File size: 2,923 Bytes
a2ee19a | 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.KernelEvaluation
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision.Kernel
lemma seed_average_bound {X Y : Type*} {Ω : Type u} [MeasurableSpace X] [MeasurableSpace Y]
[MeasurableSpace Ω] (π : Measure X) [SFinite π] (ν : Measure Y) [SFinite ν]
(ρ : Measure Ω) [IsProbabilityMeasure ρ]
{f : X × (Ω × Y) → ℝ≥0∞}
(hf : AEMeasurable f (π.prod (ρ.prod ν))) {B : ℝ≥0∞}
(hb : ∀ᵐ ω ∂ρ, (∫⁻ z : X × Y, f (z.1,(ω,z.2)) ∂π.prod ν) ≤ B) :
(∫⁻ z, f z ∂π.prod (ρ.prod ν)) ≤ B := by
have hp := ((MeasurePreserving.id π).prod
(Measure.measurePreserving_swap (μ := ν) (ν := ρ))).comp
(measurePreserving_prodAssoc π ν ρ)
let g : (X × Y) × Ω → ℝ≥0∞ := fun z => f (z.1.1,(z.2,z.1.2))
have hg : AEMeasurable g ((π.prod ν).prod ρ) :=
hf.comp_quasiMeasurePreserving hp.quasiMeasurePreserving
have he : (∫⁻ z, g z ∂(π.prod ν).prod ρ) = ∫⁻ z, f z ∂π.prod (ρ.prod ν) := by
rw [← hp.map_eq]
exact (lintegral_map' (by simpa only [hp.map_eq] using hf) hp.measurable.aemeasurable).symm
rw [← he, lintegral_prod_symm _ hg]
exact (lintegral_mono_ae hb).trans_eq (by simp)
namespace Policy
variable {d M T : ℕ}
lemma success_le_one {Ω : Type u} [MeasurableSpace Ω] (L : Ω → Policy d M T)
(ρ : Measure Ω) [IsProbabilityMeasure ρ] (π : Measure (Sphere d)) [IsProbabilityMeasure π]
(ε : ℝ) : Success L ρ π ε ≤ 1 := by
have : IsProbabilityMeasure (RowLaw d T) := by unfold RowLaw; infer_instance
unfold Success experiment
calc
_ ≤ ∫⁻ _z : Sphere d × (Ω × Rows d T), (1 : ℝ≥0∞) ∂π.prod (ρ.prod (RowLaw d T)) :=
lintegral_mono (fun z => (L z.2.1).pathValue_le_one _
((L z.2.1).outputTest_le_one (angleTest_le_one ε)) z.1 (records z.1 z.2.2))
_ = 1 := by simp
lemma chordSuccess_le_one {Ω : Type u} [MeasurableSpace Ω] (L : Ω → Policy d M T)
(ρ : Measure Ω) [IsProbabilityMeasure ρ] (π : Measure (Sphere d)) [IsProbabilityMeasure π]
(ε : ℝ) : ChordSuccess L ρ π ε ≤ 1 := by
have : IsProbabilityMeasure (RowLaw d T) := by unfold RowLaw; infer_instance
unfold ChordSuccess experiment
calc
_ ≤ ∫⁻ _z : Sphere d × (Ω × Rows d T), (1 : ℝ≥0∞) ∂π.prod (ρ.prod (RowLaw d T)) :=
lintegral_mono (fun z => (L z.2.1).pathValue_le_one _
((L z.2.1).outputTest_le_one (chordTest_le_one ε)) z.1 (records z.1 z.2.2))
_ = 1 := by simp
end Policy
end MemoryPrecision.Kernel
end
end OAI
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