| import OAI.Probability.FreeIsing.Ising.CopyCavityBounds |
| import OAI.Probability.FreeIsing.Ising.ObservationComparison |
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| namespace OAI |
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| noncomputable section |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal ENNReal |
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| namespace Problem367.IsingCopyPathBounds |
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| open IsingObservationInnovation IsingObservationFactor IsingCopyStationarity |
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| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
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| def edgeLogDifference (hbeta : 0 ≤ beta) (t : ℝ≥0) (v w : TreeVertex d) |
| (q : IsingConditionalCopyBounds.CopySpace d) : ℝ := |
| Real.log (responseAttenuation (Real.tanh beta) |
| (CavityArray.field (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta) |
| (sitePair beta t q).1 v w)) - |
| Real.log (responseAttenuation (Real.tanh beta) |
| (CavityArray.field (tanh_nonneg_of_beta_nonneg hbeta) (Real.tanh_lt_one beta) |
| (sitePair beta t q).2 v w)) |
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| def edgeLogBound (d : ℕ) (beta : ℝ) (T : ℝ≥0) : ℝ := |
| 2 * ((2 * (T : ℝ) + 2 * (d : ℝ) * beta) / (1 - Real.tanh beta)) |
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| theorem edgeLogBound_nonneg (hbeta : 0 ≤ beta) (T : ℝ≥0) : |
| 0 ≤ edgeLogBound d beta T := by |
| have hden : 0 < 1 - Real.tanh beta := sub_pos.mpr (Real.tanh_lt_one beta) |
| unfold edgeLogBound |
| positivity |
|
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| theorem measurable_edgeLogDifference (hbeta : 0 ≤ beta) (t : ℝ≥0) |
| (v w : TreeVertex d) : Measurable (edgeLogDifference hbeta t v w) := by |
| have h0 := tanh_nonneg_of_beta_nonneg hbeta |
| have h1 := Real.tanh_lt_one beta |
| have hs : Real.tanh beta ^ 2 ≤ 1 := by nlinarith |
| have hl := (lipschitzWith_log_responseAttenuation hs).continuous.measurable |
| have hc := CavityArray.measurable_field_apply h0 h1 v w |
| exact (hl.comp (hc.comp (measurable_fst.comp (measurable_sitePair beta t)))).sub |
| (hl.comp (hc.comp (measurable_snd.comp (measurable_sitePair beta t)))) |
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|
| theorem ae_all_siteDifference_bound |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| {t T : ℝ≥0} (ht : t ≤ T) : |
| ∀ᵐ q ∂copies beta μ, ∀ v : TreeVertex d, |
| |(sitePair beta t q).1 v - (sitePair beta t q).2 v| ≤ |
| 2 * (T : ℝ) + 2 * (d : ℝ) * beta := by |
| apply ae_all_iff.mpr |
| intro v |
| have : IsProbabilityMeasure ((observationLaw μ).map (innovationMap beta)) := |
| inferInstance |
| have h := IsingConditionalCopyBounds.ae_fieldDifference_bound hμ hbeta |
| kernel_graphLaw t v |
| filter_upwards [h] with q hq |
| have ht' : (t : ℝ) ≤ (T : ℝ) := ht |
| exact hq.trans (by linarith) |
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| theorem ae_edgeLogDifference_bound |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| {t T : ℝ≥0} (ht : t ≤ T) (v w : TreeVertex d) : |
| ∀ᵐ q ∂copies beta μ, |
| |edgeLogDifference hbeta t v w q| ≤ edgeLogBound d beta T := by |
| have h0 := tanh_nonneg_of_beta_nonneg hbeta |
| have h1 := Real.tanh_lt_one beta |
| have hs : Real.tanh beta ^ 2 ≤ 1 := by nlinarith |
| filter_upwards [ae_all_siteDifference_bound hμ hbeta ht] with q hq |
| have hc := CavityArray.field_sub_abs_le h0 h1 v w (hq v) (hq w) |
| have hl := (lipschitzWith_log_responseAttenuation hs).dist_le_mul |
| (CavityArray.field h0 h1 (sitePair beta t q).1 v w) |
| (CavityArray.field h0 h1 (sitePair beta t q).2 v w) |
