File size: 14,477 Bytes
2f887db | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 | import OAI.Computability.PerfectCompleteness.Foundations.HierarchicalProjectedRawMeeting
import OAI.Computability.PerfectCompleteness.Foundations.TupleIndexEmitterLemmas
namespace OAI
section
namespace PerfectCompleteness.HierarchicalProjectedRawAverage
noncomputable section
open scoped BigOperators Classical TensorProduct
open TreeSourceSpaces HierarchicalArrays UpperParameterScalars
open UniqueGamesTheorem.Foundations.Games
open UniqueGamesTheorem.Appendix.RankLevelFilter (linearMapFintype)
attribute [local instance] linearMapFintype
private theorem event_mass_le_expectation {X : Type*} [Fintype X]
(μ : FiniteDistribution X) (event : X → Bool) (value : X → ℝ) (factor : ℝ)
(hnonneg : ∀ x, 0 ≤ value x)
(hbound : ∀ x, event x = true → factor ≤ value x) :
factor * μ.probability event ≤ μ.expectation value := by
unfold FiniteDistribution.probability FiniteDistribution.expectation
rw [Finset.mul_sum]
apply Finset.sum_le_sum
intro x _
by_cases hx : event x = true
· rw [ite_eq_left hx]
calc
factor * μ.weight x = μ.weight x * factor := mul_comm _ _
_ ≤ μ.weight x * value x :=
mul_le_mul_of_nonneg_left (hbound x hx) (μ.nonnegative x)
· rw [ite_eq_right hx, mul_zero]
exact mul_nonneg (μ.nonnegative x) (hnonneg x)
variable {δ : ℚ} (plan : FixedRows.Plan δ) {branch : Nat → Nat} {t : Nat}
(slots projected : RecursiveSpaces.Slots branch plan.depth → Fin t → MixedSupport.Slot)
(projection : ∀ s j, MixedSupport.Projection (slots s j) (projected s j))
(upper lower : Nodes branch plan.depth) (lowerLevel : Nat)
{Ω : Type*} [Fintype Ω]
(original : FiniteDistribution Ω) (originalArrays : Ω → Arrays slots (FixedRows.rows plan))
(lowerEvent : Ω → Bool)
(σ : KeyStrategy.Strategy (TreeCanonical.locationCount branch plan.depth t))
(A : ManyGoodRows.RowMap (Block (FixedRows.rows plan) upper) plan.order)
(a : Block (FixedRows.rows plan) lower) (cut : OwnInputReference.Cut upper lower)
abbrev Raw := HierarchicalProjectedRawPrediction.Raw projected (FixedRows.rows plan) upper lower A
(FixedRows.repeats plan) cut
abbrev Visible := HierarchicalProjectedRawPrediction.Visible projected (FixedRows.rows plan) upper lower A
(FixedRows.repeats plan) cut
def law (externalLaw : FiniteDistribution
(OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) :
FiniteDistribution (Raw plan projected upper lower A cut) :=
OwnInputReference.rawLaw projected (FixedRows.rows plan) upper lower (LinearMap.ker A)
(FixedRows.repeats plan) cut externalLaw
abbrev J := HierarchicalProjectedRawPrediction.rawJ slots projected projection (FixedRows.rows plan) upper lower lowerLevel
original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A
(FixedRows.repeats plan) cut
abbrev prediction := HierarchicalProjectedRawPrediction.rawPrediction slots projected projection (FixedRows.rows plan) upper lower lowerLevel
original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A
(FixedRows.repeats plan) cut a
def goodRecord (externalLaw : FiniteDistribution
(OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower))
(v : Visible plan projected upper lower A cut) : Bool :=
PositiveFiberMass.goodRecord (law plan projected upper lower A cut externalLaw)
(fun sample : Raw plan projected upper lower A cut => sample.2)
(J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut)
(prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut)
(rhoPred (InitialParameters.useful δ) plan.density) v
def goodEvent (externalLaw : FiniteDistribution
(OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower))
(sample : Raw plan projected upper lower A cut) : Bool :=
J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample &&
goodRecord plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut
externalLaw sample.2
def factor : ℝ :=
(1 / (2 : ℝ) ^ plan.order) *
(tau (InitialParameters.useful δ) plan.density plan.order
(FixedRows.rows plan (Nodes.height upper)) ^ 2 /
(2 * (2 : ℝ) ^ FixedRows.rows plan (Nodes.height upper)))
theorem factor_nonneg : 0 ≤ factor plan upper := by
