| import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy |
| import OAI.Probability.FreeIsing.Energy.NormalizedInfiniteTimeEnergy |
| import OAI.Probability.FreeIsing.Ising.ObservationComparison |
| import OAI.Probability.FreeIsing.Ising.ObservationPosteriorTransport |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped NNReal ENNReal Topology |
|
|
| namespace Problem367.IsingCopyEnergy |
| open IsingObservationInnovation IsingObservationFactor |
|
|
| def responseRow {d : ℕ} (beta : ℝ) (v w : TreeVertex d) (p : ℝ × Paths d) : ℝ := |
| infiniteGibbsResponse beta v w (fun i => p.2 i p.1.toNNReal) |
|
|
| theorem measurable_responseRow {d : ℕ} (beta : ℝ) (v w : TreeVertex d) : |
| Measurable (responseRow beta v w) := by |
| apply (measurable_infiniteGibbsResponse beta v w).comp |
| apply Measurable.of_eval |
| intro i |
| exact continuous_eval.measurable.comp |
| (((measurable_pi_apply i).comp measurable_snd).prodMk |
| (measurable_real_toNNReal.comp measurable_fst)) |
|
|
| theorem observation_extended_energy_le |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (v : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) : |
| (∫⁻ p, ∑' w, ENNReal.ofReal (responseRow beta v w p ^ 2) |
| ∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ)) ≤ |
| ENNReal.ofReal (2 * ((d : ℝ) * beta + T + Real.sqrt T)) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| let : IsProbabilityMeasure (labelMeasure d) := by unfold labelMeasure; infer_instance |
| let Q := (volume.restrict (Ioc (0 : ℝ) T)).prod (μ.prod (labelMeasure d)) |
| let R := (volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ) |
| let f : ℝ × Paths d → ℝ≥0∞ := fun p => ∑' w, ENNReal.ofReal (responseRow beta v w p ^ 2) |
| have hf : Measurable f := Measurable.tsum (fun w => |
| ((measurable_responseRow beta v w).pow_const 2).ennreal_ofReal) |
| have hmp : MeasurePreserving (Prod.map id uniformObservation) Q R := |
| (MeasurePreserving.id _).prod (measurePreserving_uniformObservation μ) |
| have henergy := NormalizedInfiniteTimeEnergy.lintegral_infiniteRow_le hμ hb |
| (fun _ : TreeVertex d => uniform01) |
| (fun _ => WienerSampler.sample_isBrownianReal.toIsPreBrownianReal) |
| (fun _ t => (continuous_eval_const t).measurable.comp WienerSampler.measurable_sample) |
| (fun _ => WienerSampler.measurable_sample_eval) v hT |
| have heq : (fun p => f (Prod.map id uniformObservation p)) =ᵐ[Q] |
| (fun p => ∑' w, ENNReal.ofReal |
| (NormalizedInfiniteTimeEnergy.infiniteRow beta v |
| (fun _ t u => WienerSampler.sample u t) w p ^ 2)) := by |
| have hmem : ∀ᵐ p ∂Q, p.1 ∈ Ioc (0 : ℝ) T := |
| Measure.quasiMeasurePreserving_fst.ae (ae_restrict_mem measurableSet_Ioc) |
| filter_upwards [hmem] with p hp |
| apply tsum_congr |
| intro w |
| apply congrArg ENNReal.ofReal |
| apply congrArg (fun r : ℝ => r ^ 2) |
| change infiniteGibbsResponse beta v w (fun i => uniformObservation p.2 i p.1.toNNReal) = _ |
| apply congrArg (infiniteGibbsResponse beta v w) |
| funext i |
| change WienerSampler.sample (p.2.2 i) p.1.toNNReal + |
| (p.1.toNNReal : ℝ) * spinSign (p.2.1 i) = |
| p.1 * spinSign (p.2.1 i) + WienerSampler.sample (p.2.2 i) p.1.toNNReal |
| rw [Real.coe_toNNReal p.1 hp.1.le, add_comm] |
| calc |
| _ = ∫⁻ p, f (Prod.map id uniformObservation p) ∂Q := (hmp.lintegral_comp hf).symm |
| _ = _ := lintegral_congr_ae heq |
| _ ≤ _ := henergy |
|
|
| theorem observation_row_energy_integrable |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (v : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) : |
| (∀ᵐ p ∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ), |
| Summable (fun w => responseRow beta v w p ^ 2)) ∧ |
