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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | import OAI.Probability.FreeIsing.Estimates.FiniteBondMask
import OAI.Probability.FreeIsing.Factors.Transfer
import OAI.Probability.FreeIsing.Model
import OAI.Probability.FreeIsing.Trees.FiniteClusterLaw
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367.ClusterAssembly
theorem lintegral_finite_statistic
{Ω E : Type*} [MeasurableSpace Ω] [MeasurableSpace E]
[Fintype E] [MeasurableSingletonClass E]
(μ : Measure Ω) (A : Ω → E) (hA : Measurable A)
(weight : E → ENNReal)
(hlaw : ∀ e, μ {ω | A ω = e} = weight e) (f : E → ENNReal) :
(∫⁻ ω, f (A ω) ∂μ) = ∑ e, weight e * f e := by
rw [← lintegral_map (measurable_of_finite f) hA, lintegral_fintype]
apply Finset.sum_congr rfl
intro e he
rw [Measure.map_apply hA (measurableSet_singleton e)]
change f e * μ {ω | A ω = e} = weight e * f e
rw [hlaw, mul_comm]
theorem map_prod_apply_eq_finite_mixture
{Ω C E S : Type*} [MeasurableSpace Ω] [MeasurableSpace C]
[MeasurableSpace E] [MeasurableSpace S]
[Fintype E] [MeasurableSingletonClass E]
(μ : Measure Ω) (ν : Measure C) [SFinite ν]
(A : Ω → E) (hA : Measurable A)
(F : Ω × C → S) (hF : Measurable F)
(s : Set S) (hs : MeasurableSet s)
(weight color : E → ENNReal)
(hlaw : ∀ e, μ {ω | A ω = e} = weight e)
(hcolor : ∀ᵐ ω ∂μ, ν {c | F (ω, c) ∈ s} = color (A ω)) :
Measure.map F (μ.prod ν) s = ∑ e, weight e * color e := by
rw [Measure.map_apply hF hs, Measure.prod_apply (hs.preimage hF)]
calc
(∫⁻ ω, ν (Prod.mk ω ⁻¹' (F ⁻¹' s)) ∂μ) = ∫⁻ ω, color (A ω) ∂μ := by
apply lintegral_congr_ae
exact hcolor
_ = ∑ e, weight e * color e :=
lintegral_finite_statistic μ A hA weight hlaw color
theorem map_compProd_snd_apply_eq_finite_mixture
{Ω E S : Type*} [MeasurableSpace Ω] [MeasurableSpace E]
[MeasurableSpace S] [Fintype E] [MeasurableSingletonClass E]
(μ : Measure Ω) [SFinite μ] (κ : Kernel Ω S) [IsSFiniteKernel κ]
(A : Ω → E) (hA : Measurable A)
(s : Set S) (hs : MeasurableSet s)
(weight color : E → ENNReal)
(hlaw : ∀ e, μ {ω | A ω = e} = weight e)
(hcolor : ∀ᵐ ω ∂μ, κ ω s = color (A ω)) :
Measure.map Prod.snd (μ ⊗ₘ κ) s = ∑ e, weight e * color e := by
rw [Measure.map_apply measurable_snd hs,
Measure.compProd_apply (hs.preimage measurable_snd)]
calc
(∫⁻ ω, κ ω (Prod.mk ω ⁻¹' (Prod.snd ⁻¹' s)) ∂μ) =
∫⁻ ω, color (A ω) ∂μ := by
apply lintegral_congr_ae
exact hcolor
_ = ∑ e, weight e * color e :=
lintegral_finite_statistic μ A hA weight hlaw color
theorem cluster_mixture_sum {d : ℕ} {theta : ℝ}
(h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(D : Finset (TreeVertex d)) (eta : TreeVertex d → IsingSpin) :
(∑ A : (inducedEdges D).powerset,
ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) A.val) *
ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) A.val eta)) =
freeCylinderWeight theta D eta := by
classical
rw [Finset.sum_coe_sort (inducedEdges D).powerset (fun A =>
ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) A) *
ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) A eta))]
rw [← FiniteCluster.marginal_eq_freeCylinderWeight theta D eta]
rw [ENNReal.ofReal_sum_of_nonneg (fun A _ =>
mul_nonneg (FiniteCluster.bondWeight_nonneg h0 h1 _ _)
(FiniteCluster.colorWeight_nonneg _ _ _))]
apply Finset.sum_congr rfl
intro A hA
exact (ENNReal.ofReal_mul (FiniteCluster.bondWeight_nonneg h0 h1 _ _)).symm
lemma measurableSet_spinCylinder {d : ℕ} (D : Finset (TreeVertex d))
(eta : TreeVertex d → IsingSpin) :
MeasurableSet {x : TreeVertex d → IsingSpin | ∀ v ∈ D, x v = eta v} := by
classical
simp only [Set.ofPred_forall]
apply MeasurableSet.iInter
intro v
apply MeasurableSet.iInter
intro hv
convert (show MeasurableSet ({eta v} : Set IsingSpin) by trivial).preimage
