openAI-math-dataset / lean /OAI /Probability /FreeIsing /Posterior /FinitePosteriorVarianceInverse.lean
| import OAI.Probability.FreeIsing.Posterior.VarianceInverse | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped NNReal | |
| namespace Problem367.FinitePosteriorVariance | |
| open PlantedSymmetry | |
| local instance : Nonempty IsingSpin := ⟨.neg⟩ | |
| def spinMean {d : ℕ} (beta : ℝ) (D : Finset (TreeVertex d)) (v : D) | |
| (y : TreeVertex d → ℝ) : ℝ := | |
| FinitePosterior.fieldMean (finiteIsingEnergy beta D) | |
| (fun (eta : D → IsingSpin) w => spinSign (eta w)) | |
| (fun eta => spinSign (eta v)) (D.restrict y) | |
| theorem continuous_spinMean {d : ℕ} (beta : ℝ) | |
| (D : Finset (TreeVertex d)) (v : D) : Continuous (spinMean beta D v) := by | |
| apply (FinitePosterior.continuous_fieldMean _ _ _).comp | |
| exact continuous_pi (fun w => continuous_apply w.1) | |
| theorem spinMean_mem_Ioo {d : ℕ} (beta : ℝ) | |
| (D : Finset (TreeVertex d)) (v : D) (y : TreeVertex d → ℝ) : | |
| spinMean beta D v y ∈ Set.Ioo (-1 : ℝ) 1 := by | |
| exact FinitePosterior.fieldMean_spin_mem_Ioo _ _ _ _ | |
| (fun eta => by cases eta v <;> simp [spinSign]) | |
| ⟨fun _ => .neg, rfl⟩ ⟨fun _ => .pos, rfl⟩ | |
| theorem field_sub_observation_abs_le {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) | |
| (D : Finset (TreeVertex d)) (v : D) (y : TreeVertex d → ℝ) : | |
| |Real.artanh (spinMean beta D v y) - y v| ≤ (d : ℝ) * beta := by | |
| exact finite_tree_gibbs_field_uniform_bound D hbeta (D.restrict y) v | |
| theorem field_abs_le_noise {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) | |
| (D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) | |
| (p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) : | |
| |Real.artanh (spinMean beta D v (observation t p))| ≤ | |
| (d : ℝ) * beta + (t : ℝ) + |p.2 v| := by | |
| have ho : |observation t p v| ≤ (t : ℝ) + |p.2 v| := by | |
| calc | |
| _ ≤ |(t : ℝ) * spinSign (p.1 v)| + |p.2 v| := abs_add_le _ _ | |
| _ = _ := by | |
| cases hs : p.1 v <;> simp [spinSign, abs_of_nonneg t.coe_nonneg] | |
| calc | |
| _ ≤ |Real.artanh (spinMean beta D v (observation t p)) - observation t p v| + | |
| |observation t p v| := by | |
| simpa using abs_sub_le (Real.artanh (spinMean beta D v (observation t p))) | |
| (observation t p v) 0 | |
| _ ≤ (d : ℝ) * beta + ((t : ℝ) + |p.2 v|) := | |
| add_le_add (field_sub_observation_abs_le hbeta D v _) ho | |
| _ = _ := by ring | |
| private theorem measurable_field {d : ℕ} (beta : ℝ) | |
| (D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) : | |
| Measurable (fun p => Real.artanh (spinMean beta D v (observation t p))) := by | |
| have ha : Measurable Real.artanh := by unfold Real.artanh; fun_prop | |
| exact ha.comp ((continuous_spinMean beta D v).measurable.comp (measurable_observation t)) | |
| theorem integrable_exp_abs_field {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) | |
| (μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ] | |
| (D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) {a : ℝ} (ha : 0 ≤ a) : | |
| Integrable (fun p => Real.exp (a * |Real.artanh | |
| (spinMean beta D v (observation t p))|)) | |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := by | |
| exact integrable_exp_abs_of_gaussian_bound (measurable_field beta D v t) | |
| (PlantedInformation.noise_hasLaw μ t v) ha | |
| (Filter.Eventually.of_forall (field_abs_le_noise hbeta D v t)) | |
| theorem integral_exp_abs_field_le {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) | |
| (μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ] | |
| (D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) {a : ℝ} (ha : 0 ≤ a) : | |
