H) (hH' : Measurable H')
(hU : MemLp (fun ω => H ω - H' ω) 2 μ) :
let P := fun ω => ∏ i ∈ s, responseAttenuation theta (X i ω)
let P' := fun ω => ∏ i ∈ s, responseAttenuation theta (Y i ω)
let Z := fun ω => ∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω)))
Integrable (fun ω => (H ω - H' ω) *
(Real.tanh (H ω) * P ω ^ 2 - Real.tanh (H' ω) * P' ω ^ 2)) μ ∧
(∫ ω, |(H ω - H' ω) *
(Real.tanh (H ω) * P ω ^ 2 - Real.tanh (H' ω) * P' ω ^ 2)| ∂μ) ≤
(∫ ω, (H ω - H' ω) ^ 2 * P ω ^ 2 ∂μ) +
2 * Real.sqrt (2 * ∫ ω, (H ω - H' ω) ^ 2 ∂μ) *
Real.sqrt (∫ ω, (P ω ^ 2 + P' ω ^ 2) * Z ω ^ 2 ∂μ) := by
dsimp only
let U := fun ω => H ω - H' ω
let P := fun ω => ∏ i ∈ s, responseAttenuation theta (X i ω)
let P' := fun ω => ∏ i ∈ s, responseAttenuation theta (Y i ω)
let Z := fun ω => ∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω)))
let D := fun ω => U ω *
(Real.tanh (H ω) * P ω ^ 2 - Real.tanh (H' ω) * P' ω ^ 2)
have hP : Measurable P := measurable_prod_responseAttenuation s theta hX
have hP' : Measurable P' := measurable_prod_responseAttenuation s theta hY
have hL := (integral_mul_prod_responseAttenuation_sq_le s hθ hX hU).1
have hB := integral_abs_mul_prod_responseAttenuation_sq_sub_sq_le s hθ hX hY hInt hU
have htanh := lipschitzWith_tanh_response.continuous.measurable
have hDm : Measurable D := (hH.sub hH').mul
(((htanh.comp hH).mul (hP.pow_const 2)).sub
((htanh.comp hH').mul (hP'.pow_const 2)))
have hpoint (ω : Ω) : |D ω| ≤ U ω ^ 2 * P ω ^ 2 +
|U ω| * |P ω ^ 2 - P' ω ^ 2| :=
abs_tanh_weight_drift_le (H ω) (H' ω) (P ω) (P' ω)
have hD : Integrable D μ := by
apply (hL.add hB.1).mono' hDm.aestronglyMeasurable
filter_upwards [] with ω
exact (show ‖D ω‖ ≤ _ by simpa only [Real.norm_eq_abs, Pi.add_apply, U, P, P'] using hpoint ω)
refine ⟨hD, ?_⟩
calc
(∫ ω, |D ω| ∂μ) ≤ ∫ ω, U ω ^ 2 * P ω ^ 2 + |U ω| * |P ω ^ 2 - P' ω ^ 2| ∂μ :=
integral_mono hD.abs (hL.add hB.1) hpoint
_ = (∫ ω, U ω ^ 2 * P ω ^ 2 ∂μ) +
∫ ω, |U ω| * |P ω ^ 2 - P' ω ^ 2| ∂μ := integral_add hL hB.1
_ ≤ _ := add_le_add le_rfl hB.2
theorem integrable_series_of_summable_integral_abs
{ι : Type*} [Countable ι] {F : ι → Ω → ℝ}
(hF : ∀ i, Integrable (F i) μ)
(hs : Summable (fun i => ∫ ω, |F i ω| ∂μ)) :
(∀ᵐ ω ∂μ, Summable (fun i => |F i ω|)) ∧
Integrable (fun ω => ∑' i, F i ω) μ ∧
(∫ ω, (∑' i, F i ω) ∂μ) = ∑' i, ∫ ω, F i ω ∂μ := by
have hm (i : ι) : AEMeasurable (fun ω => ENNReal.ofReal |F i ω|) μ :=
(hF i).abs.aestronglyMeasurable.aemeasurable.ennreal_ofReal
have hfin : (∫⁻ ω, ∑' i, ENNReal.ofReal |F i ω| ∂μ) < ∞ := by
rw [lintegral_tsum hm]
calc
(∑' i, ∫⁻ ω, ENNReal.ofReal |F i ω| ∂μ) =
∑' i, ENNReal.ofReal (∫ ω, |F i ω| ∂μ) := by
apply tsum_congr
intro i
exact (ofReal_integral_eq_lintegral_ofReal (hF i).abs
(Filter.Eventually.of_forall (fun ω => abs_nonneg (F i ω)))).symm
_ < ∞ := hs.tsum_ofReal_lt_top
have hAe : ∀ᵐ ω ∂μ, Summable (fun i => |F i ω|) := by
