openAI-math-dataset / lean /OAI /Probability /FreeIsing /Conditioning /FiniteObservationInformation.lean
| 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 | |