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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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.Analysis.FiniteFactor.OrbitSampling.ProductError
import OAI.Analysis.FiniteFactor.OrbitSampling.DiagonalEvent
import OAI.Analysis.FiniteFactor.Sampling.Cancellation
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
noncomputable section
def sourceSamplingEmpiricalDensity (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (θ x : SourceCoordinate → ℝ) : ℝ :=
(∑ n ∈ sourceSamplingPowers stage, sourceSamplingProduct stage r b E G n θ x)/
((sourceSamplingPowers stage).card : ℝ)
/-- The full sampling error, including every distinct-pair and mixed term. -/
def sourceSamplingSquaredError (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (θ : SourceCoordinate → ℝ) : ℝ :=
∫ x, (sourceSamplingEmpiricalDensity stage r b E G θ x-sourceSamplingIdealDensity stage r b E G x)^2
∂Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure)
theorem sourceSampling_error_joint_integrable (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) :
Integrable (fun p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) =>
(sourceSamplingEmpiricalDensity stage r b E G p.1 p.2-sourceSamplingIdealDensity stage r b E G p.2)^2)
(sourceAngleLaw.prod (Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure))) := by
apply sampling_average_square_integrable
· intro n _ k _
exact sourceSamplingProduct_pair_integrable stage r b E G hE hG n k
· intro n _
exact sourceSamplingProduct_mixed_integrable stage r b E G hE hG n
· have hm : Measurable (fun p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) => sourceSamplingIdealDensity stage r b E G p.2) :=
(sourceSamplingIdealDensity_measurable stage r b E G hE hG).comp measurable_snd
have hi := sampling_bounded_product_integrable
(sourceAngleLaw.prod (Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure)))
(fun p => sourceSamplingIdealDensity stage r b E G p.2) (fun p => sourceSamplingIdealDensity stage r b E G p.2)
hm hm _ (fun p => sourceSamplingIdealDensity_bounds stage r b E G hE hG p.2)
(fun p => sourceSamplingIdealDensity_bounds stage r b E G hE hG p.2)
simpa only [pow_two] using hi
theorem sourceSamplingSquaredError_integrable (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) :
Integrable (sourceSamplingSquaredError stage r b E G) sourceAngleLaw :=
(sourceSampling_error_joint_integrable stage r b E G hE hG).integral_prod_left
theorem sourceSamplingSquaredError_measurable (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) :
Measurable (sourceSamplingSquaredError stage r b E G) := by
have hm : Measurable (fun p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) =>
sourceSamplingEmpiricalDensity stage r b E G p.1 p.2) :=
(Finset.measurable_sum _ (fun n _ => sourceSamplingProduct_measurable stage r b E G hE hG n)).div_const _
exact ((hm.sub ((sourceSamplingIdealDensity_measurable stage r b E G hE hG).comp measurable_snd)).pow_const 2).stronglyMeasurable.integral_prod_right.measurable
theorem sourceSamplingSquaredError_nonneg (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (θ : SourceCoordinate → ℝ) :
0 ≤ sourceSamplingSquaredError stage r b E G θ := integral_nonneg (fun _ => sq_nonneg _)
/-- Full expected squared sampling error with the source's actual sample set,
weights and ideal density. All distinct-pair and mixed assumptions are proved. -/
theorem sourceSamplingSquaredError_expectation (stage : ℕ) (hs : 200000 ≤ sourceBlockSize stage)
(r : Bool) (b : SourceCoordinate → Bool) (a c a' c' : SourceCoordinate → ℝ) :
(∫ θ, sourceSamplingSquaredError stage r b (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ ∂sourceAngleLaw) ≤
