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b317718 | 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 | import OAI.Probability.GaussianRegression.Observation
import OAI.Probability.GaussianRegression.Caps
import OAI.Probability.GaussianRegression.Execution
namespace OAI
/-!
Prefix integration and transfer between almost-everywhere equal transitions.
-/
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
open InnerProductSpace
open scoped Pointwise
noncomputable section
universe u
namespace NoiselessRegression
section StreamTransfer
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
omit [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [BorelSpace E] in
lemma measurePreserving_pi_prefix (ν : Measure E) [IsProbabilityMeasure ν]
{t T : ℕ} (ht : t ≤ T) :
MeasurePreserving (fun x : Fin T → E => fun i : Fin t => x (Fin.castLE ht i))
(Measure.pi (fun _ => ν)) (Measure.pi (fun _ => ν)) := by
refine ⟨by fun_prop, ?_⟩
apply Measure.ext_of_lintegral
intro f hf
rw [lintegral_map hf (by fun_prop)]
exact lintegral_pi_prefix ν ht f hf
lemma sphere_sample_observation_quasi [Nontrivial E] (hd : 3 ≤ Module.finrank ℝ E)
{T : ℕ} (i : Fin T) :
Measure.QuasiMeasurePreserving
(fun z : (Metric.sphere (0 : E) 1) × (Fin T → E) => (z.2 i, ⟪z.1.val,z.2 i⟫))
((sphereLaw (E := E)).prod (Measure.pi (fun _ => stdGaussian E)))
((stdGaussian E).prod volume) := by
exact (sphere_observation_quasi hd).comp
((MeasurePreserving.id (sphereLaw (E := E))).prod
(measurePreserving_eval (fun _ : Fin T => stdGaussian E) i)).quasiMeasurePreserving
omit [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
[MeasurableSpace E] [BorelSpace E] in
lemma streamRoute_congr {J : Type*} (δ δ' : ℕ → J → (E × ℝ) → J)
(n k : ℕ) (j : J) (A : Fin k → E) (b : Fin k → ℝ)
(h : ∀ t j i, δ t j (A i,b i) = δ' t j (A i,b i)) :
streamRoute δ n k j (A,b) = streamRoute δ' n k j (A,b) := by
induction k generalizing n j with
| zero => rfl
| succ k ih =>
rw [streamRoute, streamRoute, h]
exact ih (n+1) _ (Fin.tail A) (Fin.tail b) (fun t j i => h t j i.succ)
lemma prefix_route_cap_bound [Nontrivial E]
{J : Type*} [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J] [Nonempty J]
{δ : ℕ → J → (E × ℝ) → J} (hδ : ∀ n j, Measurable (δ n j)) (y : J → E)
{ε : ℝ} (hε : 0 < ε) {a q : ℕ} (ha : 2*a ≤ Module.finrank ℝ E - 1)
(hq : 2 ≤ q) (hqa : 2*q ≤ a) {k T : ℕ} (hk : k ≤ T) (j : J) :
((sphereLaw (E := E)).prod (Measure.pi (fun _ : Fin T => stdGaussian E)))
{z | dist z.1.val (y (streamRoute δ 0 k j
((fun i => z.2 (Fin.castLE hk i)),fun i => ⟪z.1.val,z.2 (Fin.castLE hk i)⟫))) ≤ ε} ≤
(ENNReal.ofReal ((3 : ℝ)*4^Module.finrank ℝ E) * ENNReal.ofReal (ε^a)) *
streamFactor (E := E) (J := J) q ^ (T/q+1) := by
have hp := (MeasurePreserving.id (sphereLaw (E := E))).prod
(measurePreserving_pi_prefix (stdGaussian E) hk)
have hs : MeasurableSet
{z : (Metric.sphere (0 : E) 1) × (Fin k → E) |
dist z.1.val (y (streamRoute δ 0 k j (z.2,fun i => ⟪z.1.val,z.2 i⟫))) ≤ ε} := by
apply measurableSet_le _ measurable_const
exact (measurable_subtype_coe.comp measurable_fst).dist
((measurable_of_finite y).comp ((measurable_streamRoute hδ 0 k j).comp (by fun_prop)))
apply (le_of_eq (hp.measure_preimage hs.nullMeasurableSet)).trans
apply (route_cap_bound hδ y hε ha hq hqa k j).trans
apply mul_le_mul' le_rfl
exact pow_le_pow_right₀ (one_le_streamFactor (by omega : 0 < q))
(Nat.add_le_add_right (Nat.div_le_div_right hk) 1)
end StreamTransfer
end NoiselessRegression
end
end OAI
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