| import OAI.Probability.FreeIsing.Estimates.MessageFunction |
| import OAI.Probability.FreeIsing.Ising.ConditionalCopyBounds |
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| namespace OAI |
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| noncomputable section |
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|
| open MeasureTheory ProbabilityTheory Set |
| open scoped NNReal |
|
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| namespace Problem367 |
|
|
| theorem abs_artanh_le_two_mul_of_abs_le_half {x : ℝ} (hx : |x| ≤ 1 / 2) : |
| |Real.artanh x| ≤ 2 * |x| := by |
| have hd (z : ℝ) (hz : z ∈ Icc (-1 / 2 : ℝ) (1 / 2)) : |
| HasDerivWithinAt Real.artanh (1 / (1 - z ^ 2)) (Icc (-1 / 2) (1 / 2)) z := |
| (hasDerivAt_artanh_interval (by constructor <;> linarith [hz.1, hz.2])).hasDerivWithinAt |
| have hb (z : ℝ) (hz : z ∈ Icc (-1 / 2 : ℝ) (1 / 2)) : |
| ‖1 / (1 - z ^ 2)‖ ≤ (2 : ℝ) := by |
| have hsq : z ^ 2 ≤ 1 / 4 := by |
| nlinarith [mul_nonneg (by linarith [hz.1] : 0 ≤ z + 1 / 2) |
| (by linarith [hz.2] : 0 ≤ 1 / 2 - z)] |
| have hden : 0 < 1 - z ^ 2 := by linarith |
| rw [Real.norm_eq_abs, abs_of_nonneg (by positivity : 0 ≤ 1 / (1 - z ^ 2))] |
| exact (div_le_iff₀ hden).mpr (by linarith) |
| have h := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le hd hb (convex_Icc _ _) |
| (by norm_num : (0 : ℝ) ∈ Icc (-1 / 2 : ℝ) (1 / 2)) |
| (show x ∈ Icc (-1 / 2 : ℝ) (1 / 2) by |
| constructor <;> linarith [(abs_le.mp hx).1, (abs_le.mp hx).2]) |
| simpa only [Real.artanh_zero, sub_zero, Real.norm_eq_abs] using h |
|
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| theorem artanh_sub_sq_le_information {x y K : ℝ} |
| (hK : |Real.artanh x - Real.artanh y| ≤ K) : |
| (Real.artanh x - Real.artanh y) ^ 2 ≤ (8 + 4 * K ^ 2) * (x ^ 2 + y ^ 2) := by |
| have hK0 : 0 ≤ K := (abs_nonneg _).trans hK |
| have hdiff : (Real.artanh x - Real.artanh y) ^ 2 ≤ K ^ 2 := by |
| simpa only [sq_abs] using (pow_le_pow_left₀ (abs_nonneg _) hK 2) |
| by_cases hx : |x| ≤ 1 / 2 |
| · by_cases hy : |y| ≤ 1 / 2 |
| · have hax : (Real.artanh x) ^ 2 ≤ 4 * x ^ 2 := by |
| have h := mul_self_le_mul_self (abs_nonneg _) (abs_artanh_le_two_mul_of_abs_le_half hx) |
| nlinarith only [h, sq_abs (Real.artanh x), sq_abs x] |
| have hay : (Real.artanh y) ^ 2 ≤ 4 * y ^ 2 := by |
| have h := mul_self_le_mul_self (abs_nonneg _) (abs_artanh_le_two_mul_of_abs_le_half hy) |
| nlinarith only [h, sq_abs (Real.artanh y), sq_abs y] |
| have hsum : 0 ≤ 4 * K ^ 2 * (x ^ 2 + y ^ 2) := by positivity |
| nlinarith [sq_nonneg (Real.artanh x + Real.artanh y)] |
| · have hy' : 1 / 2 ≤ |y| := (lt_of_not_ge hy).le |
| have hy2 := mul_self_le_mul_self (by norm_num : (0 : ℝ) ≤ 1 / 2) hy' |
| have hmul := mul_le_mul_of_nonneg_left hy2 (sq_nonneg K) |
| nlinarith [sq_abs y, sq_nonneg x, sq_nonneg y, mul_nonneg (sq_nonneg K) (sq_nonneg x)] |