filter_upwards [ae_lt_top' (AEMeasurable.tsum hm) hfin.ne] with ω hω
simpa only [ENNReal.toReal_ofReal (abs_nonneg _)] using
ENNReal.summable_toReal hω.ne
have hIabs : Integrable (fun ω => ∑' i, |F i ω|) μ := by
have h := integrable_toReal_of_lintegral_ne_top (AEMeasurable.tsum hm) hfin.ne
simpa only [ENNReal.tsum_toReal_eq (fun i => ENNReal.ofReal_ne_top),
ENNReal.toReal_ofReal (abs_nonneg _)] using h
refine ⟨hAe, ?_, ?_⟩
· apply hIabs.mono' (AEStronglyMeasurable.tsum (fun i => (hF i).aestronglyMeasurable))
filter_upwards [hAe] with ω hω
simpa only [Real.norm_eq_abs] using
norm_tsum_le_tsum_norm (show Summable (fun i => ‖F i ω‖) by
simpa only [Real.norm_eq_abs] using hω)
· exact (integral_tsum_of_summable_integral_norm hF
(by simpa only [Real.norm_eq_abs] using hs)).symm
theorem integral_series_le_of_abs_majorant
{ι : Type*} [Countable ι] {F : ι → Ω → ℝ} {B : ι → ℝ}
(hF : ∀ i, Integrable (F i) μ) (hB : Summable B)
(hbound : ∀ i, (∫ ω, |F i ω| ∂μ) ≤ B i) :
Integrable (fun ω => ∑' i, F i ω) μ ∧
|∫ ω, (∑' i, F i ω) ∂μ| ≤ ∑' i, B i := by
have hs : Summable (fun i => ∫ ω, |F i ω| ∂μ) :=
Summable.of_nonneg_of_le (fun i => integral_nonneg (fun ω => abs_nonneg _)) hbound hB
have hseries := integrable_series_of_summable_integral_abs hF hs
refine ⟨hseries.2.1, ?_⟩
rw [hseries.2.2]
have hnorm : Summable (fun i => |∫ ω, F i ω ∂μ|) :=
Summable.of_nonneg_of_le (fun i => abs_nonneg _)
(fun i => abs_integral_le_integral_abs) hs
calc
|∑' i, ∫ ω, F i ω ∂μ| ≤ ∑' i, |∫ ω, F i ω ∂μ| := by
simpa only [Real.norm_eq_abs] using
norm_tsum_le_tsum_norm (show Summable (fun i => ‖∫ ω, F i ω ∂μ‖) by
simpa only [Real.norm_eq_abs] using hnorm)
_ ≤ ∑' i, B i := hnorm.tsum_le_tsum
(fun i => abs_integral_le_integral_abs.trans (hbound i)) hB
theorem integral_response_row_density_le
{ι : Type*} [Countable ι]
{c c' : ι → Ω → ℝ} {U m m' : Ω → ℝ} {N D : ι → ℝ}
(hnoise : ∀ i, Integrable (fun ω => (c i ω - c' i ω) ^ 2) μ)
(hdrift : ∀ i, Integrable (fun ω => U ω *
(m ω * (c i ω) ^ 2 - m' ω * (c' i ω) ^ 2)) μ)
(hN : Summable N) (hD : Summable D)
(hNb : ∀ i, (∫ ω, (c i ω - c' i ω) ^ 2 ∂μ) ≤ N i)
(hDb : ∀ i, (∫ ω, |U ω *
(m ω * (c i ω) ^ 2 - m' ω * (c' i ω) ^ 2)| ∂μ) ≤ D i)
(hrows : ∀ᵐ ω ∂μ, Summable (fun i => (c i ω) ^ 2) ∧
Summable (fun i => (c' i ω) ^ 2)) :
Integrable (DiscrepancyEnergy.density c c' U m m') μ ∧
(∫ ω, DiscrepancyEnergy.density c c' U m m' ω ∂μ) ≤
(∑' i, N i) + 2 * ∑' i, D i := by
let A := fun ω => ∑' i, (c i ω - c' i ω) ^ 2
let B := fun ω => ∑' i, U ω * (m ω * (c i ω) ^ 2 - m' ω * (c' i ω) ^ 2)
have hAb : ∀ i, (∫ ω, |(c i ω - c' i ω) ^ 2| ∂μ) ≤ N i := by
simpa only [abs_sq] using hNb
have hA := integral_series_le_of_abs_majorant hnoise hN hAb
have hB := integral_series_le_of_abs_majorant hdrift hD hDb
have heq : DiscrepancyEnergy.density c c' U m m' =ᵐ[μ]
(fun ω => A ω + 2 * B ω) := by
filter_upwards [hrows] with ω hω
