| import Mathlib |
| import OAI.Probability.SKRatio.Matrices.WeightedGaussianSquares |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators ENNReal NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set Real |
| namespace SKRatio.MatrixNet |
| open SKRatioClock.Regression |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma weighted_square_product_tail {A u c : ℝ} |
| (ht : ∀ {ι : Type} [Fintype ι] (s : Finset ι) (a : ι → ℝ), |
| (∀ i∈s, |a i|≤A) → ∀ {n : ℕ}, 0<n → s.card≤n → |
| standardArrayLaw ι {z | u ≤ |(∑ i∈s, a i*(z i^2-1))/(n:ℝ)|} ≤ |
| ENNReal.ofReal (3*exp (-c*n))) |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] |
| {ι : Type} [Fintype ι] (s : Finset ι) (a : Ω → ι → ℝ) |
| (hma : ∀ i, Measurable (fun ω => a ω i)) (ha : ∀ ω i, i∈s → |a ω i|≤A) |
| {n : ℕ} (hn : 0<n) (hc : s.card≤n) : |
| (μ.prod (standardArrayLaw ι)) {p | u ≤ |(∑ i∈s, a p.1 i*(p.2 i^2-1))/(n:ℝ)|} ≤ |
| ENNReal.ofReal (3*exp (-c*n)) := by |
| have hE : MeasurableSet {p : Ω × (ι → ℝ) | |
| u ≤ |(∑ i∈s, a p.1 i*(p.2 i^2-1))/(n:ℝ)|} := by |
| apply measurableSet_le measurable_const |
| apply Measurable.abs |
| apply Measurable.div_const |
| exact Finset.measurable_sum _ (fun i _ => ((hma i).comp measurable_fst).mul |
| (((measurable_pi_apply i).comp measurable_snd).pow_const 2 |>.sub measurable_const)) |
| rw [Measure.prod_apply hE] |
| calc |
| _ ≤ ∫⁻ _ω : Ω, ENNReal.ofReal (3*exp (-c*n)) ∂μ := |
| lintegral_mono (fun ω => ht s (a ω) (ha ω) hn hc) |
| _ = _ := by simp |
|
|
| theorem weighted_square_conditioned_tail {A u : ℝ} (hA : 0≤A) (hu : 0<u) : |
| ∃ c : ℝ, 0<c ∧ ∀ {ι : Type} [Fintype ι] (s : Finset ι) |
| (a : ({i // i∉s} → ℝ) → s → ℝ), |
| (∀ i, Measurable (fun y => a y i)) → (∀ y i, |a y i|≤A) → |
| ∀ {n : ℕ}, 0<n → s.card≤n → |
| standardArrayLaw ι {z | u ≤ |(∑ i : s, |
| a (fun j => z j) i * (z i^2-1))/(n:ℝ)|} ≤ |
| ENNReal.ofReal (3*exp (-c*n)) := by |
| classical |
| obtain ⟨c,hc,ht⟩ := weighted_standard_square_tail hA hu |
| refine ⟨c,hc,?_⟩ |
| intro ι _ s a hm ha n hn hs |
| let E : Set (({i // i∉s} → ℝ) × (s → ℝ)) := |
| {p | u ≤ |(∑ i : s, a p.1 i*(p.2 i^2-1))/(n:ℝ)|} |
| have hE : MeasurableSet E := by |
| apply measurableSet_le measurable_const |
| apply Measurable.abs |
| apply Measurable.div_const |
| exact Finset.measurable_sum _ (fun i _ => ((hm i).comp measurable_fst).mul |
| (((measurable_pi_apply i).comp measurable_snd).pow_const 2 |>.sub measurable_const)) |
| have hp := measurePreserving_piEquivPiSubtypeProd |
| (fun _ : ι => gaussianReal 0 1) (fun i => i∈s) |
| have hp' := Measure.measurePreserving_swap.comp hp |
| have he : {z : ι → ℝ | u ≤ |(∑ i : s, a (fun j => z j) i*(z i^2-1))/(n:ℝ)|} = |
| (fun z => ((fun j : {i // i∉s} => z j), (fun j : s => z j))) ⁻¹' E := rfl |
| rw [he] |
| have hmap : MeasurePreserving (fun z : ι → ℝ => |
| ((fun j : {i // i∉s} => z j), (fun j : s => z j))) |
| (standardArrayLaw ι) ((standardArrayLaw {i // i∉s}).prod (standardArrayLaw s)) := by |
| convert! hp' using 1 |
| dsimp only [standardArrayLaw] |
| congr 1 |
| congr 1 |
| exact Subsingleton.elim _ _ |
|
|
| rw [hmap.measure_preimage hE.nullMeasurableSet] |
| exact weighted_square_product_tail ht (standardArrayLaw {i // i∉s}) |
| Finset.univ a hm (fun y i _ => ha y i) hn (by simpa using hs) |
|
|
| end SKRatio.MatrixNet |
|
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| end |
| end |
|
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| end OAI |
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|
