openAI-math-dataset / lean /OAI /Probability /FreeIsing /Conditioning /ComponentInformationUpper.lean
| import OAI.Probability.FreeIsing.Conditioning.PlantedInformation | |
| import OAI.Probability.FreeIsing.Energy.Attenuation | |
| import OAI.Probability.FreeIsing.Posterior.InfinitePosteriorField | |
| import OAI.Probability.FreeIsing.Posterior.SelectedPosteriorSymmetry | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped NNReal Topology | |
| namespace Problem367 | |
| open PlantedSymmetry | |
| local instance componentInformationUpperNonemptyIsingSpin : Nonempty IsingSpin := ⟨.neg⟩ | |
| theorem singletonGibbsSpinMean_eq_tanh {d : ℕ} {beta : ℝ} (hb : 0 ≤ beta) | |
| (v : TreeVertex d) (y : ({v} : Finset (TreeVertex d)) → ℝ) : | |
| FinitePosterior.fieldMean (finiteIsingEnergy beta {v}) | |
| (fun (eta : ({v} : Finset (TreeVertex d)) → IsingSpin) w => spinSign (eta w)) | |
| (fun eta => spinSign (eta ⟨v, by simp⟩)) y = Real.tanh (y ⟨v, by simp⟩) := by | |
| classical | |
| let r : ({v} : Finset (TreeVertex d)) := ⟨v, by simp⟩ | |
| have hm := FinitePosterior.fieldMean_spin_mem_Ioo (finiteIsingEnergy beta {v}) | |
| (fun (eta : ({v} : Finset (TreeVertex d)) → IsingSpin) w => spinSign (eta w)) | |
| (fun eta => spinSign (eta r)) y | |
| (fun eta => by cases eta r <;> simp [spinSign]) | |
| ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ | |
| have hfield := finite_tree_gibbs_field_bound {v} hb y r | |
| change |Real.artanh (FinitePosterior.fieldMean (finiteIsingEnergy beta {v}) | |
| (fun eta w => spinSign (eta w)) (fun eta => spinSign (eta r)) y) - y r| ≤ _ at hfield | |
| simp only [inducedEdges_singleton, Finset.filter_empty, Finset.card_empty, | |
| Nat.cast_zero, zero_mul] at hfield | |
| have heq := sub_eq_zero.mp (abs_eq_zero.mp (le_antisymm hfield (abs_nonneg _))) | |
| rw [← Real.tanh_artanh hm, heq] | |
| theorem HasFreeIsingLaw.selected_singleton_gaussian_posterior | |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) | |
| (t : ℝ≥0) (v : TreeVertex d) : | |
| selectedPosterior (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) | |
| (fun s z => (t : ℝ) * spinSign s + z) (fun _ : Unit => v) v =ᵐ[ | |
| μ.prod (gaussianNoiseLaw (TreeVertex d) t)] | |
| fun p => Real.tanh (observation t p v) := by | |
| classical | |
| have hfield : MeasurableSpace.comap (finiteObservation {v} t) inferInstance = | |
| selectedObservationField (fun s z => (t : ℝ) * spinSign s + z) | |
| (fun _ : Unit => v) := by | |
| change selectedObservationField (fun s z => (t : ℝ) * spinSign s + z) | |
| (fun w : ({v} : Finset (TreeVertex d)) => w.1) = _ | |
| apply selectedObservationField_eq_of_range_eq | |
| ext w | |
| simp | |
| have hce := hμ.finiteObservation_condExp {v} (connectedFinset_singleton v) t | |
| (fun eta => spinSign (eta ⟨v, by simp⟩)) | |
| rw [hfield] at hce | |
| filter_upwards [hce] with p hp | |
| change selectedPosterior _ _ _ _ p = _ at hp | |
| rw [hp, singletonGibbsSpinMean_eq_tanh hb] | |
| rfl | |
| theorem gaussian_observation_second_moment {d : ℕ} | |
| (μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ] | |
| (t : ℝ≥0) (v : TreeVertex d) : | |
| (∫ p, (observation t p v) ^ 2 ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) = | |
| (t : ℝ) + (t : ℝ) ^ 2 := by | |
| let ρ := μ.prod (gaussianNoiseLaw (TreeVertex d) t) | |
| have hS : MemLp (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => | |
| spinSign (p.1 v)) 2 ρ := | |
| (PlantedInformation.spinSign_memLp μ v).comp_measurePreserving measurePreserving_fst | |
| have hB := centered_gaussian_moments (PlantedInformation.noise_hasLaw μ t v) | |
| have hind := PlantedInformation.spin_noise_indep μ t v v | |
| have h := integral_observation_mul_observation hS hS hB.1 hB.1 hind hind | |
| hB.2.1 hB.2.1 (t : ℝ) | |
| have hspin : (∫ p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ), | |
| spinSign (p.1 v) * spinSign (p.1 v) ∂ρ) = 1 := by | |
| have heq : (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => | |
| spinSign (p.1 v) * spinSign (p.1 v)) = fun _ => (1 : ℝ) := by | |
| funext p | |
| cases p.1 v <;> norm_num [spinSign] | |
| rw [heq] | |
| simp [ρ] | |
| rw [hspin] at h | |
| simp only [← pow_two] at h | |
| rw [hB.2.2] at h | |
| simpa only [observation, pow_two, mul_one, add_comm] using h | |
| theorem HasFreeIsingLaw.selected_singleton_information_le | |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) | |
| (t : ℝ≥0) (v : TreeVertex d) : | |
| (∫ p, (selectedPosterior (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) | |
| (fun s z => (t : ℝ) * spinSign s + z) (fun _ : Unit => v) v p) ^ 2 | |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ (t : ℝ) + (t : ℝ) ^ 2 := by | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| let ρ := μ.prod (gaussianNoiseLaw (TreeVertex d) t) | |
| have hF : Measurable (Function.uncurry | |
| (fun (s : IsingSpin) (z : ℝ) => (t : ℝ) * spinSign s + z)) := | |
| ((PlantedInformation.measurable_spinSign.comp measurable_fst).const_mul _).add measurable_snd | |
| have hpost := integrable_selectedPosterior_abs_pow ρ hF (fun _ : Unit => v) v 2 | |
| simp only [sq_abs] at hpost | |
| have hS := (PlantedInformation.spinSign_memLp μ v).comp_measurePreserving | |
| (measurePreserving_fst (ν := gaussianNoiseLaw (TreeVertex d) t)) | |
| have hB := (centered_gaussian_moments (PlantedInformation.noise_hasLaw μ t v)).1 | |
| have hobs : Integrable (fun p => (observation t p v) ^ 2) ρ := | |
| ((hS.const_mul (t : ℝ)).add hB).integrable_sq | |
| rw [← gaussian_observation_second_moment μ t v] | |
| apply integral_mono_ae hpost hobs | |
| filter_upwards [hμ.selected_singleton_gaussian_posterior hb t v] with p hp | |
| rw [hp] | |
| have hbound := abs_tanh_le_abs_response (observation t p v) | |
| simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg _) hbound 2 | |
| end Problem367 | |
| end | |
| end OAI | |