have hs : Summable (fun i => (U ω * m ω) * (c i ω) ^ 2) :=
hω.1.mul_left (U ω * m ω)
have hs' : Summable (fun i => (U ω * m' ω) * (c' i ω) ^ 2) :=
hω.2.mul_left (U ω * m' ω)
have hBω : B ω = U ω *
(m ω * (∑' i, (c i ω) ^ 2) - m' ω * (∑' i, (c' i ω) ^ 2)) := by
calc
B ω = ∑' i, ((U ω * m ω) * (c i ω) ^ 2 -
(U ω * m' ω) * (c' i ω) ^ 2) := by
apply tsum_congr
intro i
ring
_ = _ := by rw [hs.tsum_sub hs', tsum_mul_left, tsum_mul_left]; ring
dsimp only [DiscrepancyEnergy.density, A]
rw [hBω]
ring
have hint : Integrable (DiscrepancyEnergy.density c c' U m m') μ :=
(hA.1.add (hB.1.const_mul 2)).congr heq.symm
refine ⟨hint, ?_⟩
rw [integral_congr_ae heq, integral_add hA.1 (hB.1.const_mul 2), integral_const_mul]
exact add_le_add ((le_abs_self _).trans hA.2)
(mul_le_mul_of_nonneg_left ((le_abs_self _).trans hB.2) (by norm_num))
namespace ResponseRows
def pathProduct {ι : Type*} (theta : ℝ) (s : Finset ι) (X : ι → Ω → ℝ) (ω : Ω) : ℝ :=
∏ i ∈ s, responseAttenuation theta (X i ω)
def pathLogDiff {ι : Type*} (theta : ℝ) (s : Finset ι)
(X Y : ι → Ω → ℝ) (ω : Ω) : ℝ :=
∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) -
Real.log (responseAttenuation theta (Y i ω)))
def pathNoiseEnergy {ι : Type*} (μ : Measure Ω) (theta : ℝ) (s : Finset ι)
(X Y : ι → Ω → ℝ) : ℝ :=
∫ ω, (pathProduct theta s X ω ^ 2 + pathProduct theta s Y ω ^ 2) *
pathLogDiff theta s X Y ω ^ 2 ∂μ
def pathLinearEnergy {ι : Type*} (μ : Measure Ω) (theta : ℝ) (s : Finset ι)
(X : ι → Ω → ℝ) (U : Ω → ℝ) : ℝ :=
∫ ω, U ω ^ 2 * pathProduct theta s X ω ^ 2 ∂μ
def coefficient {J ι : Type*} (theta : ℝ) (a : J → ℝ) (s : J → Finset ι)
(X : J → ι → Ω → ℝ) (j : J) (ω : Ω) : ℝ :=
a j * pathProduct theta (s j) (X j) ω
theorem integral_density_le_path_energies
{J ι : Type*} [Countable J]
(s : J → Finset ι) (a : J → ℝ) {theta : ℝ} (hθ : theta ^ 2 ≤ 1)
{X Y : J → ι → Ω → ℝ} {H H' : Ω → ℝ}
(hX : ∀ j i, i ∈ s j → Measurable (X j i))
(hY : ∀ j i, i ∈ s j → Measurable (Y j i))
(hInt : ∀ j i, i ∈ s j → Integrable (fun ω => (X j i ω - Y j i ω) ^ 2) μ)
(hH : Measurable H) (hH' : Measurable H')
(hU : MemLp (fun ω => H ω - H' ω) 2 μ)
(hQsum : Summable (fun j => (a j) ^ 2 * pathNoiseEnergy μ theta (s j) (X j) (Y j)))
(hLsum : Summable (fun j => (a j) ^ 2 *
pathLinearEnergy μ theta (s j) (X j) (fun ω => H ω - H' ω)))
(hRsum : Summable (fun j => (a j) ^ 2 *
Real.sqrt (pathNoiseEnergy μ theta (s j) (X j) (Y j))))
(hrows : ∀ᵐ ω ∂μ, Summable (fun j => (coefficient theta a s X j ω) ^ 2) ∧
Summable (fun j => (coefficient theta a s Y j ω) ^ 2)) :
Integrable (DiscrepancyEnergy.density (coefficient theta a s X)
(coefficient theta a s Y) (fun ω => H ω - H' ω)
(fun ω => Real.tanh (H ω)) (fun ω => Real.tanh (H' ω))) μ ∧
(∫ ω, DiscrepancyEnergy.density (coefficient theta a s X)
(coefficient theta a s Y) (fun ω => H ω - H' ω)
(fun ω => Real.tanh (H ω)) (fun ω => Real.tanh (H' ω)) ω ∂μ) ≤
(∑' j, (a j) ^ 2 * pathNoiseEnergy μ theta (s j) (X j) (Y j)) +
