| import OAI.Probability.FreeIsing.Ising.CanonicalContinuousFields |
| import OAI.Probability.FreeIsing.Ising.CopyResponseRowPaths |
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|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal Topology BigOperators |
|
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| namespace Problem367.IsingObservationFactor |
| open IsingObservationInnovation |
|
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| variable {d : ℕ} {beta : ℝ} |
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| private lemma continuous_tanh : Continuous Real.tanh := |
| continuous_iff_continuousAt.mpr (fun z => (hasDerivAt_tanh_response z).continuousAt) |
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|
| def siteRootResponse (hb : 0 ≤ beta) (v : TreeVertex d) |
| (H : TreeVertex d → ℝ) : ℝ := |
| Real.tanh beta ^ v.1.length * |
| ∏ i ∈ Finset.range v.1.length, responseAttenuation (Real.tanh beta) |
| (CavityArray.field (tanh_nonneg_of_beta_nonneg hb) (Real.tanh_lt_one beta) |
| H (ConnectedHull.takeVertex v (i + 1)) (ConnectedHull.takeVertex v i)) |
|
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| lemma continuous_siteRootResponse (hb : 0 ≤ beta) (v : TreeVertex d) : |
| Continuous (siteRootResponse hb v) := by |
| unfold siteRootResponse |
| apply Continuous.const_mul |
| apply continuous_finsetProd |
| intro i hi |
| exact (continuous_responseAttenuation (Real.tanh_sq_lt_one beta).le).comp |
| (continuous_fst.comp ((continuous_cavityFromSites |
| (tanh_nonneg_of_beta_nonneg hb) (Real.tanh_lt_one beta)).comp |
| ((continuous_apply _).prodMk (continuous_apply _)))) |
|
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| lemma siteRootResponse_mem_Icc (hb : 0 ≤ beta) (v : TreeVertex d) |
| (H : TreeVertex d → ℝ) : |
| 0 ≤ siteRootResponse hb v H ∧ siteRootResponse hb v H ≤ 1 := by |
| have hp0 : 0 ≤ ∏ i ∈ Finset.range v.1.length, responseAttenuation (Real.tanh beta) |
| (CavityArray.field (tanh_nonneg_of_beta_nonneg hb) (Real.tanh_lt_one beta) |
| H (ConnectedHull.takeVertex v (i + 1)) (ConnectedHull.takeVertex v i)) := |
| Finset.prod_nonneg fun i _ => (responseAttenuation_pos (Real.tanh_sq_lt_one beta).le _).le |
| have hp1 : (∏ i ∈ Finset.range v.1.length, responseAttenuation (Real.tanh beta) |
| (CavityArray.field (tanh_nonneg_of_beta_nonneg hb) (Real.tanh_lt_one beta) |
| H (ConnectedHull.takeVertex v (i + 1)) (ConnectedHull.takeVertex v i))) ≤ 1 := |
| Finset.prod_le_one₀ |
| (fun i _ => (responseAttenuation_pos (Real.tanh_sq_lt_one beta).le _).le) |
| (fun i _ => responseAttenuation_le_one (Real.tanh_sq_lt_one beta).le _) |
| have ht0 := pow_nonneg (tanh_nonneg_of_beta_nonneg hb) v.1.length |
| have ht1 : Real.tanh beta ^ v.1.length ≤ 1 := |
| pow_le_one₀ (tanh_nonneg_of_beta_nonneg hb) (Real.tanh_lt_one beta).le |
| exact ⟨mul_nonneg ht0 hp0, (mul_le_mul ht1 hp1 hp0 (by norm_num)).trans_eq (one_mul 1)⟩ |
|
|
| def continuousRootResponse (hb : 0 ≤ beta) (v : TreeVertex d) |
| (t : ℝ≥0) (x : Paths d) : ℝ := |
| siteRootResponse hb v (fun i => continuousField beta i t x) |
|
|
| lemma continuousRootResponse_progressive (hb : 0 ≤ beta) (v : TreeVertex d) : |
| IsStronglyProgressive (filtration d) (continuousRootResponse hb v) := by |
| intro t |
| let : MeasurableSpace (Paths d) := filtration d t |
| exact ((continuous_siteRootResponse hb v).measurable.comp |
| (Measurable.of_eval (fun i => (continuousField_progressive beta i t).measurable))).stronglyMeasurable |
|
|
| lemma continuousRootResponse_abs_le_one (hb : 0 ≤ beta) (v : TreeVertex d) |
| (t : ℝ≥0) (x : Paths d) : |continuousRootResponse hb v t x| ≤ 1 := by |
| obtain ⟨h0, h1⟩ := siteRootResponse_mem_Icc hb v (fun i => continuousField beta i t x) |
| exact (abs_of_nonneg h0).trans_le h1 |
|
|
| theorem continuousRootResponse_continuous_ae |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) : |
