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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.CoefficientStep
import OAI.Probability.MemoryPrecision.StochasticContinuation
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)
lemma measurable_angleTest (ε : ℝ) : Measurable (angleTest (d := d) ε) := by
unfold angleTest
exact Measurable.ite (measurableSet_le (by fun_prop) measurable_const)
measurable_const measurable_const
lemma measurable_chordTest (ε : ℝ) : Measurable (chordTest (d := d) ε) := by
apply Measurable.indicator measurable_const
exact measurableSet_le (by fun_prop) measurable_const
lemma angleTest_le_one (ε : ℝ) (x : Sphere d × Sphere d) : angleTest ε x ≤ 1 := by
unfold angleTest
split <;> simp
lemma chordTest_le_one (ε : ℝ) (x : Sphere d × Sphere d) : chordTest ε x ≤ 1 := by
exact Set.indicator_le_self' (fun _ _ => bot_le) x
lemma outputTest_measurable {f : Sphere d × Sphere d → ℝ≥0∞} (hf : Measurable f)
(t : ℕ) (j : State M) : Measurable (K.outputTest f t j) := by
exact (show Measurable (Function.uncurry (fun s v => f (s,v))) from hf).lintegral_prod_right
lemma outputTest_le_one {f : Sphere d × Sphere d → ℝ≥0∞} (hf : ∀ x, f x ≤ 1)
(t : ℕ) (j : State M) (s : Sphere d) : K.outputTest f t j s ≤ 1 := by
exact (lintegral_mono (fun v => hf (s,v))).trans_eq (by simp)
def marginal (n : ℕ) (j : State M) (obs : Vector d × ℝ) (a : State M) : ℝ≥0∞ :=
∑ b : Bool, K.transition n j obs (b,a)
lemma marginal_sum (n : ℕ) (j : State M) (obs : Vector d × ℝ) :
∑ a, K.marginal n j obs a = 1 := by
simp only [marginal]
rw [Finset.sum_comm, ← Fintype.sum_prod_type]
exact K.transition_sum n j obs
lemma marginal_measurable (hK : K.Borel) (n : ℕ) (j a : State M) :
Measurable (fun obs => K.marginal n j obs a) := by
exact Finset.measurable_sum _ (fun b _ => hK n j (b,a))
section Evaluation
variable {X : Type*} [MeasurableSpace X]
variable (w : ℕ → State M → X → ℝ≥0∞) (ι : X → Vector d)
lemma measurable_eval (hK : K.Borel) (hw : ∀ n j, Measurable (w n j))
(n k : ℕ) (j : State M) :
Measurable (fun z : X × Observations d k => K.eval w n k j z.1 z.2) := by
induction k generalizing n j with
| zero => exact (hw n j).comp measurable_fst
| succ k ih =>
apply Finset.measurable_sum _
intro a _
apply ((hK (n+1) j a).comp (by fun_prop)).mul
cases a with
| mk b a =>
cases b
· exact (ih (n+1) a).comp (measurable_fst.prodMk
(Measurable.of_eval (fun index => (measurable_pi_apply index.succ).comp measurable_snd)))
· exact (hw (n+1) a).comp measurable_fst
omit [MeasurableSpace X] in
lemma eval_le_one (hw : ∀ n j s, w n j s ≤ 1) (n k : ℕ) (j : State M)
(s : X) (obs : Observations d k) : K.eval w n k j s obs ≤ 1 := by
induction k generalizing n j with
| zero => exact hw n j s
| succ k ih =>
apply (Finset.sum_le_sum (fun a _ => mul_le_of_le_one_right' ?_)).trans_eq
(K.transition_sum (n+1) j (obs 0))