| Integrable (fun p => ∑' w, responseRow beta v w p ^ 2) |
| ((volume.restrict (Ioc 0 T)).prod (observationLaw μ)) ∧ |
| (∫ p, ∑' w, responseRow beta v w p ^ 2 |
| ∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ)) ≤ |
| 2 * ((d : ℝ) * beta + T + Real.sqrt T) := by |
| let P := (volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ) |
| have he := observation_extended_energy_le hμ hb v hT |
| rw [lintegral_tsum (fun w => |
| ((measurable_responseRow beta v w).pow_const 2).ennreal_ofReal.aemeasurable)] at he |
| have hm (w : TreeVertex d) : AEStronglyMeasurable (responseRow beta v w) P := |
| (measurable_responseRow beta v w).aestronglyMeasurable |
| exact ⟨ae_summable_sq_of_tsum_lintegral_sq_ne_top P hm |
| (ne_top_of_le_ne_top ENNReal.ofReal_ne_top he), |
| integral_tsum_sq_le_of_tsum_lintegral_sq_le P hm (by positivity) he⟩ |
|
|
| abbrev Joint (d : ℕ) := Paths d × (Paths d × Paths d) |
|
|
| def projection {d : ℕ} (j : Bool) (q : Joint d) : Paths d := |
| if j then q.2.1 else q.2.2 |
|
|
| private theorem projection_measurePreserving |
| {d : ℕ} (beta : ℝ) (μ : Measure (TreeVertex d → IsingSpin)) |
| [IsProbabilityMeasure μ] (j : Bool) : |
| MeasurePreserving (projection (d := d) j) (copies beta μ) (observationLaw μ) := by |
| obtain ⟨hl, hr⟩ := IsingConditionalCopyBounds.measurePreserving_observation_copies |
| (beta := beta) (μ := μ) |
| (rho := (observationLaw μ).map (innovationMap beta)) (eta := kernel beta μ) kernel_graphLaw |
| cases j |
| · exact hr |
| · exact hl |
|
|
| def timeProjection {d : ℕ} (j : Bool) (p : ℝ × Joint d) : ℝ × Paths d := |
| (p.1, projection j p.2) |
|
|
| private theorem timeProjection_measurePreserving |
| {d : ℕ} (beta : ℝ) (μ : Measure (TreeVertex d → IsingSpin)) |
| [IsProbabilityMeasure μ] (j : Bool) (T : ℝ) : |
| MeasurePreserving (timeProjection (d := d) j) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod (copies beta μ)) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) := |
| (MeasurePreserving.id _).prod (projection_measurePreserving beta μ j) |
|
|
| theorem copy_row_energy_integrable |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (v : TreeVertex d) (j : Bool) {T : ℝ} (hT : 0 ≤ T) : |
| (∀ᵐ p ∂(volume.restrict (Ioc 0 T)).prod (copies beta μ), |
| Summable (fun w => responseRow beta v w (timeProjection j p) ^ 2)) ∧ |
| Integrable (fun p => ∑' w, responseRow beta v w (timeProjection j p) ^ 2) |
| ((volume.restrict (Ioc 0 T)).prod (copies beta μ)) ∧ |
| (∫ p, ∑' w, responseRow beta v w (timeProjection j p) ^ 2 |
| ∂(volume.restrict (Ioc 0 T)).prod (copies beta μ)) ≤ |
| 2 * ((d : ℝ) * beta + T + Real.sqrt T) := by |
| have ho := observation_row_energy_integrable hμ hb v hT |
| have hp := timeProjection_measurePreserving beta μ j T |
| refine ⟨hp.quasiMeasurePreserving.ae ho.1, hp.integrable_comp_of_integrable ho.2.1, ?_⟩ |
| simpa only [Function.comp_def] using |
| (hp.hasLaw.integral_comp ho.2.1.aestronglyMeasurable).le.trans ho.2.2 |
|
|
| def density {d : ℕ} (beta : ℝ) (v : TreeVertex d) (t : ℝ) (q : Joint d) : ℝ := |
| DiscrepancyEnergy.density |
| (fun w q => responseRow beta v w (t, q.2.1)) |
| (fun w q => responseRow beta v w (t, q.2.2)) |
| (fun q => field beta v t.toNNReal q.2.1 - field beta v t.toNNReal q.2.2) |
| (fun q => posterior beta v t.toNNReal q.2.1) |
| (fun q => posterior beta v t.toNNReal q.2.2) q |
|
|
| theorem density_integrable |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (v : TreeVertex d) (T : ℝ≥0) : |
| Integrable (fun p : ℝ × Joint d => density beta v p.1 p.2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod (copies beta μ)) := by |
| have hl := copy_row_energy_integrable hμ hb v true T.coe_nonneg |