(measurable_pi_apply v : Measurable (fun x : TreeVertex d → IsingSpin => x v)) using 1
ext x
simp only [Set.mem_preimage, Set.mem_singleton_iff, Set.mem_ofPred_eq]
theorem hasFreeIsingLaw_map_prod_of_cluster_conditionals
{d : ℕ} {Ω C : Type*} [MeasurableSpace Ω] [MeasurableSpace C]
(μ : Measure Ω) (ν : Measure C) [SFinite ν]
{theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(F : Ω × C → TreeVertex d → IsingSpin) (hF : Measurable F)
(A : (D : Finset (TreeVertex d)) → Ω → (inducedEdges D).powerset)
(hA : ∀ D, Measurable (A D))
(hmask : ∀ D, ConnectedFinset D → ∀ a : (inducedEdges D).powerset,
μ {ω | A D ω = a} =
ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) a.val))
(hcolor : ∀ D, ConnectedFinset D → ∀ eta : TreeVertex d → IsingSpin,
∀ᵐ ω ∂μ, ν {c | ∀ v ∈ D, F (ω, c) v = eta v} =
ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) (A D ω).val eta)) :
HasFreeIsingLaw theta (Measure.map F (μ.prod ν)) := by
intro D hD eta
rw [map_prod_apply_eq_finite_mixture μ ν (A D) (hA D) F hF _
(measurableSet_spinCylinder D eta)
(fun a => ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) a.val))
(fun a => ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) a.val eta))
(hmask D hD) (hcolor D hD eta)]
exact cluster_mixture_sum h0 h1 D eta
theorem hasFreeIsingLaw_compProd_of_cluster_conditionals
{d : ℕ} {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [SFinite μ]
(κ : Kernel Ω (TreeVertex d → IsingSpin)) [IsSFiniteKernel κ]
{theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(A : (D : Finset (TreeVertex d)) → Ω → (inducedEdges D).powerset)
(hA : ∀ D, Measurable (A D))
(hmask : ∀ D, ConnectedFinset D → ∀ a : (inducedEdges D).powerset,
μ {ω | A D ω = a} =
ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) a.val))
(hcolor : ∀ D, ConnectedFinset D → ∀ eta : TreeVertex d → IsingSpin,
∀ᵐ ω ∂μ, κ ω {x | ∀ v ∈ D, x v = eta v} =
ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) (A D ω).val eta)) :
HasFreeIsingLaw theta (Measure.map Prod.snd (μ ⊗ₘ κ)) := by
intro D hD eta
rw [map_compProd_snd_apply_eq_finite_mixture μ κ (A D) (hA D) _
(measurableSet_spinCylinder D eta)
(fun a => ENNReal.ofReal (FiniteCluster.bondWeight theta (inducedEdges D) a.val))
(fun a => ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D) a.val eta))
(hmask D hD) (hcolor D hD eta)]
exact cluster_mixture_sum h0 h1 D eta
theorem hasFreeIsingLaw_map_prod_of_independent_bonds
{d : ℕ} {Ω C : Type*} [MeasurableSpace Ω] [MeasurableSpace C]
(μ : Measure Ω) (ν : Measure C) [SFinite ν]
{theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta ≤ 1)
(X : (TreeVertex d × TreeVertex d) → Ω → Bool)
(hX : ∀ e, Measurable (X e))
(hindep : ∀ D : Finset (TreeVertex d),
iIndepFun (fun e : inducedEdges D => X e.val) μ)
(htrue : ∀ D : Finset (TreeVertex d), ∀ e ∈ inducedEdges D,
μ {ω | X e ω = true} = ENNReal.ofReal theta)
(hfalse : ∀ D : Finset (TreeVertex d), ∀ e ∈ inducedEdges D,
μ {ω | X e ω = false} = ENNReal.ofReal (1 - theta))
(F : Ω × C → TreeVertex d → IsingSpin) (hF : Measurable F)
(hcolor : ∀ D, ConnectedFinset D → ∀ eta : TreeVertex d → IsingSpin,
∀ᵐ ω ∂μ, ν {c | ∀ v ∈ D, F (ω, c) v = eta v} =
ENNReal.ofReal (FiniteCluster.colorWeight (inducedEdges D)
(openMask (inducedEdges D) X ω).val eta)) :
HasFreeIsingLaw theta (Measure.map F (μ.prod ν)) := by
classical
apply hasFreeIsingLaw_map_prod_of_cluster_conditionals μ ν h0 h1 F hF
(fun D => openMask (inducedEdges D) X)
(fun D => measurable_openMask _ _ (fun e _ => hX e)) _ hcolor
intro D hD A
exact openMask_atom_measure_ofReal_restricted μ (inducedEdges D) X
(hindep D) h0 h1 (htrue D) (hfalse D) A
end Problem367.ClusterAssembly
end
end OAI
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