| ∀ᵐ x ∂observationLaw μ, Continuous (fun t => continuousRootResponse hb v t x) := by |
| have hc (i : TreeVertex d) := (continuousField_modification hd hμ hb hcrit i).1 |
| filter_upwards [ae_all_iff.mpr hc] with x hx |
| exact (continuous_siteRootResponse hb v).comp (continuous_pi hx) |
|
|
| theorem continuousRootResponse_eq_response_ae |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| (v : TreeVertex d) (t : ℝ≥0) : |
| continuousRootResponse hb v t =ᵐ[observationLaw μ] |
| fun x => infiniteGibbsResponse beta (ConnectedHull.root d) v (fun i => x i t) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have he (i : TreeVertex d) := (continuousField_modification hd hμ hb hcrit i).2 t |
| filter_upwards [ae_all_iff.mpr he, |
| IsingCopyResponseRowPaths.row_eq_product_ae hμ hb t v] with x hx hr |
| change siteRootResponse hb v (fun i => continuousField beta i t x) = _ |
| rw [show (fun i => continuousField beta i t x) = (fun i => field beta i t x) |
| from funext hx] |
| exact hr.symm |
|
|
| def continuousRootCovariance (hb : 0 ≤ beta) (v : TreeVertex d) |
| (t : ℝ≥0) (x : Paths d) : ℝ := |
| (1 - Real.tanh (continuousField beta (ConnectedHull.root d) t x) ^ 2) * |
| continuousRootResponse hb v t x |
|
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| lemma continuousRootCovariance_progressive (hb : 0 ≤ beta) (v : TreeVertex d) : |
| IsStronglyProgressive (filtration d) (continuousRootCovariance hb v) := by |
| intro t |
| have hf := continuousField_progressive beta (ConnectedHull.root d) t |
| exact (stronglyMeasurable_const.sub |
| ((continuous_tanh.comp_stronglyMeasurable hf).pow 2)).mul |
| (continuousRootResponse_progressive hb v t) |
|
|
| lemma continuousRootCovariance_abs_le_one (hb : 0 ≤ beta) (v : TreeVertex d) |
| (t : ℝ≥0) (x : Paths d) : |continuousRootCovariance hb v t x| ≤ 1 := by |
| have h0 := (sub_pos.mpr (Real.tanh_sq_lt_one |
| (continuousField beta (ConnectedHull.root d) t x))).le |
| have h1 : 1 - Real.tanh (continuousField beta (ConnectedHull.root d) t x) ^ 2 ≤ 1 := |
| sub_le_self _ (sq_nonneg _) |
| rw [continuousRootCovariance, abs_mul, abs_of_nonneg h0] |
| exact (mul_le_mul h1 (continuousRootResponse_abs_le_one hb v t x) |
| (abs_nonneg _) (by norm_num)).trans_eq (one_mul 1) |
|
|
| theorem continuousRootCovariance_continuous_ae |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (v : TreeVertex d) : |
| ∀ᵐ x ∂observationLaw μ, Continuous (fun t => continuousRootCovariance hb v t x) := by |
| filter_upwards [(continuousField_modification hd hμ hb hcrit (ConnectedHull.root d)).1, |
| continuousRootResponse_continuous_ae hd hμ hb hcrit v] with x hf hr |
| exact (continuous_const.sub ((continuous_tanh.comp hf).pow 2)).mul hr |
|
|
| theorem continuousRootCovariance_eq_covariance_ae |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| (v : TreeVertex d) (t : ℝ≥0) : |
| continuousRootCovariance hb v t =ᵐ[observationLaw μ] |
| fun x => infiniteGibbsCovariance beta (ConnectedHull.root d) v (fun i => x i t) := by |
| filter_upwards [(continuousField_modification hd hμ hb hcrit (ConnectedHull.root d)).2 t, |
| continuousRootResponse_eq_response_ae hd hμ hb hcrit v t, |
| tanh_field_eq_posterior_ae hμ hb (ConnectedHull.root d) t, |
| posterior_mem_Ioo_ae hμ hb (ConnectedHull.root d) t] with x hf hr hm hi |
| have hpos : 0 < 1 - posterior beta (ConnectedHull.root d) t x ^ 2 := by |
| nlinarith [hi.1, hi.2] |
| rw [continuousRootCovariance, hf, hm, hr, infiniteGibbsResponse] |
| change (1 - posterior beta (ConnectedHull.root d) t x ^ 2) * |
| (_ / (1 - posterior beta (ConnectedHull.root d) t x ^ 2)) = _ |
| exact mul_div_cancel₀ _ hpos.ne' |
|
|
| end Problem367.IsingObservationFactor |
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| end |
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| end OAI |
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