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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 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 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.CoefficientStep
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
namespace Policy
variable {d M T : ℕ} (K : Policy d M T)
private def versionCoordinates (hK : K.Completed) (t : ℕ) (j : State M)
(obs : Vector d × ℝ) (a : Action M) : ℝ≥0∞ :=
(hK t j a).mk (fun obs => K.transition t j obs a) obs
private lemma measurable_versionCoordinates (hK : K.Completed) (t : ℕ) (j : State M) (a : Action M) :
Measurable (fun obs => K.versionCoordinates hK t j obs a) :=
(hK t j a).measurable_mk
private lemma versionCoordinates_ae (hK : K.Completed) (t : ℕ) (j : State M) :
∀ᵐ obs ∂ObservationReference d, ∀ a, K.versionCoordinates hK t j obs a = K.transition t j obs a := by
apply ae_all_iff.mpr
intro a
exact (hK t j a).ae_eq_mk.symm
def version (hK : K.Completed) : Policy d M T where
initial := K.initial
initial_sum := K.initial_sum
transition := fun t j obs a => if ∑ b, K.versionCoordinates hK t j obs b = 1
then K.versionCoordinates hK t j obs a else if a = (false,0) then 1 else 0
transition_sum := by
intro t j obs
split_ifs with h
· exact h
· simp
output := K.output
output_probability := K.output_probability
lemma version_borel (hK : K.Completed) : (K.version hK).Borel := by
intro t j a
exact Measurable.ite
(measurableSet_eq_fun (Finset.measurable_sum _ (fun b _ =>
K.measurable_versionCoordinates hK t j b)) measurable_const)
(K.measurable_versionCoordinates hK t j a) measurable_const
lemma version_transition_ae (hK : K.Completed) (t : ℕ) (j : State M) :
∀ᵐ obs ∂ObservationReference d, ∀ a, (K.version hK).transition t j obs a = K.transition t j obs a := by
filter_upwards [K.versionCoordinates_ae hK t j] with obs h
have he : ∑ a, K.versionCoordinates hK t j obs a = 1 := by
simpa only [h] using K.transition_sum t j obs
intro a
simp only [version, h, K.transition_sum, ite_true]
lemma eval_eq_of_rules {L : Policy d M T} {X : Type*}
(w : ℕ → State M → X → ℝ≥0∞) (n k : ℕ) (j : State M) (s : X)
(obs : Observations d k)
(h : ∀ t j i a, K.transition t j (obs i) a = L.transition t j (obs i) a) :
K.eval w n k j s obs = L.eval w n k j s obs := by
induction k generalizing n j with
| zero => rfl
| succ k ih =>
simp only [eval, h]
congr 1
funext a
congr 1
split
· rfl
· exact ih (n+1) a.2 (Fin.tail obs) (fun t j i a => h t j i.succ a)
lemma version_pathValue_ae {X : Type*} [MeasurableSpace X] (μ : Measure X)
(hK : K.Completed) (s : X → Sphere d) (obs : X → Observations d T)
(ho : ∀ i : Fin T, Measure.QuasiMeasurePreserving (fun z => obs z i) μ (ObservationReference d))
(f : Sphere d × Sphere d → ℝ≥0∞) :
(fun z => (K.version hK).pathValue ((K.version hK).outputTest f) (s z) (obs z)) =ᵐ[μ]
(fun z => K.pathValue (K.outputTest f) (s z) (obs z)) := by
have hall : ∀ᵐ z ∂μ, ∀ i t j a,
(K.version hK).transition t j (obs z i) a = K.transition t j (obs z i) a := by
apply ae_all_iff.mpr
intro i
apply ae_all_iff.mpr
intro t
apply ae_all_iff.mpr
intro j
exact (ho i).ae (K.version_transition_ae hK t j)
filter_upwards [hall] with z hz
change (∑ a : Action M, K.initial a * if a.1 then K.outputTest f 0 a.2 (s z)
else (K.version hK).eval (K.outputTest f) 0 T a.2 (s z) (obs z)) = _
simp only [pathValue]
congr 1
funext a
congr 1
split
· rfl
· exact (K.version hK).eval_eq_of_rules (K.outputTest f) 0 T a.2 (s z) (obs z)
(fun t j i a => hz i t j a)
end Policy
end MemoryPrecision.Kernel
end
end OAI
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