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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.Hierarchy.CascadeRecursion
import OAI.Probability.SKBarriers.Scalar.ThermalMoments
namespace OAI
section
section
noncomputable section
open scoped BigOperators Topology
open MeasureTheory ProbabilityTheory Filter
namespace SK.Analytic
section CascadeProbability
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
omit [NormedAddCommGroup E] [NormedSpace ℝ E] in
theorem cascadeMoment_nonneg (n : ℕ) (m : Fin n → ℝ)
(f g : CascadeSpace E n → ℝ) (hg : ∀ z, 0 ≤ g z) (x : E) :
0 ≤ cascadeMoment n m f g x := by
induction n with
| zero => exact hg x
| succ n ih =>
apply ih
intro z
exact integral_nonneg (fun y => hg (z,y))
theorem cascadeMoment_const (n : ℕ) (m : Fin n → ℝ)
(f : CascadeSpace E n → ℝ) (hf : BoundedDerivs f) (a : ℝ) :
cascadeMoment n m f (fun _ => a) = fun _ => a := by
induction n with
| zero => rfl
| succ n ih =>
have h : gaussianAverage (m (Fin.last n)) f (fun _ => a) = fun _ => a := by
funext z
let := gaussianStepLaw_probability hf (m (Fin.last n)) z
simp only [gaussianAverage,integral_const,probReal_univ,one_smul]
rw [cascadeMoment,h]
exact ih _ _ (hf.gaussianStep _)
omit [NormedAddCommGroup E] [NormedSpace ℝ E] in
theorem cascadeMoment_const_mul (n : ℕ) (m : Fin n → ℝ)
(f g : CascadeSpace E n → ℝ) (a : ℝ) :
cascadeMoment n m f (fun z => a*g z) = fun x => a*cascadeMoment n m f g x := by
induction n with
| zero => rfl
| succ n ih =>
have h : gaussianAverage (m (Fin.last n)) f (fun z => a*g z) =
fun z => a*gaussianAverage (m (Fin.last n)) f g z := by
funext z; exact integral_const_mul a _
rw [cascadeMoment,h,ih]
rfl
theorem cascadeMoment_sum {T : Type} [Fintype T] (n : ℕ) (m : Fin n → ℝ)
(f : CascadeSpace E n → ℝ) (hf : BoundedDerivs f)
(g : T → CascadeSpace E n → ℝ) (hg : ∀ t, Continuous (g t))
(C : T → ℝ) (hC : ∀ t, 0 ≤ C t) (hb : ∀ t z, ‖g t z‖ ≤ C t) :
cascadeMoment n m f (fun z => ∑ t, g t z) =
fun x => ∑ t, cascadeMoment n m f (g t) x := by
induction n with
| zero => rfl
| succ n ih =>
have h : gaussianAverage (m (Fin.last n)) f (fun z => ∑ t, g t z) =
fun z => ∑ t, gaussianAverage (m (Fin.last n)) f (g t) z := by
funext z
let := gaussianStepLaw_probability hf (m (Fin.last n)) z
apply integral_finsetSum
intro t _
exact Integrable.of_bound ((hg t).comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable
(C t) (ae_of_all _ (fun y => hb t (z,y)))
rw [cascadeMoment,h]
exact ih _ _ (hf.gaussianStep _) _
(fun t => gaussianAverage_continuous hf _ (hg t) (hC t) (hb t))
(fun t z => gaussianAverage_norm_le hf _ (hb t) z)
variable {S : Type} [Fintype S] [Nonempty S]
def cascadeSpinWeight (n : ℕ) (m : Fin n → ℝ)
(c : S → ℝ) (L : S → CascadeSpace E n →L[ℝ] ℝ) (x : E) (s : S) : ℝ :=
cascadeMoment n m (affineLogPartition c L) (fun z => affineGibbs c L z s) x
