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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 | import OAI.Probability.FreeIsing.Conditioning.FiniteObservationInformation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace Problem367
theorem tanh_add_eq_binaryFusion (x y : ℝ) :
Real.tanh (x + y) = binaryFusion (Real.tanh x) (Real.tanh y) := by
unfold binaryFusion
simp only [Real.tanh_eq_sinh_div_cosh, Real.sinh_add, Real.cosh_add]
have hx := (Real.cosh_pos x).ne'
have hy := (Real.cosh_pos y).ne'
field_simp
theorem artanh_binaryFusion {u v : ℝ} (hu : |u| < 1) (hv : |v| < 1) :
Real.artanh (binaryFusion u v) = Real.artanh u + Real.artanh v := by
have h := tanh_add_eq_binaryFusion (Real.artanh u) (Real.artanh v)
rw [Real.tanh_artanh (abs_lt.mp hu), Real.tanh_artanh (abs_lt.mp hv)] at h
rw [← h, Real.artanh_tanh]
theorem artanh_condExp_spin_sup
{Ω : Type*} {m₁ m₂ mΩ : MeasurableSpace Ω} {rho : Measure Ω}
[IsProbabilityMeasure rho] (hm₁ : m₁ ≤ mΩ) (hm₂ : m₂ ≤ mΩ)
{S : Ω → IsingSpin} (hS : Measurable S)
(h₁ : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₁] x| < 1)
(h₂ : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m₂] x| < 1)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hInd : ∀ a : IsingSpin, Indep m₁ m₂ (cond rho {x | S x = a})) :
(fun x => Real.artanh (rho[fun y => spinSign (S y) | m₁ ⊔ m₂] x)) =ᵐ[rho]
fun x => Real.artanh (rho[fun y => spinSign (S y) | m₁] x) +
Real.artanh (rho[fun y => spinSign (S y) | m₂] x) := by
filter_upwards [condExp_spin_sup_eq_binaryFusion hm₁ hm₂ hS h₁ h₂ hfair hInd,
h₁, h₂] with x hx hx₁ hx₂
rw [hx, artanh_binaryFusion hx₁ hx₂]
theorem artanh_condExp_spin_finiteSignalField
{Ω ι : Type*} {mΩ : MeasurableSpace Ω} {rho : Measure Ω}
[IsProbabilityMeasure rho] {m : ι → MeasurableSpace Ω}
(hm : ∀ i, m i ≤ mΩ) {S : Ω → IsingSpin} (hS : Measurable S)
(hstrict : ∀ i, ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | m i] x| < 1)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hInd : ∀ a : IsingSpin, iIndep m (cond rho {x | S x = a}))
(I : Finset ι) (hI : I.Nonempty) :
(fun x => Real.artanh
(rho[fun y => spinSign (S y) | finiteSignalField m I] x)) =ᵐ[rho]
fun x => ∑ i ∈ I, Real.artanh (rho[fun y => spinSign (S y) | m i] x) := by
classical
induction I using Finset.induction_on with
| empty => simp at hI
| @insert a I ha ih =>
by_cases hIn : I.Nonempty
· have hi := ih hIn
have hf := artanh_condExp_spin_sup (finiteSignalField_le hm I) (hm a) hS
(abs_condExp_spin_finiteSignalField_lt_one hm hS hstrict hfair hInd I hIn)
(hstrict a) hfair
(fun s => indep_finiteSignalField_of_not_mem hm (hInd s) ha)
rw [finiteSignalField_insert, sup_comm]
filter_upwards [hi, hf] with x hx hfx
rw [hfx, hx, Finset.sum_insert ha]
exact add_comm _ _
· have hIe : I = ∅ := Finset.not_nonempty_iff_eq_empty.mp hIn
subst I
simp
theorem artanh_condExp_finite_observation
{Ω ι : Type*} {A : ι → Type*} [mΩ : MeasurableSpace Ω]
[StandardBorelSpace Ω] [mA : ∀ i, MeasurableSpace (A i)] [Fintype ι] [Nonempty ι]
{rho : Measure Ω} [IsProbabilityMeasure rho]
{S : Ω → IsingSpin} (hS : Measurable S)
{X : ∀ i, Ω → A i} (hX : ∀ i, Measurable (X i))
