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f61e547 | 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 | import OAI.Probability.FreeIsing.Conditioning.ObservationCopyStationarity
import OAI.Probability.FreeIsing.Martingales.InnovationMartingale
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped NNReal
namespace Problem367
theorem observations_ae_eq_of_common_innovation_and_field_zero
{O : Type*} [MeasurableSpace O] {mu : Measure O}
{h h' : ℝ≥0 → O → ℝ} {X Y W : ℕ → O → ℝ}
(hc : ∀ᵐ p ∂mu, Continuous (fun t => h t p))
(hc' : ∀ᵐ p ∂mu, Continuous (fun t => h' t p))
(hInt : ∀ t, Integrable (fun p => (h t p - h' t p) ^ 2) mu)
(hzero : ∀ t, (∫ p, (h t p - h' t p) ^ 2 ∂mu) = 0)
(hX : ∀ n, ∀ᵐ p ∂mu,
X n p = W n p + nnrealTimePrimitive (fun t p => Real.tanh (h t p)) n p)
(hY : ∀ n, ∀ᵐ p ∂mu,
Y n p = W n p + nnrealTimePrimitive (fun t p => Real.tanh (h' t p)) n p) :
∀ n, X n =ᵐ[mu] Y n := by
have hfield : (fun p t => h t p) =ᵐ[mu] (fun p t => h' t p) :=
ae_eq_of_continuous_of_integral_sq_sub_eq_zero hc hc' hInt hzero
intro n
filter_upwards [hfield, hX n, hY n] with p hp hx hy
rw [hx, hy]
congr 1
unfold nnrealTimePrimitive
apply intervalIntegral.integral_congr
intro t _
exact congrArg Real.tanh (congrFun hp t.toNNReal)
theorem observations_ae_eq_of_common_innovation_and_measurable_fields
{O : Type*} [MeasurableSpace O] {mu : Measure O} [SFinite mu]
{h h' : ℝ≥0 → O → ℝ} {X Y W : ℕ → O → ℝ}
(hm : Measurable (fun q : ℝ × O => Real.tanh (h q.1.toNNReal q.2)))
(hm' : Measurable (fun q : ℝ × O => Real.tanh (h' q.1.toNNReal q.2)))
(heq : ∀ t, h t =ᵐ[mu] h' t)
(hX : ∀ n, ∀ᵐ p ∂mu,
X n p = W n p + nnrealTimePrimitive (fun t p => Real.tanh (h t p)) n p)
(hY : ∀ n, ∀ᵐ p ∂mu,
Y n p = W n p + nnrealTimePrimitive (fun t p => Real.tanh (h' t p)) n p) :
∀ n, X n =ᵐ[mu] Y n := by
have htime : ∀ᵐ t ∂(volume : Measure ℝ), ∀ᵐ p ∂mu,
Real.tanh (h t.toNNReal p) = Real.tanh (h' t.toNNReal p) := by
apply Filter.Eventually.of_forall
intro t
filter_upwards [heq t.toNNReal] with p hp
rw [hp]
have hsample : ∀ᵐ p ∂mu, ∀ᵐ t ∂(volume : Measure ℝ),
Real.tanh (h t.toNNReal p) = Real.tanh (h' t.toNNReal p) :=
(Measure.ae_ae_comm (μ := (volume : Measure ℝ)) (ν := mu)
(p := fun t p => Real.tanh (h t.toNNReal p) = Real.tanh (h' t.toNNReal p))
(measurableSet_eq_fun hm hm')).1 htime
intro n
filter_upwards [hsample, hX n, hY n] with p hp hx hy
rw [hx, hy]
congr 1
unfold nnrealTimePrimitive
apply intervalIntegral.integral_congr_ae
exact hp.mono (fun _ ht _ => ht)
theorem observations_ae_eq_of_common_innovation_and_measurable_field_zero
{O : Type*} [MeasurableSpace O] {mu : Measure O} [SFinite mu]
{h h' : ℝ≥0 → O → ℝ} {X Y W : ℕ → O → ℝ}
(hm : Measurable (fun q : ℝ × O => Real.tanh (h q.1.toNNReal q.2)))
(hm' : Measurable (fun q : ℝ × O => Real.tanh (h' q.1.toNNReal q.2)))
(hInt : ∀ t, Integrable (fun p => (h t p - h' t p) ^ 2) mu)
(hzero : ∀ t, (∫ p, (h t p - h' t p) ^ 2 ∂mu) = 0)
(hX : ∀ n, ∀ᵐ p ∂mu,
X n p = W n p + nnrealTimePrimitive (fun t p => Real.tanh (h t p)) n p)
(hY : ∀ n, ∀ᵐ p ∂mu,
Y n p = W n p + nnrealTimePrimitive (fun t p => Real.tanh (h' t p)) n p) :
∀ n, X n =ᵐ[mu] Y n := by
apply observations_ae_eq_of_common_innovation_and_measurable_fields hm hm' _ hX hY
exact fun t => ae_eq_of_integral_sq_sub_eq_zero (hInt t) (hzero t)