(∫ θ, sourceSamplingDiagonalContribution stage r b (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ ∂sourceAngleLaw)+
3*sourceSamplingCorrelationBudget stage := by
let E := fun i => Set.Icc (a i) (c i)
let G := fun i => Set.Icc (a' i) (c' i)
have hE : ∀ i, MeasurableSet (E i) := fun _ => measurableSet_Icc
have hG : ∀ i, MeasurableSet (G i) := fun _ => measurableSet_Icc
let μ := sourceAngleLaw.prod (Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure))
let S := sourceSamplingPowers stage
let f (n : ℤ) (p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)) := sourceSamplingProduct stage r b E G n p.1 p.2
let g (p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)) := sourceSamplingIdealDensity stage r b E G p.2
have hpair (n k : ℤ) : Integrable (fun p => f n p*f k p) μ :=
sourceSamplingProduct_pair_integrable stage r b E G hE hG n k
have hmix (n : ℤ) : Integrable (fun p => f n p*g p) μ :=
sourceSamplingProduct_mixed_integrable stage r b E G hE hG n
have hg : Integrable (fun p => g p^2) μ := by
have hm : Measurable (fun p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) => sourceSamplingIdealDensity stage r b E G p.2) :=
(sourceSamplingIdealDensity_measurable stage r b E G hE hG).comp measurable_snd
simpa only [pow_two] using sampling_bounded_product_integrable μ g g hm hm _
(fun p => sourceSamplingIdealDensity_bounds stage r b E G hE hG p.2)
(fun p => sourceSamplingIdealDensity_bounds stage r b E G hE hG p.2)
have hA : (∫ p, g p^2 ∂μ) = ∫ x, sourceSamplingIdealDensity stage r b E G x^2
∂Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure) := by
rw [integral_prod _ hg]
simp [g]
have hB (n k : ℤ) : (∫ p, f n p*f k p ∂μ) =
∫ θ, ∫ x, sourceSamplingProduct stage r b E G n θ x*sourceSamplingProduct stage r b E G k θ x
∂Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure) ∂sourceAngleLaw := integral_prod _ (hpair n k)
have hM (n : ℤ) : (∫ p, f n p*g p ∂μ) =
∫ θ, ∫ x, sourceSamplingProduct stage r b E G n θ x*sourceSamplingIdealDensity stage r b E G x
∂Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure) ∂sourceAngleLaw := integral_prod _ (hmix n)
have hS : S.Nonempty := by
have hP : (0 : ℝ) < (sourceSamplingPowers stage).card :=
(Real.rpow_pos_of_pos (sourceBlockSize_pos stage) _).trans_le (sourceSamplingPowers_spec stage hs).2.1
exact Finset.card_pos.mp (by exact_mod_cast hP)
have hc := sampling_finite_cancellation S hS
(fun n k => ∫ p, f n p*f k p ∂μ) (fun n => ∫ p, f n p*g p ∂μ)
(∫ p, g p^2 ∂μ) (sourceSamplingCorrelationBudget stage)
(integral_nonneg (fun _ => sq_nonneg _)) (by unfold sourceSamplingCorrelationBudget; positivity)
(fun n hn k hk hne => by rw [hB,hA]; exact sourceSamplingProduct_pair_error stage hs r b a c a' c' n k hn hk hne)
(fun n hn => by rw [hM,hA]; exact sourceSamplingProduct_mixed_error stage hs r b E G hE hG n hn)
have hex := sampling_average_square_expansion μ S f g (fun n _ k _ => hpair n k) (fun n _ => hmix n) hg
rw [← hex] at hc
have hErr : (∫ p, ((∑ n ∈ S, f n p)/(S.card : ℝ)-g p)^2 ∂μ) =
∫ θ, sourceSamplingSquaredError stage r b E G θ ∂sourceAngleLaw :=
integral_prod _ (sourceSampling_error_joint_integrable stage r b E G hE hG)
have hDiag : (∑ n ∈ S, ∫ p, f n p*f n p ∂μ)/(S.card : ℝ)^2 =
∫ θ, sourceSamplingDiagonalContribution stage r b E G θ ∂sourceAngleLaw := by
unfold sourceSamplingDiagonalContribution
rw [integral_div,integral_finsetSum _ (fun n _ =>
(sourceSamplingProduct_square_integrable stage r b E G hE hG n).integral_prod_left)]
simp only [hB,pow_two,f,S]
rw [hErr,hDiag] at hc
exact hc
end
end FiniteFactor
end OAI
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