(hCI : iCondIndepFun (MeasurableSpace.comap S inferInstance) hS.comap_le X rho)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hstrict : ∀ i, ∀ᵐ x ∂rho,
|rho[fun y => spinSign (S y) | (mA i).comap (X i)] x| < 1) :
(fun x => Real.artanh (rho[fun y => spinSign (S y) |
(inferInstance : MeasurableSpace (∀ i, A i)).comap (fun x i => X i x)] x)) =ᵐ[rho]
fun x => ∑ i, Real.artanh
(rho[fun y => spinSign (S y) | (mA i).comap (X i)] x) := by
have h := artanh_condExp_spin_finiteSignalField (fun i => (hX i).comap_le)
hS hstrict hfair (iIndep_observationFields_cond_spin hS hX hCI hfair)
Finset.univ Finset.univ_nonempty
simpa only [finiteSignalField, Finset.mem_univ, iSup_pos,
MeasurableSpace.comap_process_pi] using h
theorem artanh_condExp_own_and_finite_observation
{Ω ι : Type*} {A : ι → Type*} [mΩ : MeasurableSpace Ω]
[StandardBorelSpace Ω] [mA : ∀ i, MeasurableSpace (A i)] [Fintype ι] [Nonempty ι]
{rho : Measure Ω} [IsProbabilityMeasure rho]
{S : Ω → IsingSpin} (hS : Measurable S)
{X : ∀ i, Ω → A i} (hX : ∀ i, Measurable (X i))
(hCI : iCondIndepFun (MeasurableSpace.comap S inferInstance) hS.comap_le X rho)
(hfair : ∀ a : IsingSpin, rho {x | S x = a} = (1 / 2 : ENNReal))
(hstrict : ∀ i, ∀ᵐ x ∂rho,
|rho[fun y => spinSign (S y) | (mA i).comap (X i)] x| < 1)
{B : Type*} [mB : MeasurableSpace B] {Y : Ω → B} (hY : Measurable Y)
(hOwn : CondIndepFun (MeasurableSpace.comap S inferInstance) hS.comap_le
Y (fun x i => X i x) rho)
(hOwnStrict : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | mB.comap Y] x| < 1) :
(fun x => Real.artanh (rho[fun y => spinSign (S y) |
(inferInstance : MeasurableSpace (B × (∀ i, A i))).comap
(fun x => (Y x, fun i => X i x))] x)) =ᵐ[rho]
fun x => Real.artanh (rho[fun y => spinSign (S y) | mB.comap Y] x) +
∑ i, Real.artanh (rho[fun y => spinSign (S y) | (mA i).comap (X i)] x) := by
let mAll : MeasurableSpace Ω :=
(inferInstance : MeasurableSpace (∀ i, A i)).comap (fun x i => X i x)
let : MeasurableSpace Ω := mΩ
have hAll : Measurable (fun x i => X i x) := Measurable.of_eval hX
have hFields := iIndep_observationFields_cond_spin hS hX hCI hfair
have hf : finiteSignalField (fun i => (mA i).comap (X i)) Finset.univ = mAll := by
simp [mAll, finiteSignalField, MeasurableSpace.comap_process_pi]
have hAllStrict : ∀ᵐ x ∂rho, |rho[fun y => spinSign (S y) | mAll] x| < 1 := by
rw [← hf]
exact abs_condExp_spin_finiteSignalField_lt_one (fun i => (hX i).comap_le)
hS hstrict hfair hFields Finset.univ Finset.univ_nonempty
have hPairInd (s : IsingSpin) : Indep (mB.comap Y) mAll (cond rho {x | S x = s}) :=
indepFun_cond_fiber_of_condIndepFun hS hY hAll hOwn s
have hPair := artanh_condExp_spin_sup hY.comap_le hAll.comap_le hS
hOwnStrict hAllStrict hfair hPairInd
have hFamily := artanh_condExp_finite_observation hS hX hCI hfair hstrict
change (fun x => Real.artanh (rho[fun y => spinSign (S y) |
(mB.prod MeasurableSpace.pi).comap (fun x => (Y x, fun i => X i x))] x)) =ᵐ[rho] _
rw [MeasurableSpace.comap_prodMk]
filter_upwards [hPair, hFamily] with x hp hfam
exact hp.trans (congrArg (fun z =>
Real.artanh (rho[fun y => spinSign (S y) | mB.comap Y] x) + z) hfam)
end Problem367
end
end OAI
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