File size: 6,669 Bytes
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 | import OAI.Probability.FreeIsing.Conditioning.DiscreteConditioning
import OAI.Probability.FreeIsing.Gaussian.FiniteSignalInformation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace Problem367
theorem iIndep_observationFields_cond_spin
{Ω ι : Type*} {A : ι → Type*} [mΩ : MeasurableSpace Ω]
[StandardBorelSpace Ω] [mA : ∀ i, MeasurableSpace (A i)]
{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))
(a : IsingSpin) :
iIndep (fun i => (mA i).comap (X i)) (cond rho {x | S x = a}) := by
have hne : rho (S ⁻¹' {a}) ≠ 0 := by
change rho {x | S x = a} ≠ 0
rw [hfair]
norm_num
exact (iIndepFun_cond_fiber_of_iCondIndepFun hS hX hCI a hne).iIndep
section Observations
variable {Ω ι : Type*} {A : ι → Type*} [mΩ : MeasurableSpace Ω]
[StandardBorelSpace Ω] [mA : ∀ i, MeasurableSpace (A i)] [Fintype 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)
(hsym : ∀ i, Measure.map (fun u : ℝ => -u)
(Measure.map (rho[fun y => spinSign (S y) | (mA i).comap (X i)]) rho) =
Measure.map (rho[fun y => spinSign (S y) | (mA i).comap (X i)]) rho)
include hS hX hCI hfair hstrict hsym
theorem condExp_finite_observation_secondMoment_le_sum [Nonempty ι] :
(∫ x, (rho[fun y => spinSign (S y) |
(inferInstance : MeasurableSpace (∀ i, A i)).comap (fun x i => X i x)] x) ^ 2 ∂rho) ≤
∑ i, ∫ x, (rho[fun y => spinSign (S y) | (mA i).comap (X i)] x) ^ 2 ∂rho := by
have h := spinInformation_finiteSignalField_le_sum (fun i => (hX i).comap_le)
hS hstrict hfair (iIndep_observationFields_cond_spin hS hX hCI hfair) hsym
Finset.univ Finset.univ_nonempty
have hf : finiteSignalField (fun i => (mA i).comap (X i)) Finset.univ =
(inferInstance : MeasurableSpace (∀ i, A i)).comap (fun x i => X i x) := by
simp [finiteSignalField, MeasurableSpace.comap_process_pi]
rw [hf] at h
simpa only [spinInformation] using h
theorem condExp_finite_observation_secondMoment_le_sum_sub_loss
{a b : ι} (hab : a ≠ b) {theta : ℝ}
(haTheta : ∀ᵐ x ∂rho,
|rho[fun y => spinSign (S y) | (mA a).comap (X a)] x| ≤ theta)
(hbTheta : ∀ᵐ x ∂rho,
|rho[fun y => spinSign (S y) | (mA b).comap (X b)] x| ≤ theta)
(htheta : theta < 1) :
(∫ x, (rho[fun y => spinSign (S y) |
(inferInstance : MeasurableSpace (∀ i, A i)).comap (fun x i => X i x)] x) ^ 2 ∂rho) ≤
(∑ i, ∫ x, (rho[fun y => spinSign (S y) | (mA i).comap (X i)] x) ^ 2 ∂rho) -
2 * (1 - theta ^ 2) *
(∫ x, (rho[fun y => spinSign (S y) | (mA a).comap (X a)] x) ^ 2 ∂rho) *
(∫ x, (rho[fun y => spinSign (S y) | (mA b).comap (X b)] x) ^ 2 ∂rho) := by
have h := spinInformation_finiteSignalField_le_sum_sub_loss (fun i => (hX i).comap_le)
hS hstrict hfair (iIndep_observationFields_cond_spin hS hX hCI hfair) hsym
Finset.univ hab (Finset.mem_univ a) (Finset.mem_univ b) haTheta hbTheta htheta
have hf : finiteSignalField (fun i => (mA i).comap (X i)) Finset.univ =
(inferInstance : MeasurableSpace (∀ i, A i)).comap (fun x i => X i x) := by
simp [finiteSignalField, MeasurableSpace.comap_process_pi]
rw [hf] at h
simpa only [spinInformation] using h
theorem condExp_own_and_finite_observation_secondMoment_le_sum_sub_loss
{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)
(hOwnSym : Measure.map (fun u : ℝ => -u)
(Measure.map (rho[fun y => spinSign (S y) | mB.comap Y]) rho) =
Measure.map (rho[fun y => spinSign (S y) | mB.comap Y]) rho)
{a b : ι} (hab : a ≠ b) {theta : ℝ}
(haTheta : ∀ᵐ x ∂rho,
|rho[fun y => spinSign (S y) | (mA a).comap (X a)] x| ≤ theta)
(hbTheta : ∀ᵐ x ∂rho,
|rho[fun y => spinSign (S y) | (mA b).comap (X b)] x| ≤ theta)
(htheta : theta < 1) :
(∫ x, (rho[fun y => spinSign (S y) |
(inferInstance : MeasurableSpace (B × (∀ i, A i))).comap
(fun x => (Y x, fun i => X i x))] x) ^ 2 ∂rho) ≤
(∫ x, (rho[fun y => spinSign (S y) | mB.comap Y] x) ^ 2 ∂rho) +
(∑ i, ∫ x, (rho[fun y => spinSign (S y) | (mA i).comap (X i)] x) ^ 2 ∂rho) -
2 * (1 - theta ^ 2) *
(∫ x, (rho[fun y => spinSign (S y) | (mA a).comap (X a)] x) ^ 2 ∂rho) *
(∫ x, (rho[fun y => spinSign (S y) | (mA b).comap (X b)] x) ^ 2 ∂rho) := 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 ⟨a, Finset.mem_univ a⟩
have hPairInd (s : IsingSpin) : Indep mAll (mB.comap Y) (cond rho {x | S x = s}) := by
exact (indepFun_cond_fiber_of_condIndepFun hS hY hAll hOwn s).symm
have hPair := spinInformation_sup_le_add hAll.comap_le hY.comap_le hS
hAllStrict hOwnStrict hfair hPairInd hOwnSym
have hFamily := condExp_finite_observation_secondMoment_le_sum_sub_loss
hS hX hCI hfair hstrict hsym hab haTheta hbTheta htheta
change spinInformation rho S
((mB.prod MeasurableSpace.pi).comap (fun x => (Y x, fun i => X i x))) ≤ _
rw [MeasurableSpace.comap_prodMk, sup_comm]
change spinInformation rho S (mAll ⊔ mB.comap Y) ≤ _
unfold spinInformation at hPair ⊢
linarith
end Observations
end Problem367
end
end OAI
|