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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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.Probability.GaussianReplacement.CompletedKernel
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
universe u
namespace CurrentProjection
namespace StreamKernel
open NoiselessRegression unitInterval
structure CompletedLearner (d M T : ℕ) where
initial : Measure (Bool × State M)
initial_probability : IsProbabilityMeasure initial
transition : ℕ → State M → (NoiselessRegression.Vector d × ℝ) → Measure (Bool × State M)
transition_probability : ∀ n a o, IsProbabilityMeasure (transition n a o)
transition_completed : ∀ n a, AEMeasurable (transition n a) (observationReference d)
output : ℕ → State M → Measure (UnitSphere d)
output_probability : ∀ n a, IsProbabilityMeasure (output n a)
instance runStatus_singletons (M : ℕ) :
MeasurableSingletonClass (NoiselessRegression.Learner.RunStatus M) where
measurableSet_singleton x := by
rw [measurableSet_sum_iff]
cases x with
| inl a =>
constructor
· simp
· have h : (Sum.inr : ℕ × State M → NoiselessRegression.Learner.RunStatus M) ⁻¹' {Sum.inl a} = ∅ := by
ext z
simp
rw [h]
exact MeasurableSet.empty
| inr a =>
constructor
· simp
· have h : (Sum.inr : ℕ × State M → NoiselessRegression.Learner.RunStatus M) ⁻¹' {Sum.inr a} = {a} := by
ext z
simp
rw [h]
exact measurableSet_singleton a
namespace CompletedLearner
variable {d M T : ℕ} (K : CompletedLearner d M T)
def startStatus (a : Bool × State M) : NoiselessRegression.Learner.RunStatus M :=
if a.1 then Sum.inr (0,a.2) else Sum.inl a.2
def choiceStatus (n : ℕ) (a : Bool × State M) : NoiselessRegression.Learner.RunStatus M :=
if a.1 then Sum.inr (n,a.2) else Sum.inl a.2
def nextLaw (s : UnitSphere d) (x : Samples d T) (n : ℕ) :
NoiselessRegression.Learner.RunStatus M → Measure (NoiselessRegression.Learner.RunStatus M)
| .inr done => Measure.dirac (.inr done)
| .inl a => if hn : n < T then
(K.transition (n+1) a (x ⟨n,hn⟩, ⟪x ⟨n,hn⟩,s.val⟫)).map (choiceStatus (n+1))
else Measure.dirac (.inl a)
def runLaw (s : UnitSphere d) (x : Samples d T) :
ℕ → Measure (NoiselessRegression.Learner.RunStatus M)
| 0 => K.initial.map startStatus
| n+1 => (runLaw s x n).bind (K.nextLaw s x n)
def terminalStatus : NoiselessRegression.Learner.RunStatus M → ℕ × State M
| .inl a => (T,a)
| .inr done => done
def estimateLaw (s : UnitSphere d) (x : Samples d T) : Measure (UnitSphere d) :=
((K.runLaw s x T).map (terminalStatus (T:=T))).bind
(fun a => K.output a.1 a.2)
def uniformSuccess (ε : ℝ) : ℝ≥0∞ :=
∫⁻ z : UnitSphere d × Samples d T,
K.estimateLaw z.1 z.2 {v | NoiselessRegression.Learner.angularError z.1 v ≤ ε}
∂(NoiselessRegression.uniformSphere d).prod (sampleLaw d T)
def borel : CompletedLearner d M T where
initial := K.initial
initial_probability := K.initial_probability
transition := fun n a => borelKernel (K.transition n a) (K.transition_completed n a)
transition_probability := fun _ _ _ => inferInstance
transition_completed := fun n a =>
(borelKernel (K.transition n a) (K.transition_completed n a)).measurable.aemeasurable
output := K.output
output_probability := K.output_probability
lemma borel_transition_ae (n : ℕ) (a : State M) :
K.borel.transition n a =ᵐ[observationReference d] K.transition n a :=
borelKernel_ae _ _ (K.transition_probability n a)
def transitionKernel : Kernel ((ℕ × State M) × (NoiselessRegression.Vector d × ℝ))
(Bool × State M) :=
⟨fun p => K.borel.transition p.1.1 p.1.2 p.2,
measurable_from_prod_countable_right fun a =>
(borelKernel (K.transition a.1 a.2) (K.transition_completed a.1 a.2)).measurable⟩
instance transitionKernel_markov : IsMarkovKernel K.transitionKernel :=
⟨fun _ => K.borel.transition_probability _ _ _⟩
def outputKernel : Kernel (ℕ × State M) (UnitSphere d) :=
⟨fun a => K.output a.1 a.2, measurable_of_countable _⟩
instance outputKernel_markov : IsMarkovKernel K.outputKernel :=
⟨fun _ => K.output_probability _ _⟩
end CompletedLearner
end StreamKernel
end CurrentProjection
end
end OAI
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