| · have hx' : 1 / 2 ≤ |x| := (lt_of_not_ge hx).le |
| have hx2 := mul_self_le_mul_self (by norm_num : (0 : ℝ) ≤ 1 / 2) hx' |
| have hmul := mul_le_mul_of_nonneg_left hx2 (sq_nonneg K) |
| nlinarith [sq_abs x, sq_nonneg x, sq_nonneg y, mul_nonneg (sq_nonneg K) (sq_nonneg y)] |
|
|
| namespace IsingConditionalCopyBounds |
|
|
| open IsingObservationInnovation |
|
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| def posteriorInformation {d : ℕ} (beta : ℝ) (μ : Measure (TreeVertex d → IsingSpin)) |
| (v : TreeVertex d) (t : ℝ≥0) : ℝ := |
| ∫ x, (IsingObservationInnovation.posterior beta v t x) ^ 2 ∂observationLaw μ |
|
|
| theorem posteriorInformation_eq_gaussian |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (t : ℝ≥0) (v : TreeVertex d) : |
| posteriorInformation beta μ v t = |
| ∫ p, (PlantedSymmetry.posteriorMagnetization (μ.prod (PlantedSymmetry.gaussianNoiseLaw |
| (TreeVertex d) t)) t v p) ^ 2 ∂μ.prod |
| (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have heval := hasLaw_observationLaw_eval μ t |
| have hobs : HasLaw (PlantedSymmetry.observation t) |
| ((μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)).map |
| (PlantedSymmetry.observation t)) |
| (μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t)) := |
| ⟨(PlantedSymmetry.measurable_observation t).aemeasurable, rfl⟩ |
| have hsame := (heval.identDistrib hobs).comp |
| ((measurable_infiniteGibbsSpinMean beta v).pow_const 2) |
| calc |
| _ = ∫ p, (infiniteGibbsSpinMean beta v (PlantedSymmetry.observation t p)) ^ 2 |
| ∂μ.prod (PlantedSymmetry.gaussianNoiseLaw (TreeVertex d) t) := hsame.integral_eq |
| _ = _ := integral_congr_ae |
| ((hμ.infiniteGibbsSpinMean_eq_posterior t v).fun_comp (fun z : ℝ => z ^ 2)) |
|
|
| theorem fieldDifference_secondMoment_le_information |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| {rho : Measure (Paths d)} [IsProbabilityMeasure rho] |
| {eta : Kernel (Paths d) (Paths d)} [IsMarkovKernel eta] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (hjoint : rho ⊗ₘ eta = graphLaw beta μ) (t : ℝ≥0) (v : TreeVertex d) : |
| (∫ q, (fieldDifference beta v t q) ^ 2 ∂rho ⊗ₘ (eta ×ₖ eta)) ≤ |
| 2 * (8 + 4 * (2 * (t : ℝ) + 2 * (d : ℝ) * beta) ^ 2) * |
| posteriorInformation beta μ v t := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| let : IsProbabilityMeasure (labelMeasure d) := by unfold labelMeasure; infer_instance |
| let : IsProbabilityMeasure (observationLaw μ) := by |
| unfold observationLaw |
| infer_instance |
| let m : Paths d → ℝ := IsingObservationInnovation.posterior beta v t |
| let C : ℝ := 8 + 4 * (2 * (t : ℝ) + 2 * (d : ℝ) * beta) ^ 2 |
| have hm : Measurable m := |
| (((posterior_progressive beta v).stronglyAdapted t).mono ((filtration d).le t)).measurable |
| have hmLp : MemLp m 2 (observationLaw μ) := |
