openAI-math-dataset / lean /OAI /Probability /FreeIsing /Conditioning /ObservationCopyStationarity.lean
| import OAI.Probability.FreeIsing.Conditioning.CopyInvariance | |
| import OAI.Probability.FreeIsing.Posterior.ObservationFactorRecovery | |
| import OAI.Probability.FreeIsing.Trees.Transitivity | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367.ObservationCopyStationarity | |
| def arrayRelabelEquiv {d : ℕ} {A : Type*} [MeasurableSpace A] | |
| (g : TreeAut d) : (TreeVertex d → A) ≃ᵐ (TreeVertex d → A) where | |
| toFun := relabel g | |
| invFun := fun x v => x (g.toEquiv v) | |
| left_inv := by intro x; funext v; simp [relabel] | |
| right_inv := by intro x; funext v; simp [relabel] | |
| measurable_toFun := measurable_relabel g | |
| measurable_invFun := Measurable.of_eval (fun _ => measurable_pi_apply _) | |
| theorem canonical_conditionalCopies_invariant | |
| {d : ℕ} {A : Type*} [MeasurableSpace A] | |
| (nu : Measure A) [IsProbabilityMeasure nu] | |
| (rho : Measure ((TreeVertex d → A) × (TreeVertex d → ℕ → ℝ))) | |
| [IsProbabilityMeasure rho] | |
| (hfirst : rho.fst = Measure.infinitePi (fun _ : TreeVertex d => nu)) | |
| (g : TreeAut d) | |
| (hjoint : rho.map (Prod.map (relabel g) (relabel g)) = rho) : | |
| (rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)).map | |
| (Prod.map (relabel g) (Prod.map (relabel g) (relabel g))) = | |
| rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel) := by | |
| have hT : MeasurePreserving (arrayRelabelEquiv (A := A) g) rho.fst rho.fst := by | |
| refine ⟨measurable_relabel g, ?_⟩ | |
| rw [hfirst] | |
| exact Measure.map_infinitePi_infinitePi_of_inj g.toEquiv.symm.injective | |
| have hdis : rho.fst ⊗ₘ rho.condKernel = rho := | |
| Measure.disintegrate rho rho.condKernel | |
| apply conditionalCopies_invariant rho.fst rho.condKernel (arrayRelabelEquiv g) | |
| (measurable_relabel g) hT | |
| simpa [hdis, arrayRelabelEquiv] using hjoint | |
| theorem ae_eq_arrays_of_reference_vertex | |
| {d : ℕ} {A : Type*} [MeasurableSpace A] | |
| (mu : Measure ((TreeVertex d → A) × | |
| ((TreeVertex d → ℕ → ℝ) × (TreeVertex d → ℕ → ℝ)))) | |
| (o : TreeVertex d) | |
| (hinvariant : ∀ g : TreeAut d, | |
| mu.map (Prod.map (relabel g) (Prod.map (relabel g) (relabel g))) = mu) | |
| (hroot : ∀ n, ∀ᵐ p ∂mu, p.2.1 o n = p.2.2 o n) : | |
| ∀ᵐ p ∂mu, p.2.1 = p.2.2 := by | |
| have hcoord : ∀ v n, ∀ᵐ p ∂mu, p.2.1 v n = p.2.2 v n := by | |
| intro v n | |
| obtain ⟨g, hg⟩ := exists_treeAut_map v o | |
| have hgo : g.toEquiv.symm o = v := by | |
| rw [← hg, Equiv.symm_apply_apply] | |
| have hpres : MeasurePreserving | |
| (Prod.map (relabel g) (Prod.map (relabel g) (relabel g))) mu mu := | |
| ⟨(measurable_relabel g).prodMap | |
| ((measurable_relabel g).prodMap (measurable_relabel g)), hinvariant g⟩ | |
| filter_upwards [hpres.quasiMeasurePreserving.ae (hroot n)] with p hp | |
| change p.2.1 (g.toEquiv.symm o) n = p.2.2 (g.toEquiv.symm o) n at hp | |
| simpa only [hgo] using hp | |
| have hall : ∀ᵐ p ∂mu, ∀ v n, p.2.1 v n = p.2.2 v n := | |
| ae_all_iff.2 (fun v => ae_all_iff.2 (hcoord v)) | |
| filter_upwards [hall] with p hp | |
| exact funext (fun v => funext (hp v)) | |
| theorem isFactorOfIID_of_joint_observation_reference_collapse | |
| {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) | |
| (hroot : ∀ n, ∀ᵐ p ∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel), | |
| p.2.1 o n = p.2.2 o n) : | |
| IsFactorOfIID d theta := by | |
| have hcopies := ae_eq_arrays_of_reference_vertex | |
| (rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)) o | |
| (fun g => canonical_conditionalCopies_invariant nu rho hfirst g (hjoint g)) hroot | |
| have hdis : (Measure.infinitePi (fun _ : TreeVertex d => nu)) ⊗ₘ | |
| rho.condKernel = rho := by | |
| rw [← hfirst] | |
| exact Measure.disintegrate rho rho.condKernel | |
| apply isFactorOfIID_of_equal_observation_conditionalCopies nu rho.condKernel | |
| · simpa only [hdis] using hLaw | |
| · intro g | |
| rw [hdis] | |
| exact (show MeasurePreserving (Prod.map (relabel g) (relabel g)) rho rho from | |
| ⟨(measurable_relabel g).prodMap (measurable_relabel g), hjoint g⟩).quasiMeasurePreserving | |
| · simpa only [hfirst] using hcopies | |
| theorem isFactorOfIID_of_joint_observation_reference_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) | |
| (hInt : ∀ n, Integrable (fun p => (p.2.1 o n - p.2.2 o n) ^ 2) | |
| (rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel))) | |
| (hzero : ∀ n, (∫ p, (p.2.1 o n - p.2.2 o n) ^ 2 | |
| ∂rho.fst ⊗ₘ (rho.condKernel ×ₖ rho.condKernel)) = 0) : | |
| IsFactorOfIID d theta := by | |
| apply isFactorOfIID_of_joint_observation_reference_collapse nu rho o hfirst hLaw hjoint | |
| exact fun n => ae_eq_of_integral_sq_sub_eq_zero (hInt n) (hzero n) | |
| end Problem367.ObservationCopyStationarity | |
| end | |
| end OAI | |