| import OAI.Probability.FreeIsing.Gaussian.FiniteGaussianPlanted |
| import OAI.Probability.FreeIsing.Ising.GaussianPosterior |
| import OAI.Probability.FreeIsing.Posterior.Limit |
| import OAI.Probability.FreeIsing.Posterior.Symmetry |
| import OAI.Probability.FreeIsing.Trees.ConnectedExhaustion |
| import OAI.Probability.FreeIsing.Trees.GibbsField |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| namespace Problem367 |
|
|
| open PlantedSymmetry |
|
|
| local instance infinitePosteriorFieldNonemptySpin : Nonempty IsingSpin := ⟨.neg⟩ |
|
|
| def finiteObservation {d : ℕ} (D : Finset (TreeVertex d)) (t : ℝ≥0) |
| (p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) : D → ℝ := |
| D.restrict (observation t p) |
|
|
| theorem measurable_finiteObservation {d : ℕ} (D : Finset (TreeVertex d)) (t : ℝ≥0) : |
| Measurable (finiteObservation D t) := |
| D.measurable_restrict.comp (measurable_observation t) |
|
|
| theorem gaussianPlanted_restrict_joint_law {d : ℕ} |
| (μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ] |
| (D : Finset (TreeVertex d)) (t : ℝ≥0) : |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)).map |
| (fun p => (D.restrict p.1, finiteObservation D t p)) = |
| (μ.map D.restrict) ⊗ₘ finiteGaussianObservationKernel t |
| (fun (eta : D → IsingSpin) v => spinSign (eta v)) := by |
| let R : ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) → |
| (D → IsingSpin) × (D → ℝ) := Prod.map D.restrict D.restrict |
| have hR : Measurable R := D.measurable_restrict.prodMap D.measurable_restrict |
| have hRlaw : (μ.prod (gaussianNoiseLaw (TreeVertex d) t)).map R = |
| (μ.map D.restrict).prod (Measure.pi (fun _ : D => gaussianReal 0 t)) := by |
| simpa only [R, gaussianNoiseLaw, Measure.infinitePi_map_restrict] using |
| (Measure.map_prod_map μ (gaussianNoiseLaw (TreeVertex d) t) |
| D.measurable_restrict D.measurable_restrict).symm |
| let F : (D → IsingSpin) × (D → ℝ) → (D → IsingSpin) × (D → ℝ) := |
| fun p => (p.1, fun v => (t : ℝ) * spinSign (p.1 v) + p.2 v) |
| have hF : Measurable F := by |
| apply measurable_fst.prodMk |
| apply Measurable.of_eval |
| intro v |
| exact (measurable_const.mul |
| ((measurable_of_finite spinSign).comp ((measurable_pi_apply v).comp measurable_fst))).add |
| ((measurable_pi_apply v).comp measurable_snd) |
| have : IsProbabilityMeasure (μ.map D.restrict) := inferInstance |
| calc |
| _ = ((μ.prod (gaussianNoiseLaw (TreeVertex d) t)).map R).map F := by |
| rw [Measure.map_map hF hR] |
| rfl |
| _ = _ := by |
| rw [hRlaw] |
| exact finiteGaussianPlanted_joint_law (μ.map D.restrict) t |
| (fun (eta : D → IsingSpin) v => spinSign (eta v)) |
|
|
| theorem HasFreeIsingLaw.finiteObservation_condExp {d : ℕ} {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (t : ℝ≥0) |
| (q : (D → IsingSpin) → ℝ) : |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t))[ |
| fun p => q (D.restrict p.1) | |
| MeasurableSpace.comap (finiteObservation D t) inferInstance] =ᵐ[ |
| μ.prod (gaussianNoiseLaw (TreeVertex d) t)] |
| fun p => FinitePosterior.fieldMean (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) v => spinSign (eta v)) q (finiteObservation D t p) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| let : IsProbabilityMeasure (μ.map D.restrict) := inferInstance |
| exact FiniteBayes.condExp_state_eq_of_posterior_integral |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) (μ.map D.restrict) |
| (finiteGaussianObservationKernel t (fun (eta : D → IsingSpin) v => spinSign (eta v))) |
| (fun p => D.restrict p.1) (finiteObservation D t) |
| (D.measurable_restrict.comp measurable_fst) (measurable_finiteObservation D t) |
| (gaussianPlanted_restrict_joint_law μ D t) q _ |
| (hμ.gaussian_restrict_posterior_mean D hD t q) |
|
|
| theorem finiteIsingPosterior_tanh_bounds {d : ℕ} {beta : ℝ} |
| (hbeta : 0 ≤ beta) (D : Finset (TreeVertex d)) (y : D → ℝ) (v : D) : |
| Real.tanh (y v - (d : ℝ) * beta) ≤ |
| FinitePosterior.fieldMean (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y ∧ |
| FinitePosterior.fieldMean (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y ≤ Real.tanh (y v + (d : ℝ) * beta) := by |
| have hmem := FinitePosterior.fieldMean_spin_mem_Ioo (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y |
| (fun eta => by cases eta v <;> simp [spinSign]) |
| ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ |
| have hb := finite_tree_gibbs_field_uniform_bound D hbeta y v |
| change |Real.artanh (FinitePosterior.fieldMean (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y) - y v| ≤ (d : ℝ) * beta at hb |
| have hmono : Monotone Real.tanh := by |
| apply monotone_of_deriv_nonneg |
| · intro x |
| exact (hasDerivAt_tanh_response x).differentiableAt |
| · intro x |
| rw [(hasDerivAt_tanh_response x).deriv] |
| exact (sub_pos.mpr (Real.tanh_sq_lt_one x)).le |