| MemLp.of_bound hm.aestronglyMeasurable 1 |
| (Filter.Eventually.of_forall (fun x => posterior_norm_le_one beta v t x)) |
| obtain ⟨hl, hr⟩ := measurePreserving_observation_copies hjoint |
| have hml := hmLp.comp_measurePreserving hl |
| have hmr := hmLp.comp_measurePreserving hr |
| have hdiff := (fieldDifference_secondMoment_bounds hμ hbeta hjoint t v).1 |
| have hlint := hl.hasLaw.integral_comp (f := fun x => (m x) ^ 2) |
| (hm.pow_const 2).aestronglyMeasurable |
| have hrint := hr.hasLaw.integral_comp (f := fun x => (m x) ^ 2) |
| (hm.pow_const 2).aestronglyMeasurable |
| simp only [Function.comp_def] at hlint hrint |
| calc |
| (∫ q, (fieldDifference beta v t q) ^ 2 ∂rho ⊗ₘ (eta ×ₖ eta)) ≤ |
| ∫ q : CopySpace d, C * ((m q.2.1) ^ 2 + (m q.2.2) ^ 2) |
| ∂rho ⊗ₘ (eta ×ₖ eta) := by |
| apply integral_mono_ae hdiff.integrable_sq |
| ((hml.integrable_sq.add hmr.integrable_sq).const_mul C) |
| filter_upwards [ae_fieldDifference_bound hμ hbeta hjoint t v] with q hq |
| exact artanh_sub_sq_le_information hq |
| _ = C * ((∫ q : CopySpace d, (m q.2.1) ^ 2 ∂rho ⊗ₘ (eta ×ₖ eta)) + |
| ∫ q : CopySpace d, (m q.2.2) ^ 2 ∂rho ⊗ₘ (eta ×ₖ eta)) := by |
| rw [integral_const_mul] |
| congr 1 |
| exact integral_add hml.integrable_sq hmr.integrable_sq |
| _ = 2 * (8 + 4 * (2 * (t : ℝ) + 2 * (d : ℝ) * beta) ^ 2) * |
| posteriorInformation beta μ v t := by |
| rw [hlint, hrint] |
| change C * (posteriorInformation beta μ v t + posteriorInformation beta μ v t) = _ |
| dsimp [C] |
| ring |
|
|
| theorem fieldDifference_secondMoment_le_horizon_information |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| {rho : Measure (Paths d)} [IsProbabilityMeasure rho] |
| {eta : Kernel (Paths d) (Paths d)} [IsMarkovKernel eta] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (hjoint : rho ⊗ₘ eta = graphLaw beta μ) {T t : ℝ≥0} (ht : t ≤ T) |
| (v : TreeVertex d) : |
| (∫ q, (fieldDifference beta v t q) ^ 2 ∂rho ⊗ₘ (eta ×ₖ eta)) ≤ |
| 2 * (8 + 4 * (2 * (T : ℝ) + 2 * (d : ℝ) * beta) ^ 2) * |
| posteriorInformation beta μ v t := by |
| apply (fieldDifference_secondMoment_le_information hμ hbeta hjoint t v).trans |
| have hI : 0 ≤ posteriorInformation beta μ v t := integral_nonneg (fun _ => sq_nonneg _) |
| apply mul_le_mul_of_nonneg_right _ hI |
| have htR : (t : ℝ) ≤ (T : ℝ) := ht |
| have hK : 0 ≤ 2 * (t : ℝ) + 2 * (d : ℝ) * beta := by positivity |
| have hsq := pow_le_pow_left₀ hK |
| (show 2 * (t : ℝ) + 2 * (d : ℝ) * beta ≤ 2 * (T : ℝ) + 2 * (d : ℝ) * beta by linarith) 2 |
| linarith |
|
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| end IsingConditionalCopyBounds |
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| end Problem367 |
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| end |
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| end OAI |
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