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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 | import OAI.Probability.ProjectionMoments.BorelRules
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open MeasureTheory ProbabilityTheory Set Filter unitInterval
open scoped ENNReal NNReal RealInnerProductSpace
namespace NoiselessRegression
lemma finite_kernel_borel_version {X J : Type*} [MeasurableSpace X]
[Fintype J] [Nonempty J] [MeasurableSpace J] [MeasurableSingletonClass J]
(μ : Measure X) (P : X → Measure J) (hP : ∀ x, IsProbabilityMeasure (P x))
(hm : ∀ j, AEMeasurable (fun x => P x {j}) μ) :
∃ κ : Kernel X J, IsMarkovKernel κ ∧ (fun x => κ x) =ᵐ[μ] P := by
classical
let v : X → Measure J := fun x => ∑ j, ((hm j).mk _ x) • Measure.dirac j
have hv : Measurable v := by
apply Finset.measurable_sum
intro j _
exact (hm j).measurable_mk.smul_measure (Measure.dirac j)
have he : v =ᵐ[μ] P := by
have hh : ∀ᵐ x ∂μ, ∀ j, (hm j).mk _ x = P x {j} := by
simp only [ae_all_iff]
intro j
exact (hm j).ae_eq_mk.symm
filter_upwards [hh] with x hx
simp only [v,hx,← Measure.sum_fintype,Measure.sum_smul_dirac]
let κ : Kernel X J := ⟨fun x => if v x Set.univ = 1 then v x else Measure.dirac (Classical.choice ‹Nonempty J›),
hv.ite (measurableSet_eq_fun ((Measure.measurable_coe MeasurableSet.univ).comp hv) measurable_const)
measurable_const⟩
refine ⟨κ,⟨fun x => ⟨?_⟩⟩,?_⟩
· dsimp [κ]
split_ifs with h
· exact h
· simp
· filter_upwards [he] with x hx
have hx1 : v x Set.univ = 1 := by rw [hx]; exact (hP x).measure_univ
dsimp only [κ,Kernel.coe_mk]
rw [ite_eq_left hx1,hx]
namespace FiniteKernelLearner
variable {d M T : ℕ}
structure BorelRules (K : FiniteKernelLearner d M T) : Prop where
transition : ∀ t j, Measurable (K.transition t j)
lemma exists_borel_version (K : FiniteKernelLearner d M T) (hK : K.CompletedRules) :
∃ K' : FiniteKernelLearner d M T, K'.BorelRules ∧ K'.initial = K.initial ∧
K'.output = K.output ∧ ∀ t j, K'.transition t j =ᵐ[observationReference d] K.transition t j := by
classical
choose κ hκ he using fun t j => finite_kernel_borel_version (observationReference d)
(K.transition t j) (K.transition_probability t j) (hK.transition t j)
let K' : FiniteKernelLearner d M T := ⟨K.initial,K.initial_probability,
fun t j obs => κ t j obs,fun t j obs => (hκ t j).isProbabilityMeasure obs,
K.output,K.output_probability⟩
exact ⟨K',⟨fun t j => (κ t j).measurable⟩,rfl,rfl,he⟩
end FiniteKernelLearner
variable {J O : Type*} [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J]
[MeasurableSpace O]
def coinRun (F : ℕ → J → O → I → J) :
(n : ℕ) → J → (Fin n → O) → (Fin n → I) → J
| 0, j, _, _ => j
| n+1, j, obs, u => F (n+1) (coinRun F n j (Fin.init obs) (Fin.init u)) (obs (Fin.last n)) (u (Fin.last n))
omit [MeasurableSpace O] in
lemma measurable_coinRun [MeasurableSpace O] {F : ℕ → J → O → I → J}
(hF : ∀ t j obs, Measurable (F t j obs)) (n : ℕ) (j : J) (obs : Fin n → O) :
Measurable (coinRun F n j obs) := by
induction n with
| zero => exact measurable_const
| succ n ih =>
change Measurable (fun u : Fin (n+1) → I => F (n+1) (coinRun F n j (Fin.init obs) (Fin.init u)) (obs (Fin.last n)) (u (Fin.last n)))
have hi : Measurable (fun u : Fin (n+1) → I => Fin.init u) := by
change Measurable (fun u : Fin (n+1) → I => fun i : Fin n => u i.castSucc)
fun_prop
have hg : Measurable (fun u : Fin (n+1) → I => coinRun F n j (Fin.init obs) (Fin.init u)) :=
(ih (Fin.init obs)).comp hi
exact measurable_select (h := fun a (u : Fin (n+1) → I) => F (n+1) a (obs (Fin.last n)) (u (Fin.last n)))
(fun a => (hF (n+1) a (obs (Fin.last n))).comp (measurable_pi_apply _)) hg
lemma lintegral_coinRun {F : ℕ → J → O → I → J} {P : ℕ → J → O → Measure J}
(hF : ∀ t j obs, Measurable (F t j obs))
(hP : ∀ t j obs, volume.map (F t j obs) = P t j obs)
(n : ℕ) (j : J) (obs : Fin n → O) (f : J → ℝ≥0∞) :
(∫⁻ u, f (coinRun F n j obs u) ∂Measure.pi (fun _ => (volume : Measure I))) =
∫⁻ a, f a ∂kernelRun P n j obs := by
induction n generalizing f with
| zero => simp [coinRun,kernelRun]
| succ n ih =>
rw [lintegral_pi_snoc volume (fun u => f (coinRun F (n+1) j obs u))
((measurable_of_finite f).comp (measurable_coinRun hF (n+1) j obs))]
simp only [coinRun,Fin.init_snoc,Fin.snoc_last]
have hh (a : J) : (∫⁻ u, f (F (n+1) a (obs (Fin.last n)) u) ∂volume) =
∫⁻ b, f b ∂P (n+1) a (obs (Fin.last n)) := by
rw [←hP]
exact (lintegral_map (measurable_of_finite f) (hF _ _ _)).symm
simp_rw [hh]
rw [ih (Fin.init obs) (fun a => ∫⁻ b, f b ∂P (n+1) a (obs (Fin.last n)))]
exact (Measure.lintegral_bind (measurable_of_finite _).aemeasurable
(measurable_of_finite f).aemeasurable).symm
lemma coinRun_law {F : ℕ → J → O → I → J} {P : ℕ → J → O → Measure J}
(hF : ∀ t j obs, Measurable (F t j obs))
(hP : ∀ t j obs, volume.map (F t j obs) = P t j obs)
(n : ℕ) (j : J) (obs : Fin n → O) :
(Measure.pi (fun _ => (volume : Measure I))).map (coinRun F n j obs) = kernelRun P n j obs := by
apply Measure.ext
intro s hs
have hh := lintegral_coinRun hF hP n j obs (s.indicator (fun _ => (1 : ℝ≥0∞)))
rw [← lintegral_map (by fun_prop) (measurable_coinRun hF n j obs)] at hh
simpa only [lintegral_indicator_const hs,one_mul] using hh
omit [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J] [MeasurableSpace O] in
lemma kernelRun_congr [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J]
[MeasurableSpace O] {P P' : ℕ → J → O → Measure J} (n : ℕ) (j : J) (obs : Fin n → O)
(h : ∀ t a i, P t a (obs i) = P' t a (obs i)) :
kernelRun P n j obs = kernelRun P' n j obs := by
induction n with
| zero => rfl
| succ n ih =>
simp only [kernelRun]
rw [ih (Fin.init obs) (fun t a i => h t a i.castSucc)]
congr 1
funext a
exact h _ _ _
end NoiselessRegression
end
end OAI
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