| import OAI.Analysis.FiniteFactor.OrbitSampling.RationalTests |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical ENNReal Topology |
| noncomputable section |
|
|
| /-- The elementary probability-space Cauchy--Schwarz bound, proved by |
| integrating the nonnegative square centered at the mean. -/ |
| theorem sampling_integral_square_le {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (f : Ω → ℝ) |
| (hf : Integrable f μ) (hf2 : Integrable (fun x => f x^2) μ) : |
| (∫ x, f x ∂μ)^2 ≤ ∫ x, f x^2 ∂μ := by |
| let c := ∫ x, f x ∂μ |
| have he : (fun x => (f x-c)^2) = (fun x => f x^2-2*c*f x+c^2) := by funext x; ring |
| have hh : 0 ≤ ∫ x, (f x-c)^2 ∂μ := integral_nonneg (fun x => sq_nonneg _) |
| have hi : Integrable (fun x => f x^2-2*c*f x) μ := hf2.sub (hf.const_mul (2*c)) |
| have hc : Integrable (fun _x : Ω => c^2) μ := integrable_const _ |
| rw [he,integral_add hi hc,integral_sub hf2 (hf.const_mul (2*c)),integral_const_mul] at hh |
| simp only [integral_const,measureReal_def,measure_univ,ENNReal.toReal_one,smul_eq_mul,one_mul] at hh |
| change c^2 ≤ _ |
| change 0 ≤ (∫ x, f x^2 ∂μ)-2*c*c+c^2 at hh |
| nlinarith |
|
|
| /-- Actual rectangle mass equals the integral of the first tested empirical |
| density at every admissible stage. -/ |
| theorem sourceSamplingUntrimmed_rectangle_mass (I : Finset SourceCoordinate) (stage : ℕ) |
| (E G : SourceCoordinate → Set ℝ) |
| (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) |
| (houtside : ∀ i ∉ sourceSamplingPrefix stage, |
| Set.Ico (0 : ℝ) 1 ⊆ E i ∧ Set.Ico (0 : ℝ) 1 ⊆ G i) |
| (θ : SourceCoordinate → ℝ) : |
| ((sourceSamplingUntrimmed I stage θ) (sourceSamplingRectangle E G)).toReal = |
| ∫ x, sourceSamplingEmpiricalDensity stage false (fun i => decide (i ∈ I)) E G θ x |
| ∂sourceSamplingHaar := by |
| let z := sourceSamplingEmpiricalDensity stage false (fun i => decide (i ∈ I)) E G θ |
| have hm := sourceSamplingEmpiricalDensity_measurable stage false (fun i => decide (i ∈ I)) E G hE hG θ |
| have hb := sourceSamplingEmpiricalDensity_bounds stage false (fun i => decide (i ∈ I)) E G θ |
| have hi : Integrable z sourceSamplingHaar := |
| sampling_integrable_bounded sourceSamplingHaar z hm _ (fun x => by |
| rw [abs_of_nonneg (hb x).1]; exact (hb x).2) |
| have hh := congrArg (fun ν : Measure (SourceCoordinate → ℝ) => ν Set.univ) |
| (sourceSamplingUntrimmed_tested_marginal I stage false E G hE hG houtside θ) |
| rw [Measure.map_apply (sourceSamplingEndpoint_measurable false) MeasurableSet.univ, |
| Set.preimage_univ,Measure.restrict_apply MeasurableSet.univ,Set.univ_inter, |
| withDensity_apply _ MeasurableSet.univ,Measure.restrict_univ] at hh |
| rw [hh,← ofReal_integral_eq_lintegral_ofReal hi (Eventually.of_forall (fun x => (hb x).1)), |
| ENNReal.toReal_ofReal (integral_nonneg (fun x => (hb x).1))] |
|
|
| /-- Full squared sampling error bounds the squared discrepancy of the actual |
| rectangle mass and the ideal product integral. -/ |
| theorem sourceSamplingUntrimmed_rectangle_error (I : Finset SourceCoordinate) (stage : ℕ) |
| (E G : SourceCoordinate → Set ℝ) |
| (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) |
| (houtside : ∀ i ∉ sourceSamplingPrefix stage, |
| Set.Ico (0 : ℝ) 1 ⊆ E i ∧ Set.Ico (0 : ℝ) 1 ⊆ G i) |
| (θ : SourceCoordinate → ℝ) : |
| (((sourceSamplingUntrimmed I stage θ) (sourceSamplingRectangle E G)).toReal - |
| ∫ x, sourceSamplingIdealDensity stage false (fun i => decide (i ∈ I)) E G x ∂sourceSamplingHaar)^2 ≤ |
| sourceSamplingSquaredError stage false (fun i => decide (i ∈ I)) E G θ := by |
| rw [sourceSamplingUntrimmed_rectangle_mass I stage E G hE hG houtside θ] |
| let z := sourceSamplingEmpiricalDensity stage false (fun i => decide (i ∈ I)) E G θ |
| let g := sourceSamplingIdealDensity stage false (fun i => decide (i ∈ I)) E G |
| let C := (max 1 (1/(sourceSamplingWidth stage*sourceSamplingCutoff stage)))^(sourceSamplingPrefix stage).card |
| have hzm := sourceSamplingEmpiricalDensity_measurable stage false (fun i => decide (i ∈ I)) E G hE hG θ |
| have hgm := sourceSamplingIdealDensity_measurable stage false (fun i => decide (i ∈ I)) E G hE hG |
| have hzb := sourceSamplingEmpiricalDensity_bounds stage false (fun i => decide (i ∈ I)) E G θ |
| have hgb := sourceSamplingIdealDensity_bounds stage false (fun i => decide (i ∈ I)) E G hE hG |
| have hzi : Integrable z sourceSamplingHaar := sampling_integrable_bounded sourceSamplingHaar z hzm C |
| (fun x => by rw [abs_of_nonneg (hzb x).1]; exact (hzb x).2) |
| have hgi : Integrable g sourceSamplingHaar := sampling_integrable_bounded sourceSamplingHaar g hgm C |
| (fun x => by rw [abs_of_nonneg (hgb x).1]; exact (hgb x).2) |
| have hei : Integrable (fun x => (z x-g x)^2) sourceSamplingHaar := by |
| apply sampling_integrable_bounded sourceSamplingHaar _ ((hzm.sub hgm).pow_const 2) (C^2) |
| intro x |
| rw [abs_of_nonneg (sq_nonneg _)] |
| have hz' : 0 ≤ z x ∧ z x ≤ C := hzb x |
| have hg' : 0 ≤ g x ∧ g x ≤ C := hgb x |
| change (z x-g x)^2 ≤ C^2 |
| nlinarith [sq_nonneg (z x+g x),mul_nonneg (show 0 ≤ C-(z x-g x) by linarith) |
| (show 0 ≤ C+(z x-g x) by linarith)] |
| have hh := sampling_integral_square_le sourceSamplingHaar (fun x => z x-g x) (hzi.sub hgi) hei |
| rw [integral_sub hzi hgi] at hh |
| exact hh |
|
|
| end |
| end FiniteFactor |
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| end OAI |
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