2 * (∑' j, (a j) ^ 2 * pathLinearEnergy μ theta (s j) (X j) (fun ω => H ω - H' ω)) +
4 * Real.sqrt (2 * ∫ ω, (H ω - H' ω) ^ 2 ∂μ) *
(∑' j, (a j) ^ 2 * Real.sqrt (pathNoiseEnergy μ theta (s j) (X j) (Y j))) := by
let U := fun ω => H ω - H' ω
let P := fun j => pathProduct theta (s j) (X j)
let P' := fun j => pathProduct theta (s j) (Y j)
let Q := fun j => pathNoiseEnergy μ theta (s j) (X j) (Y j)
let L := fun j => pathLinearEnergy μ theta (s j) (X j) U
let V := ∫ ω, U ω ^ 2 ∂μ
let K := 2 * Real.sqrt (2 * V)
let F := fun j ω => U ω *
(Real.tanh (H ω) * P j ω ^ 2 - Real.tanh (H' ω) * P' j ω ^ 2)
have hfiniteN (j : J) := integral_prod_responseAttenuation_sub_sq_le
(s j) hθ (hX j) (hY j) (hInt j)
have hfiniteD (j : J) := integral_abs_tanh_path_drift_le
(s j) hθ (hX j) (hY j) (hInt j) hH hH' hU
have hnoiseEq (j : J) (ω : Ω) :
(coefficient theta a s X j ω - coefficient theta a s Y j ω) ^ 2 =
(a j) ^ 2 * (P j ω - P' j ω) ^ 2 := by
dsimp only [coefficient, P, P']
ring
have hdriftEq (j : J) (ω : Ω) : U ω *
(Real.tanh (H ω) * coefficient theta a s X j ω ^ 2 -
Real.tanh (H' ω) * coefficient theta a s Y j ω ^ 2) =
(a j) ^ 2 * F j ω := by
dsimp only [coefficient, F, P, P']
ring
have hnoise (j : J) : Integrable (fun ω =>
(coefficient theta a s X j ω - coefficient theta a s Y j ω) ^ 2) μ := by
simp_rw [hnoiseEq]
exact (hfiniteN j).2.1.const_mul _
have hdrift (j : J) : Integrable (fun ω => U ω *
(Real.tanh (H ω) * coefficient theta a s X j ω ^ 2 -
Real.tanh (H' ω) * coefficient theta a s Y j ω ^ 2)) μ := by
simp_rw [hdriftEq]
exact (hfiniteD j).1.const_mul _
have hNb (j : J) : (∫ ω,
(coefficient theta a s X j ω - coefficient theta a s Y j ω) ^ 2 ∂μ) ≤
(a j) ^ 2 * Q j := by
simp_rw [hnoiseEq]
rw [integral_const_mul]
exact mul_le_mul_of_nonneg_left (hfiniteN j).2.2 (sq_nonneg _)
have hDb (j : J) : (∫ ω, |U ω *
(Real.tanh (H ω) * coefficient theta a s X j ω ^ 2 -
Real.tanh (H' ω) * coefficient theta a s Y j ω ^ 2)| ∂μ) ≤
(a j) ^ 2 * (L j + K * Real.sqrt (Q j)) := by
simp_rw [hdriftEq, abs_mul, abs_sq]
rw [integral_const_mul]
exact mul_le_mul_of_nonneg_left (hfiniteD j).2 (sq_nonneg _)
have hDsum : Summable (fun j => (a j) ^ 2 * (L j + K * Real.sqrt (Q j))) := by
convert hLsum.add (hRsum.mul_left K) using 1
funext j
dsimp only [L, Q, U]
ring
have hb := integral_response_row_density_le hnoise hdrift hQsum hDsum hNb hDb hrows
refine ⟨hb.1, hb.2.trans_eq ?_⟩
have hsumD : (∑' j, (a j) ^ 2 * (L j + K * Real.sqrt (Q j))) =
(∑' j, (a j) ^ 2 * L j) + K * ∑' j, (a j) ^ 2 * Real.sqrt (Q j) := by
calc
_ = ∑' j, ((a j) ^ 2 * L j + K * ((a j) ^ 2 * Real.sqrt (Q j))) := by
apply tsum_congr
intro j
ring
_ = _ := by rw [hLsum.tsum_add (hRsum.mul_left K), tsum_mul_left]
rw [hsumD]
dsimp only [K, V, Q, L, U]
ring
end ResponseRows
end Probability
end Problem367
end
end OAI
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