theorem isFactorOfIID_of_joint_innovation_field_zero
{d : ℕ} {theta : ℝ} {A : Type*} [MeasurableSpace A]
[StandardBorelSpace A] [Nonempty A]
(nu : Measure A) [IsProbabilityMeasure nu]
(rho : Measure ((TreeVertex d → A) × (TreeVertex d → ℕ → ℝ)))
[IsProbabilityMeasure rho] (o : TreeVertex d)
(hfirst : rho.fst = Measure.infinitePi (fun _ : TreeVertex d => nu))
(hLaw : HasFreeIsingLaw theta
(Measure.map (fun p => SpinRecovery.decodeArray p.2) rho))
(hjoint : ∀ g : TreeAut d, rho.map (Prod.map (relabel g) (relabel g)) = rho)
(h h' : ℝ≥0 →
((TreeVertex d → A) × ((TreeVertex d → ℕ → ℝ) × (TreeVertex d → ℕ → ℝ))) → ℝ)
(W : ℕ → (TreeVertex d → A) → ℝ)
(hc : ∀ᵐ p ∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel),
Continuous (fun t => h t p))
(hc' : ∀ᵐ p ∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel),
Continuous (fun t => h' t p))
(hInt : ∀ t, Integrable (fun p => (h t p - h' t p) ^ 2)
(rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)))
(hzero : ∀ t, (∫ p, (h t p - h' t p) ^ 2
∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)) = 0)
(hX : ∀ n, ∀ᵐ p ∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel),
p.2.1 o n = W n p.1 + nnrealTimePrimitive (fun t p => Real.tanh (h t p)) n p)
(hY : ∀ n, ∀ᵐ p ∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel),
p.2.2 o n = W n p.1 + nnrealTimePrimitive (fun t p => Real.tanh (h' t p)) n p) :
IsFactorOfIID d theta := by
apply ObservationCopyStationarity.isFactorOfIID_of_joint_observation_reference_collapse
nu rho o hfirst hLaw hjoint
exact observations_ae_eq_of_common_innovation_and_field_zero hc hc' hInt hzero hX hY
theorem isFactorOfIID_of_joint_innovation_measurable_field_zero
{d : ℕ} {theta : ℝ} {A : Type*} [MeasurableSpace A]
[StandardBorelSpace A] [Nonempty A]
(nu : Measure A) [IsProbabilityMeasure nu]
(rho : Measure ((TreeVertex d → A) × (TreeVertex d → ℕ → ℝ)))
[IsProbabilityMeasure rho] (o : TreeVertex d)
(hfirst : rho.fst = Measure.infinitePi (fun _ : TreeVertex d => nu))
(hLaw : HasFreeIsingLaw theta
(Measure.map (fun p => SpinRecovery.decodeArray p.2) rho))
(hjoint : ∀ g : TreeAut d, rho.map (Prod.map (relabel g) (relabel g)) = rho)
(h h' : ℝ≥0 →
((TreeVertex d → A) × ((TreeVertex d → ℕ → ℝ) × (TreeVertex d → ℕ → ℝ))) → ℝ)
(W : ℕ → (TreeVertex d → A) → ℝ)
(hm : Measurable (Function.uncurry (fun (t : ℝ) p => Real.tanh (h t.toNNReal p))))
(hm' : Measurable (Function.uncurry (fun (t : ℝ) p => Real.tanh (h' t.toNNReal p))))
(hInt : ∀ t, Integrable (fun p => (h t p - h' t p) ^ 2)
(rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)))
(hzero : ∀ t, (∫ p, (h t p - h' t p) ^ 2
∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)) = 0)
(hX : ∀ n, ∀ᵐ p ∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel),
p.2.1 o n = W n p.1 + nnrealTimePrimitive (fun t p => Real.tanh (h t p)) n p)
(hY : ∀ n, ∀ᵐ p ∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel),
p.2.2 o n = W n p.1 + nnrealTimePrimitive (fun t p => Real.tanh (h' t p)) n p) :
IsFactorOfIID d theta := by
apply ObservationCopyStationarity.isFactorOfIID_of_joint_observation_reference_collapse
nu rho o hfirst hLaw hjoint
exact observations_ae_eq_of_common_innovation_and_measurable_field_zero
hm hm' hInt hzero hX hY
end Problem367
end
end OAI
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