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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | import OAI.Probability.GaussianReplacement.CompletedModel
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology
noncomputable section
universe u
namespace CurrentProjection
namespace StreamKernel
open NoiselessRegression unitInterval
namespace CompletedLearner
variable {d M T : ℕ} (K : CompletedLearner d M T)
structure Sampling where
initial : I → Bool × State M
measurable_initial : Measurable initial
initial_law : (volume : Measure I).map initial = K.initial
transition : ((ℕ × State M) × (NoiselessRegression.Vector d × ℝ)) → I → Bool × State M
measurable_transition : Measurable (Function.uncurry transition)
transition_law : ∀ p, (volume : Measure I).map (transition p) = K.borel.transition p.1.1 p.1.2 p.2
output : (ℕ × State M) → I → UnitSphere d
measurable_output : Measurable (Function.uncurry output)
output_law : ∀ p, (volume : Measure I).map (output p) = K.output p.1 p.2
lemma exists_sampling : Nonempty K.Sampling := by
let : IsProbabilityMeasure K.initial := K.initial_probability
let : IsProbabilityMeasure (K.output 0 0) := K.output_probability 0 0
let : Nonempty (UnitSphere d) := nonempty_of_isProbabilityMeasure (K.output 0 0)
obtain ⟨i, hi, hei⟩ := K.initial.exists_measurable_map_eq
obtain ⟨t, ht, het⟩ := K.transitionKernel.exists_measurable_map_eq_unitInterval
obtain ⟨o, ho, heo⟩ := K.outputKernel.exists_measurable_map_eq_unitInterval
exact ⟨⟨i,hi,hei,t,ht,het,o,ho,heo⟩⟩
def sampling : K.Sampling := Classical.choice K.exists_sampling
def coinIndex (T n : ℕ) : Fin (T+2) := if h : n < T+2 then ⟨n,h⟩ else 0
def Sampling.program (R : K.Sampling) : NoiselessRegression.Learner d M T (Coins (T+1)) where
initialChoice := fun z => R.initial (z 0)
transition := fun n z a o => R.transition ((n,a),o) (z (coinIndex T n))
output := fun n z a => R.output (n,a) (z (Fin.last (T+1)))
lemma Sampling.measurable_initialChoice (R : K.Sampling) :
Measurable R.program.initialChoice := R.measurable_initial.comp (measurable_pi_apply 0)
lemma Sampling.measurable_program_transition (R : K.Sampling) (n : ℕ) :
Measurable (fun p : (Coins (T+1) × State M) × (NoiselessRegression.Vector d × ℝ) =>
R.program.transition n p.1.1 p.1.2 p.2) := by
exact R.measurable_transition.comp
(((measurable_const (a:=n)).prodMk measurable_fst.snd |>.prodMk measurable_snd).prodMk
((measurable_pi_apply (coinIndex T n)).comp measurable_fst.fst))
lemma Sampling.measurable_program_output (R : K.Sampling) :
Measurable (fun p : Coins (T+1) × (ℕ × State M) =>
R.program.output p.2.1 p.1 p.2.2) := by
exact R.measurable_output.comp
(measurable_snd.prodMk ((measurable_pi_apply (Fin.last (T+1))).comp measurable_fst))
end CompletedLearner
section BorelProgram
variable {d M T : ℕ} {Ω : Type*} [MeasurableSpace Ω]
(L : NoiselessRegression.Learner d M T Ω)
(hi : Measurable L.initialChoice)
(ht : ∀ n, Measurable (fun p : (Ω × State M) × (NoiselessRegression.Vector d × ℝ) =>
L.transition n p.1.1 p.1.2 p.2))
(ho : Measurable (fun p : Ω × (ℕ × State M) => L.output p.2.1 p.1 p.2.2))
include hi ht in
lemma measurable_program_run (n : ℕ) :
Measurable (fun z : UnitSphere d × (Ω × Samples d T) => L.run z.1 z.2.1 z.2.2 n) := by
induction n with
| zero =>
exact (measurable_of_countable (CompletedLearner.startStatus (M:=M))).comp
(hi.comp measurable_snd.fst)
| succ n ih =>
let F : NoiselessRegression.Learner.RunStatus M × (UnitSphere d × (Ω × Samples d T)) →
NoiselessRegression.Learner.RunStatus M := fun p =>
match p.1 with
| .inr done => .inr done
| .inl a => if hn : n < T then
CompletedLearner.choiceStatus (n+1)
(L.transition (n+1) p.2.2.1 a
(p.2.2.2 ⟨n,hn⟩, ⟪p.2.2.2 ⟨n,hn⟩,p.2.1.val⟫))
else .inl a
have hF : Measurable F := by
apply measurable_from_prod_countable_right
intro a
cases a with
| inr done => exact measurable_const
| inl a =>
by_cases hn : n < T
· simp only [F, dite_eq_left hn]
apply (measurable_of_countable (CompletedLearner.choiceStatus (M:=M) (n+1))).comp
have hr : Measurable (fun z : UnitSphere d × (Ω × Samples d T) => z.2.2 ⟨n,hn⟩) :=
(measurable_pi_apply _).comp measurable_snd.snd
exact (ht (n+1)).comp
((measurable_snd.fst.prodMk measurable_const).prodMk
(hr.prodMk (hr.inner measurable_fst.subtype_val)))
· simpa only [F, dite_eq_right hn] using
(measurable_const : Measurable (fun _ : UnitSphere d × (Ω × Samples d T) =>
(Sum.inl a : NoiselessRegression.Learner.RunStatus M)))
exact hF.comp (ih.prodMk measurable_id)
include hi ht ho in
lemma measurable_program_estimate :
Measurable (fun z : UnitSphere d × (Ω × Samples d T) => L.estimate z.1 z.2.1 z.2.2) := by
exact ho.comp (measurable_snd.fst.prodMk
((measurable_of_countable (CompletedLearner.terminalStatus (M:=M) (T:=T))).comp
(measurable_program_run L hi ht T)))
include hi ht ho in
lemma admissible_of_borel (ρ : Measure Ω) : L.Admissible ρ := by
have he := measurable_program_estimate L hi ht ho
refine ⟨hi.aemeasurable, ?_, ?_, ?_, he.aemeasurable⟩
· intro n a
exact ((ht n).comp
((measurable_fst.prodMk measurable_const).prodMk measurable_snd)).aemeasurable
· intro n a
exact (ho.comp (measurable_id.prodMk (measurable_const (a:=(n,a))))).aemeasurable
· intro s
exact (he.comp (measurable_const.prodMk measurable_id)).aemeasurable
end BorelProgram
namespace CompletedLearner
variable {d M T : ℕ} {K : CompletedLearner d M T}
lemma Sampling.admissible (R : K.Sampling) : R.program.Admissible (coinLaw (T+1)) :=
admissible_of_borel R.program R.measurable_initialChoice R.measurable_program_transition
R.measurable_program_output _
end CompletedLearner
end StreamKernel
end CurrentProjection
end
end OAI
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