| (∫ p, Real.exp (a * |Real.artanh (spinMean beta D v (observation t p))|) | |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ | |
| 2 * Real.exp (a * ((d : ℝ) * beta + t) + (t : ℝ) * a ^ 2 / 2) := by | |
| exact integral_exp_abs_le_of_gaussian_bound (measurable_field beta D v t) | |
| (PlantedInformation.noise_hasLaw μ t v) ha | |
| (Filter.Eventually.of_forall (field_abs_le_noise hbeta D v t)) | |
| theorem inverse_variance_pow_le_noise {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) | |
| (D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) (k : ℕ) | |
| (p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ)) : | |
| ((1 - (spinMean beta D v (observation t p)) ^ 2)⁻¹) ^ k ≤ | |
| Real.exp ((2 * (k : ℝ)) * ((d : ℝ) * beta + (t : ℝ) + |p.2 v|)) := by | |
| apply (inverse_spin_variance_pow_le_exp (spinMean_mem_Ioo beta D v _) k).trans | |
| exact Real.exp_le_exp.mpr | |
| (mul_le_mul_of_nonneg_left (field_abs_le_noise hbeta D v t p) (by positivity)) | |
| theorem integrable_inverse_variance_pow {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) | |
| (μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ] | |
| (D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) (k : ℕ) : | |
| Integrable (fun p => ((1 - (spinMean beta D v (observation t p)) ^ 2)⁻¹) ^ k) | |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := by | |
| apply (integrable_exp_abs_field hbeta μ D v t | |
| (a := 2 * (k : ℝ)) (by positivity)).mono' | |
| · exact ((measurable_const.sub (((continuous_spinMean beta D v).measurable.comp | |
| (measurable_observation t)).pow_const 2)).inv.pow_const k).aestronglyMeasurable | |
| · apply Filter.Eventually.of_forall | |
| intro p | |
| have hm := spinMean_mem_Ioo beta D v (observation t p) | |
| have hpos : 0 < 1 - (spinMean beta D v (observation t p)) ^ 2 := by | |
| nlinarith [mul_pos (sub_pos.mpr hm.2) (show 0 < 1 + spinMean beta D v | |
| (observation t p) by linarith [hm.1])] | |
| rw [Real.norm_of_nonneg (pow_nonneg (inv_nonneg.mpr hpos.le) _)] | |
| exact inverse_spin_variance_pow_le_exp hm k | |
| theorem integral_inverse_variance_pow_le {d : ℕ} {beta : ℝ} (hbeta : 0 ≤ beta) | |
| (μ : Measure (TreeVertex d → IsingSpin)) [IsProbabilityMeasure μ] | |
| (D : Finset (TreeVertex d)) (v : D) (t : ℝ≥0) (k : ℕ) : | |
| (∫ p, ((1 - (spinMean beta D v (observation t p)) ^ 2)⁻¹) ^ k | |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t)) ≤ | |
| 2 * Real.exp ((2 * (k : ℝ)) * ((d : ℝ) * beta + t) + | |
| (t : ℝ) * (2 * (k : ℝ)) ^ 2 / 2) := by | |
| calc | |
| _ ≤ ∫ p, Real.exp ((2 * (k : ℝ)) * |Real.artanh | |
| (spinMean beta D v (observation t p))|) | |
| ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t) := by | |
| apply integral_mono (integrable_inverse_variance_pow hbeta μ D v t k) | |
| (integrable_exp_abs_field hbeta μ D v t (by positivity)) | |
| intro p | |
| exact inverse_spin_variance_pow_le_exp (spinMean_mem_Ioo beta D v _) k | |
| _ ≤ _ := integral_exp_abs_field_le hbeta μ D v t (by positivity) | |
| theorem condExp_eq_spinMean {d : ℕ} {beta : ℝ} | |
| {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) | |
| (D : Finset (TreeVertex d)) (hD : ConnectedFinset D) (v : D) (t : ℝ≥0) : | |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t))[ | |
| fun p => spinSign (p.1 v) | MeasurableSpace.comap (finiteObservation D t) inferInstance] | |
| =ᵐ[μ.prod (gaussianNoiseLaw (TreeVertex d) t)] | |
| fun p => spinMean beta D v (observation t p) := | |
| hμ.finiteObservation_condExp D hD t (fun eta => spinSign (eta v)) | |
| end Problem367.FinitePosteriorVariance | |
| end | |
| end OAI | |