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9eeb9c7 | 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 | import OAI.Analysis.IrrationalRotation.Model.OneCoordinateConditional
import OAI.Analysis.IrrationalRotation.Analysis.FiniteKernelMixture
import OAI.Analysis.IrrationalRotation.Product.OneCoordinateDensity
namespace OAI
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal ProbabilityTheory
namespace Invariant233.OneCoordinateCoupling
open CouplingMixture CouplingMarginalBounds FiniteKernelMixture
private lemma finite_total (q : ℝ) (p : Fin 5 → ℝ) (u : Fin 5 → ℝ → ℝ) (x : ℝ) :
ENNReal.ofReal q + ∑ i, ENNReal.ofReal (p i) * ENNReal.ofReal (u i x) ≠ ∞ := by
apply ENNReal.add_ne_top.mpr
refine ⟨ENNReal.ofReal_ne_top, ENNReal.sum_ne_top.mpr ?_⟩
intro i _
exact ENNReal.mul_ne_top ENNReal.ofReal_ne_top ENNReal.ofReal_ne_top
section SourceParameters
variable {κ m a b t : ℝ}
(hκ0 : 0 ≤ κ) (hκ : κ ≤ 1 / 100) (hm : 100 ≤ m)
(ha : 1 / m ≤ a) (ha' : a ≤ 1 / 4)
(hb : 1 / m ≤ b) (hb' : b ≤ 1 / 4)
(ht0 : 0 ≤ t) (ht : t ≤ 1)
include hκ0 hκ hm ha ha' hb hb' ht0 ht
lemma rightGivenLeft_isMarkov : IsMarkovKernel (rightGivenLeft κ m a b t) := by
have hm0 : 0 < m := by linarith
have : ∀ i, IsProbabilityMeasure (rightLaw m a b i) :=
rightLaw_isProbabilityMeasure (by linarith) (width_pos hm0 ha) ha' (width_pos hm0 hb) hb'
apply diagonalReplacementKernel_isMarkov
· apply ENNReal.ofReal_ne_zero_iff.mpr
have := (diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht).1
linarith
· exact finite_total _ _ _
lemma leftGivenRight_isMarkov : IsMarkovKernel (leftGivenRight κ m a b t) := by
have hm0 : 0 < m := by linarith
have : ∀ i, IsProbabilityMeasure (leftLaw m a b i) :=
leftLaw_isProbabilityMeasure hm0 (width_pos hm0 ha) ha' (width_pos hm0 hb) hb'
apply diagonalReplacementKernel_isMarkov
· apply ENNReal.ofReal_ne_zero_iff.mpr
have := (diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht).1
linarith
· exact finite_total _ _ _
lemma rightGivenLeft_disintegration :
(pairLaw κ m a b t).fst ⊗ₘ rightGivenLeft κ m a b t = pairLaw κ m a b t := by
have hm0 : 0 < m := by linarith
have ha0 := width_pos hm0 ha
have hb0 := width_pos hm0 hb
have : ∀ i, IsProbabilityMeasure (rightLaw m a b i) :=
rightLaw_isProbabilityMeasure (by linarith) ha0 ha' hb0 hb'
have := rightGivenLeft_isMarkov hκ0 hκ hm ha ha' hb hb' ht0 ht
have hq : ENNReal.ofReal (diagonalMass κ m a b t) ≠ 0 := by
apply ENNReal.ofReal_ne_zero_iff.mpr
have := (diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht).1
linarith
let d : ℝ → ℝ≥0∞ := fun x => ENNReal.ofReal (diagonalMass κ m a b t) +
∑ i, ENNReal.ofReal (componentProbabilities κ m a b t i) *
ENNReal.ofReal (leftDensity m a b i x)
have hpair : (baseLaw.withDensity d) ⊗ₘ rightGivenLeft κ m a b t = pairLaw κ m a b t := by
unfold rightGivenLeft d
rw [diagonalReplacementKernel_disintegration _ _ hq]
· unfold pairLaw mixture CouplingMixture.diagonal
congr 1
apply Finset.sum_congr rfl
intro i _
rw [leftLaw_eq_withDensity (by linarith) ha0 ha' hb0 hb']
· exact finite_total _ _ _
have hfst := congrArg Measure.fst hpair
rw [Measure.fst_compProd] at hfst
rw [← hfst]
exact hpair
lemma leftGivenRight_disintegration :
(pairLaw κ m a b t).snd ⊗ₘ leftGivenRight κ m a b t =
(pairLaw κ m a b t).map Prod.swap := by
have hm0 : 0 < m := by linarith
have ha0 := width_pos hm0 ha
have hb0 := width_pos hm0 hb
have : ∀ i, IsProbabilityMeasure (leftLaw m a b i) :=
leftLaw_isProbabilityMeasure hm0 ha0 ha' hb0 hb'
have : ∀ i, IsProbabilityMeasure (rightLaw m a b i) :=
rightLaw_isProbabilityMeasure (by linarith) ha0 ha' hb0 hb'
have := leftGivenRight_isMarkov hκ0 hκ hm ha ha' hb hb' ht0 ht
have hq : ENNReal.ofReal (diagonalMass κ m a b t) ≠ 0 := by
apply ENNReal.ofReal_ne_zero_iff.mpr
have := (diagonalMass_bounds hκ0 hκ hm ha ha' hb hb' ht0 ht).1
linarith
have hswap : (pairLaw κ m a b t).map Prod.swap =
ENNReal.ofReal (diagonalMass κ m a b t) • CouplingMixture.diagonal baseLaw +
∑ i, ENNReal.ofReal (componentProbabilities κ m a b t i) •
(rightLaw m a b i).prod (leftLaw m a b i) := by
unfold pairLaw mixture
rw [Measure.map_add _ _ measurable_swap, map_finsetSum _ _ measurable_swap]
simp only [Measure.map_smul _ measurable_swap.aemeasurable, Measure.prod_swap]
congr 1
unfold CouplingMixture.diagonal
rw [Measure.map_map measurable_swap (show Measurable (fun x : ℝ => (x, x)) from
measurable_id.prodMk measurable_id)]
rfl
let d : ℝ → ℝ≥0∞ := fun x => ENNReal.ofReal (diagonalMass κ m a b t) +
∑ i, ENNReal.ofReal (componentProbabilities κ m a b t i) *
ENNReal.ofReal (rightDensity m a b i x)
have hpair : (baseLaw.withDensity d) ⊗ₘ leftGivenRight κ m a b t =
(pairLaw κ m a b t).map Prod.swap := by
unfold leftGivenRight d
rw [diagonalReplacementKernel_disintegration _ _ hq]
· rw [hswap]
unfold CouplingMixture.diagonal
congr 1
apply Finset.sum_congr rfl
intro i _
rw [rightLaw_eq_withDensity (by linarith) ha0 ha' hb0 hb']
· exact finite_total _ _ _
have hfst := congrArg Measure.fst hpair
rw [Measure.fst_compProd, Measure.fst_map_swap] at hfst
rw [← hfst]
exact hpair
end SourceParameters
end Invariant233.OneCoordinateCoupling
end OAI
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