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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | 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
end OAI
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