openAI-math-dataset / lean /OAI /Probability /FreeIsing /Stochastic /PlantedBrownianLinearCombination.lean
| import OAI.Probability.FreeIsing.Brownian.ArrayMoments | |
| import OAI.Probability.FreeIsing.Brownian.LinearCombination | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped NNReal | |
| namespace Problem367 | |
| lemma iIndepFun_comp_measurePreserving | |
| {ι Ω Ω' E : Type*} [MeasurableSpace Ω] [MeasurableSpace Ω'] [MeasurableSpace E] | |
| {μ : Measure Ω} {ν : Measure Ω'} [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] | |
| {r : Ω → Ω'} (hr : MeasurePreserving r μ ν) | |
| {f : ι → Ω' → E} (hf : ∀ i, Measurable (f i)) (hind : iIndepFun f ν) : | |
| iIndepFun (fun i => f i ∘ r) μ := by | |
| apply (iIndepFun_iff_map_fun_eq_infinitePi_map | |
| (fun i => (hf i).comp hr.measurable)).2 | |
| have h := hind.map_fun_eq_infinitePi_map hf | |
| rw [← hr.map_eq] at h | |
| simpa only [Measure.map_map (Measurable.of_eval hf) hr.measurable, | |
| Measure.map_map (hf _) hr.measurable, Function.comp_def] using h | |
| theorem planted_coordinate_paths_iIndepFun | |
| {ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω] | |
| (ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν] | |
| [∀ i, IsProbabilityMeasure (μ i)] | |
| {B : ι → ℝ≥0 → Ω → ℝ} (hmB : ∀ i t, Measurable (B i t)) : | |
| iIndepFun (fun i (q : S × (ι → Ω)) t => B i t (q.2 i)) | |
| (ν.prod (Measure.infinitePi μ)) := by | |
| have hm (i : ι) : Measurable (fun ω t => B i t ω) := | |
| Measurable.of_eval (hmB i) | |
| exact iIndepFun_comp_measurePreserving measurePreserving_snd | |
| (fun i => (hm i).comp (measurable_pi_apply i)) (iIndepFun_infinitePi hm) | |
| theorem planted_finite_projection_preBrownian | |
| {ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω] | |
| (ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν] | |
| {B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ i, IsPreBrownianReal (B i) (μ i)) | |
| (hmB : ∀ i t, Measurable (B i t)) (I : Finset ι) (a : ι → ℝ) | |
| (ha : ∑ i ∈ I, a i ^ 2 = 1) : | |
| IsPreBrownianReal (fun t (q : S × (ι → Ω)) => ∑ i ∈ I, a i * B i t (q.2 i)) | |
| (ν.prod (Measure.infinitePi μ)) := by | |
| classical | |
| have (i : ι) : IsProbabilityMeasure (μ i) := | |
| (hB i).isGaussianProcess.isProbabilityMeasure | |
| have hind := (planted_coordinate_paths_iIndepFun ν μ hmB).precomp | |
| (show Function.Injective (fun i : I => (i : ι)) from Subtype.val_injective) | |
| have h := BrownianLinearCombination.isPreBrownianReal_sum | |
| (fun i : I => planted_coordinate_preBrownian ν μ hB i) hind | |
| (fun i : I => a i) (by rw [Finset.sum_coe_sort I (fun i => a i ^ 2)]; exact ha) | |
| convert h using 1 | |
| funext t q | |
| exact (Finset.sum_coe_sort I (fun i => a i * B i t (q.2 i))).symm | |
| theorem planted_finite_projection_normalized_preBrownian | |
| {ι S Ω : Type*} [MeasurableSpace S] [MeasurableSpace Ω] | |
| (ν : Measure S) (μ : ι → Measure Ω) [IsProbabilityMeasure ν] | |
| {B : ι → ℝ≥0 → Ω → ℝ} (hB : ∀ i, IsPreBrownianReal (B i) (μ i)) | |
| (hmB : ∀ i t, Measurable (B i t)) (I : Finset ι) (a : ι → ℝ) | |
| (ha : 0 < ∑ i ∈ I, a i ^ 2) : | |
| IsPreBrownianReal (fun t (q : S × (ι → Ω)) => | |
| (Real.sqrt (∑ i ∈ I, a i ^ 2))⁻¹ * ∑ i ∈ I, a i * B i t (q.2 i)) | |
| (ν.prod (Measure.infinitePi μ)) := by | |
| classical | |
| have (i : ι) : IsProbabilityMeasure (μ i) := | |
| (hB i).isGaussianProcess.isProbabilityMeasure | |
| have hind := (planted_coordinate_paths_iIndepFun ν μ hmB).precomp | |
| (show Function.Injective (fun i : I => (i : ι)) from Subtype.val_injective) | |
| have h := BrownianLinearCombination.isPreBrownianReal_normalized_sum | |
| (fun i : I => planted_coordinate_preBrownian ν μ hB i) hind | |
| (fun i : I => a i) (by rw [Finset.sum_coe_sort I (fun i => a i ^ 2)]; exact ha) | |
| convert h using 1 | |
| funext t q | |
| rw [Finset.sum_coe_sort I (fun i => a i ^ 2), | |
| Finset.sum_coe_sort I (fun i => a i * B i t (q.2 i))] | |
| end Problem367 | |
| end | |
| end OAI | |