split
· exact hw _ _ _
· exact ih _ _ _
omit [MeasurableSpace X] in
lemma pathValue_le_one (hw : ∀ n j s, w n j s ≤ 1) (s : X) (obs : Observations d T) :
K.pathValue w s obs ≤ 1 := by
apply (Finset.sum_le_sum (fun a _ => mul_le_of_le_one_right' ?_)).trans_eq K.initial_sum
split
· exact hw _ _ _
· exact K.eval_le_one w hw _ _ _ _ _
def cont (n : ℕ) : ℕ → State M → X → ℝ≥0∞
| 0, j, s => w n j s
| k+1, j, s => ∫⁻ x, ∑ a : Action M, K.transition (n+1) j (x,⟪ι s,x⟫) a *
if a.1 then w (n+1) a.2 s else cont (n+1) k a.2 s ∂stdGaussian (Vector d)
lemma measurable_cont (hK : K.Borel) (hw : ∀ n j, Measurable (w n j))
(hι : Measurable ι) (n k : ℕ) (j : State M) : Measurable (K.cont w ι n k j) := by
induction k generalizing n j with
| zero => exact hw n j
| succ k ih =>
apply Measurable.lintegral_prod_left
apply Finset.measurable_sum _
intro a _
apply ((hK (n+1) j a).comp (by fun_prop)).mul
rcases a with ⟨b,a⟩
cases b
· exact (ih (n+1) a).comp measurable_snd
· exact (hw (n+1) a).comp measurable_snd
omit [MeasurableSpace X] in
lemma cont_le_forced_sum (hK : K.Borel) (n k : ℕ) (j : State M) (s : X) :
K.cont w ι n k j s ≤ ∑ r ∈ Finset.range (k+1),
MemoryPrecision.stochCont K.marginal w ι n r j s := by
classical
induction k generalizing n j with
| zero => simp [cont, MemoryPrecision.stochCont]
| succ k ih =>
let V (a : State M) := ∑ r ∈ Finset.range (k+1),
MemoryPrecision.stochCont K.marginal w ι (n+1) r a s
have hwV (a : State M) : w (n+1) a s ≤ V a := by
simpa [MemoryPrecision.stochCont] using
(Finset.single_le_sum (fun r _ => bot_le (a := MemoryPrecision.stochCont K.marginal w ι (n+1) r a s))
(show 0 ∈ Finset.range (k+1) by simp))
calc
_ ≤ ∫⁻ x, ∑ a : Action M, K.transition (n+1) j (x,⟪ι s,x⟫) a * V a.2
∂stdGaussian (Vector d) := lintegral_mono (fun x => Finset.sum_le_sum (fun a _ => by
apply mul_le_mul' le_rfl
split
· exact hwV a.2
· exact ih (n+1) a.2))
_ = ∑ r ∈ Finset.range (k+1), MemoryPrecision.stochCont K.marginal w ι n (r+1) j s := by
simp only [MemoryPrecision.stochCont]
have hm (r : ℕ) : Measurable (fun x : Vector d => ∑ a,
K.marginal (n+1) j (x,⟪ι s,x⟫) a *
MemoryPrecision.stochCont K.marginal w ι (n+1) r a s) :=
Finset.measurable_sum _ (fun a _ =>
((K.marginal_measurable hK (n+1) j a).comp (by fun_prop)).mul measurable_const)
rw [← lintegral_finsetSum _ (fun r _ => hm r)]
apply lintegral_congr
intro x
simp only [V, Finset.mul_sum, Fintype.sum_prod_type, marginal, Finset.sum_mul]
rw [Finset.sum_comm]
conv_lhs =>
arg 2
ext a
rw [Finset.sum_comm]
rw [Finset.sum_comm]
_ ≤ ∑ r ∈ Finset.range (k+1+1), MemoryPrecision.stochCont K.marginal w ι n r j s := by
conv_rhs => rw [Finset.sum_range_succ']
exact le_add_of_nonneg_right (show (0 : ℝ≥0∞) ≤ _ from zero_le)
end Evaluation
end Policy
end MemoryPrecision.Kernel
end
end OAI
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