| apply (show |edgeLogDifference hbeta t v w q| ≤ |
| 2 * |CavityArray.field h0 h1 (sitePair beta t q).1 v w - |
| CavityArray.field h0 h1 (sitePair beta t q).2 v w| from by |
| simpa only [edgeLogDifference, Real.dist_eq, NNReal.coe_ofNat] using hl).trans |
| exact mul_le_mul_of_nonneg_left hc (by norm_num) |
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| def pathLogDifference {ι : Type*} (hbeta : 0 ≤ beta) (t : ℝ≥0) |
| (s : Finset ι) (e : ι → TreeVertex d × TreeVertex d) |
| (q : IsingConditionalCopyBounds.CopySpace d) : ℝ := |
| ∑ i ∈ s, edgeLogDifference hbeta t (e i).1 (e i).2 q |
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| theorem measurable_pathLogDifference {ι : Type*} |
| (hbeta : 0 ≤ beta) (t : ℝ≥0) (s : Finset ι) |
| (e : ι → TreeVertex d × TreeVertex d) : |
| Measurable (pathLogDifference hbeta t s e) := |
| Finset.measurable_fun_sum s (fun i _ => |
| measurable_edgeLogDifference hbeta t (e i).1 (e i).2) |
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| theorem ae_pathLogDifference_bound {ι : Type*} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| {t T : ℝ≥0} (ht : t ≤ T) (s : Finset ι) |
| (e : ι → TreeVertex d × TreeVertex d) : |
| ∀ᵐ q ∂copies beta μ, |
| |pathLogDifference hbeta t s e q| ≤ (s.card : ℝ) * edgeLogBound d beta T := by |
| classical |
| have hall : ∀ᵐ q ∂copies beta μ, ∀ i ∈ s, |
| |edgeLogDifference hbeta t (e i).1 (e i).2 q| ≤ edgeLogBound d beta T := |
| (ae_ball_iff s.finite_toSet.countable).mpr (fun i _ => |
| ae_edgeLogDifference_bound hμ hbeta ht (e i).1 (e i).2) |
| filter_upwards [hall] with q hq |
| calc |
| |pathLogDifference hbeta t s e q| ≤ |
| ∑ i ∈ s, |edgeLogDifference hbeta t (e i).1 (e i).2 q| := |
| Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑ _i ∈ s, edgeLogBound d beta T := Finset.sum_le_sum hq |
| _ = (s.card : ℝ) * edgeLogBound d beta T := by simp |
|
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| theorem ae_pathLogDifference_sq_bound {ι : Type*} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| {t T : ℝ≥0} (ht : t ≤ T) (s : Finset ι) |
| (e : ι → TreeVertex d × TreeVertex d) : |
| ∀ᵐ q ∂copies beta μ, |
| pathLogDifference hbeta t s e q ^ 2 ≤ |
| (s.card : ℝ) ^ 2 * edgeLogBound d beta T ^ 2 := by |
| filter_upwards [ae_pathLogDifference_bound hμ hbeta ht s e] with q hq |
| have hb : 0 ≤ (s.card : ℝ) * edgeLogBound d beta T := |
| mul_nonneg (Nat.cast_nonneg _) (edgeLogBound_nonneg hbeta T) |
| simpa only [sq_abs, mul_pow] using (sq_le_sq₀ (abs_nonneg _) hb).mpr hq |
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| theorem memLp_pathLogDifference {ι : Type*} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (s : Finset ι) (e : ι → TreeVertex d × TreeVertex d) |
| (p : ℝ≥0∞) : MemLp (pathLogDifference hbeta t s e) p (copies beta μ) := by |
| apply MemLp.of_bound (measurable_pathLogDifference hbeta t s e).aestronglyMeasurable |
| ((s.card : ℝ) * edgeLogBound d beta t) |
| simpa only [Real.norm_eq_abs] using ae_pathLogDifference_bound hμ hbeta le_rfl s e |
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| theorem integrable_pathLogDifference_sq {ι : Type*} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hbeta : 0 ≤ beta) |
| (t : ℝ≥0) (s : Finset ι) (e : ι → TreeVertex d × TreeVertex d) : |
| Integrable (fun q => pathLogDifference hbeta t s e q ^ 2) (copies beta μ) := |
| (memLp_pathLogDifference hμ hbeta t s e 2).integrable_sq |
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| end Problem367.IsingCopyPathBounds |
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| end |
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| end OAI |
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