| have hlo := hmono (show y v - (d : ℝ) * beta ≤ |
| Real.artanh (FinitePosterior.fieldMean (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y) by linarith [(abs_le.mp hb).1]) |
| have hhi := hmono (show Real.artanh (FinitePosterior.fieldMean (finiteIsingEnergy beta D) |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) |
| (fun eta => spinSign (eta v)) y) ≤ y v + (d : ℝ) * beta by |
| linarith [(abs_le.mp hb).2]) |
| rw [Real.tanh_artanh hmem] at hlo hhi |
| exact ⟨hlo, hhi⟩ |
|
|
| theorem HasFreeIsingLaw.finiteObservation_spin_tanh_bounds {d : ℕ} {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (t : ℝ≥0) (v : D) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| Real.tanh (observation t p v.1 - (d : ℝ) * beta) ≤ |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t))[ |
| fun p => spinSign (p.1 v.1) | |
| MeasurableSpace.comap (finiteObservation D t) inferInstance] p ∧ |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t))[ |
| fun p => spinSign (p.1 v.1) | |
| MeasurableSpace.comap (finiteObservation D t) inferInstance] p ≤ |
| Real.tanh (observation t p v.1 + (d : ℝ) * beta) := by |
| filter_upwards [hμ.finiteObservation_condExp D hD t (fun eta => spinSign (eta v))] |
| with p hp |
| change (μ.prod (gaussianNoiseLaw (TreeVertex d) t))[ |
| fun p => spinSign (p.1 v.1) | |
| MeasurableSpace.comap (finiteObservation D t) inferInstance] p = _ at hp |
| rw [hp] |
| exact finiteIsingPosterior_tanh_bounds hbeta D (finiteObservation D t p) v |
|
|
| def finiteObservationFiltration {d : ℕ} (D : ℕ → Finset (TreeVertex d)) |
| (hD : Monotone D) (t : ℝ≥0) : |
| Filtration ℕ (inferInstance : MeasurableSpace |
| ((TreeVertex d → IsingSpin) × (TreeVertex d → ℝ))) where |
| seq n := MeasurableSpace.comap (finiteObservation (D n) t) inferInstance |
| mono' n m h := by |
| change MeasurableSpace.comap (fun p (v : D n) => observation t p v.1) inferInstance ≤ |
| MeasurableSpace.comap (fun p (v : D m) => observation t p v.1) inferInstance |
| rw [MeasurableSpace.comap_process_pi, MeasurableSpace.comap_process_pi] |
| apply iSup_le |
| intro v |
| exact le_iSup_of_le (⟨v.1, hD h v.2⟩ : D m) le_rfl |
| le' n := (measurable_finiteObservation (D n) t).comap_le |
|
|
| theorem finiteObservationFiltration_iSup {d : ℕ} |
| (D : ℕ → Finset (TreeVertex d)) (hD : Monotone D) |
| (hex : ∀ v, ∃ n, v ∈ D n) (t : ℝ≥0) : |
| (⨆ n, finiteObservationFiltration D hD t n) = observationSigma (TreeVertex d) t := by |
| change (⨆ n, MeasurableSpace.comap |
| (fun p (v : D n) => observation t p v.1) inferInstance) = |
| MeasurableSpace.comap (fun p (v : TreeVertex d) => observation t p v) inferInstance |
| simp only [MeasurableSpace.pi, MeasurableSpace.comap_iSup, |
| MeasurableSpace.comap_comp, Function.comp_def] |
| apply le_antisymm |
| · apply iSup_le |
| intro n |
| apply iSup_le |
| intro v |
| exact le_iSup_of_le v.1 le_rfl |
| · apply iSup_le |
| intro v |
| obtain ⟨n, hn⟩ := hex v |
| exact le_iSup_of_le n (le_iSup_of_le (⟨v, hn⟩ : D n) le_rfl) |
|
|
| theorem HasFreeIsingLaw.posterior_field_bound_of_exhaustion {d : ℕ} {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (v : TreeVertex d) (D : ℕ → Finset (TreeVertex d)) (hD : Monotone D) |
| (hconn : ∀ n, ConnectedFinset (D n)) (hv : ∀ n, v ∈ D n) |
| (hex : ∀ w, ∃ n, w ∈ D n) (t : ℝ≥0) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| posteriorMagnetization (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) t v p ∈ |
| Set.Ioo (-1 : ℝ) 1 ∧ |
| |Real.artanh (posteriorMagnetization |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) t v p) - observation t p v| ≤ |
| (d : ℝ) * beta := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have hb (n : ℕ) := hμ.finiteObservation_spin_tanh_bounds hbeta |
| (D n) (hconn n) t ⟨v, hv n⟩ |
| have hlim := tendsto_posterior_fields_ae |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) (finiteObservationFiltration D hD t) |
| (fun p => spinSign (p.1 v)) (fun p => observation t p v) ((d : ℝ) * beta) hb |
| rw [finiteObservationFiltration_iSup D hD hex t] at hlim |
| filter_upwards [hlim] with p hp |
| exact ⟨hp.1, hp.2.1⟩ |
|
|
| theorem HasFreeIsingLaw.gaussian_posterior_field_bound {d : ℕ} {beta : ℝ} |
| {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (v : TreeVertex d) : |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), |
| posteriorMagnetization (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) t v p ∈ |
| Set.Ioo (-1 : ℝ) 1 ∧ |
| |Real.artanh (posteriorMagnetization |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) t v p) - observation t p v| ≤ |
| (d : ℝ) * beta := by |
| obtain ⟨D, hD, hconn, hex⟩ := exists_connected_exhaustion v |
| exact hμ.posterior_field_bound_of_exhaustion hbeta v D hD |
| (fun n => (hconn n).1) (fun n => (hconn n).2) hex t |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