| have hr := copy_row_energy_integrable hμ hb v false T.coe_nonneg |
| have hmap (j : Bool) : Measurable (fun p : ℝ × Joint d => |
| (p.1.toNNReal, projection j p.2)) := |
| (measurable_real_toNNReal.comp measurable_fst).prodMk |
| ((projection_measurePreserving beta μ j).measurable.comp measurable_snd) |
| have hbound := (IsingConditionalCopyBounds.fieldDifference_joint_bounds |
| hμ hb kernel_graphLaw T v).1 |
| apply DiscrepancyEnergy.integrable_noise_add_drift |
| (c := fun w p => responseRow beta v w (timeProjection true p)) |
| (c' := fun w p => responseRow beta v w (timeProjection false p)) |
| (H := 2 * (T : ℝ) + 2 * (d : ℝ) * beta) |
| (fun w => ((measurable_responseRow beta v w).comp |
| (timeProjection_measurePreserving beta μ true T).measurable).aestronglyMeasurable) |
| (fun w => ((measurable_responseRow beta v w).comp |
| (timeProjection_measurePreserving beta μ false T).measurable).aestronglyMeasurable) |
| (((measurable_field_joint beta v).comp (hmap true)).sub |
| ((measurable_field_joint beta v).comp (hmap false))).aestronglyMeasurable |
| (((measurable_posterior_joint beta v).comp (hmap true)).aestronglyMeasurable) |
| (((measurable_posterior_joint beta v).comp (hmap false)).aestronglyMeasurable) |
| hl.2.1 hr.2.1 (hl.1.and hr.1) |
| filter_upwards [hbound] with p hp |
| exact ⟨hp, posterior_norm_le_one beta v p.1.toNNReal p.2.2.1, |
| posterior_norm_le_one beta v p.1.toNNReal p.2.2.2⟩ |
|
|
| theorem ae_integral_density_eq |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} |
| [IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (v : TreeVertex d) (T : ℝ≥0) : |
| ∀ᵐ t ∂volume.restrict (Icc (0 : ℝ) T), |
| (∫ q, density beta v t q ∂copies beta μ) = |
| (∑' w, ∫ q, (responseRow beta v w (t, q.2.1) - |
| responseRow beta v w (t, q.2.2)) ^ 2 ∂copies beta μ) + |
| 2 * ∫ q, (field beta v t.toNNReal q.2.1 - field beta v t.toNNReal q.2.2) * |
| (posterior beta v t.toNNReal q.2.1 * (∑' w, responseRow beta v w (t, q.2.1) ^ 2) - |
| posterior beta v t.toNNReal q.2.2 * (∑' w, responseRow beta v w (t, q.2.2) ^ 2)) |
| ∂copies beta μ := by |
| have hl := copy_row_energy_integrable hμ hb v true T.coe_nonneg |
| have hr := copy_row_energy_integrable hμ hb v false T.coe_nonneg |
| rw [← restrict_Ioc_eq_restrict_Icc] |
| filter_upwards [hl.2.1.prod_right_ae, hr.2.1.prod_right_ae, |
| Measure.ae_ae_of_ae_prod hl.1, Measure.ae_ae_of_ae_prod hr.1] with t hAl hAr hsl hsr |
| have hmap (j : Bool) : Measurable (fun q : Joint d => (t.toNNReal, projection j q)) := |
| measurable_const.prodMk (projection_measurePreserving beta μ j).measurable |
| have hrow (j : Bool) (w : TreeVertex d) : |
| Measurable (fun q : Joint d => responseRow beta v w (t, projection j q)) := |
| (measurable_responseRow beta v w).comp |
| (measurable_const.prodMk (projection_measurePreserving beta μ j).measurable) |
| apply DiscrepancyEnergy.integral_noise_add_drift |
| (H := 2 * (t.toNNReal : ℝ) + 2 * (d : ℝ) * beta) |
| (fun w => (hrow true w).aestronglyMeasurable) |
| (fun w => (hrow false w).aestronglyMeasurable) |
| (((measurable_field_joint beta v).comp (hmap true)).sub |
| ((measurable_field_joint beta v).comp (hmap false))).aestronglyMeasurable |
| (((measurable_posterior_joint beta v).comp (hmap true)).aestronglyMeasurable) |
| (((measurable_posterior_joint beta v).comp (hmap false)).aestronglyMeasurable) |
| hAl hAr (hsl.and hsr) |
| filter_upwards [IsingConditionalCopyBounds.ae_fieldDifference_bound |
| hμ hb kernel_graphLaw t.toNNReal v] with q hq |
| exact ⟨hq, posterior_norm_le_one beta v t.toNNReal q.2.1, |
| posterior_norm_le_one beta v t.toNNReal q.2.2⟩ |
|
|
| end Problem367.IsingCopyEnergy |
|
|
| end |
|
|
| end OAI |
|
|