theorem affineGibbs_continuous (c : S → ℝ) (L : S → E →L[ℝ] ℝ) (s : S) :
Continuous (fun x => affineGibbs c L x s) := by
exact (Real.continuous_exp.comp (continuous_const.add (L s).continuous)).div
(continuous_finsetSum _ (fun t _ => Real.continuous_exp.comp (continuous_const.add (L t).continuous)))
(fun x => (affinePartition_pos c L x).ne')
theorem affineGibbs_le_one (c : S → ℝ) (L : S → E →L[ℝ] ℝ) (x : E) (s : S) :
affineGibbs c L x s ≤ 1 := by
rw [← affineGibbs_sum c L x]
exact Finset.single_le_sum (fun t _ => (affineGibbs_pos c L x t).le) (Finset.mem_univ s)
theorem affineGibbs_norm_le_one (c : S → ℝ) (L : S → E →L[ℝ] ℝ) (x : E) (s : S) :
‖affineGibbs c L x s‖ ≤ 1 := by
rw [Real.norm_eq_abs,abs_of_pos (affineGibbs_pos c L x s)]
exact affineGibbs_le_one c L x s
theorem cascadeSpinWeight_nonneg (n : ℕ) (m : Fin n → ℝ)
(c : S → ℝ) (L : S → CascadeSpace E n →L[ℝ] ℝ) (x : E) (s : S) :
0 ≤ cascadeSpinWeight n m c L x s :=
cascadeMoment_nonneg n m _ _ (fun z => (affineGibbs_pos c L z s).le) x
theorem cascadeSpinWeight_sum (n : ℕ) (m : Fin n → ℝ)
(c : S → ℝ) (L : S → CascadeSpace E n →L[ℝ] ℝ) (x : E) :
∑ s, cascadeSpinWeight n m c L x s = 1 := by
have H := congrFun (cascadeMoment_sum n m (affineLogPartition c L)
(affineLogPartition_boundedDerivs c L) (fun s z => affineGibbs c L z s)
(affineGibbs_continuous c L) (fun _ => 1) (by intro; norm_num)
(fun s z => affineGibbs_norm_le_one c L z s)) x
simp_rw [affineGibbs_sum] at H
rw [cascadeMoment_const n m _ (affineLogPartition_boundedDerivs c L)] at H
exact H.symm
theorem cascadeMoment_affineMoment (n : ℕ) (m : Fin n → ℝ)
(c : S → ℝ) (L : S → CascadeSpace E n →L[ℝ] ℝ) (g : S → ℝ) (x : E) :
cascadeMoment n m (affineLogPartition c L) (affineMoment c L g) x =
∑ s, cascadeSpinWeight n m c L x s*g s := by
have H := congrFun (cascadeMoment_sum n m (affineLogPartition c L)
(affineLogPartition_boundedDerivs c L) (fun s z => g s*affineGibbs c L z s)
(fun s => continuous_const.mul (affineGibbs_continuous c L s)) (fun s => ‖g s‖)
(fun s => norm_nonneg _) (fun s z => by
rw [norm_mul]; exact (mul_le_mul_of_nonneg_left (affineGibbs_norm_le_one c L z s) (norm_nonneg _)).trans_eq (mul_one _))) x
simp_rw [cascadeMoment_const_mul] at H
change cascadeMoment n m (affineLogPartition c L) (fun z => ∑ s, affineGibbs c L z s*g s) x = _
simpa only [cascadeSpinWeight,mul_comm] using H
theorem cascadeSpinWeight_regular (n : ℕ) (m : Fin n → ℝ)
(c : S → ℝ) (L : S → CascadeSpace E n →L[ℝ] ℝ) (s : S) :
Continuous (fun x => cascadeSpinWeight n m c L x s) ∧
∀ x, ‖cascadeSpinWeight n m c L x s‖ ≤ 1 :=
cascadeMoment_bounded_continuous n m _ (affineLogPartition_boundedDerivs c L) _
(affineGibbs_continuous c L s) (by norm_num) (fun x => affineGibbs_norm_le_one c L x s)
end CascadeProbability
end SK.Analytic
end
end
end
end OAI
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