unfold factor
positivity
abbrev meetingProbability :=
HierarchicalProjectedRawMeeting.meetingProbability plan slots projected projection upper lower lowerLevel
original originalArrays lowerEvent σ A a cut
def expectedMeeting (externalLaw : FiniteDistribution
(OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) : ℝ :=
(law plan projected upper lower A cut externalLaw).expectation
(fun sample => meetingProbability plan slots projected projection upper lower lowerLevel
original originalArrays lowerEvent σ A a cut sample.2)
theorem meeting_lower_of_goodRecord (hδ : 0 < δ)
(externalLaw : FiniteDistribution
(OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower))
(v : Visible plan projected upper lower A cut)
(hgood : goodRecord plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ
A a cut externalLaw v = true) :
factor plan upper ≤
meetingProbability plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut v := by
have hpositive := (PositiveFiberMass.goodRecord_iff
(law plan projected upper lower A cut externalLaw)
(fun sample : Raw plan projected upper lower A cut => sample.2)
(J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut)
(prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut)
(rhoPred (InitialParameters.useful δ) plan.density) v).mp hgood
let originalGiven : Raw plan projected upper lower A cut → Bool := fun sample =>
J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample &&
@decide (sample.2 = v) (Classical.propDecidable _)
have hgiven : originalGiven =
HierarchicalProjectedRawPrediction.given slots projected projection (FixedRows.rows plan) upper lower lowerLevel
original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A
(FixedRows.repeats plan) cut v := by
funext sample
unfold originalGiven HierarchicalProjectedRawPrediction.given
apply congrArg (fun flag : Bool =>
J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample && flag)
exact (@decide_eq_decide _ _ _ _).mpr Iff.rfl
have positive : 0 < (law plan projected upper lower A cut externalLaw).probability
(HierarchicalProjectedRawPrediction.given slots projected projection (FixedRows.rows plan) upper lower lowerLevel
original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A
(FixedRows.repeats plan) cut v) := by
rw [← hgiven]
exact hpositive.1
have conditional : rhoPred (InitialParameters.useful δ) plan.density ≤
((law plan projected upper lower A cut externalLaw).condition
(HierarchicalProjectedRawPrediction.given slots projected projection (FixedRows.rows plan) upper lower lowerLevel
original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A
(FixedRows.repeats plan) cut v) positive).probability
(prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut) := by
have hprediction := hpositive.2
have hmatch : (fun sample : Raw plan projected upper lower A cut =>
originalGiven sample &&
prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut sample) =
(fun sample =>
(J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample &&
prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut sample) &&
@decide (sample.2 = v) (Classical.propDecidable _)) := by
funext sample
dsimp only [originalGiven]
cases J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample <;>
cases prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut sample <;>
cases @decide (sample.2 = v) (Classical.propDecidable _) <;> rfl
change rhoPred (InitialParameters.useful δ) plan.density ≤
(law plan projected upper lower A cut externalLaw).probability
(fun sample =>
(J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample &&
prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut sample) &&
@decide (sample.2 = v) (Classical.propDecidable _)) /
(law plan projected upper lower A cut externalLaw).probability originalGiven at hprediction
rw [← hmatch, hgiven] at hprediction
exact hprediction.trans_eq
(FiniteDistribution.probability_condition (law plan projected upper lower A cut externalLaw)
(HierarchicalProjectedRawPrediction.given slots projected projection (FixedRows.rows plan) upper lower lowerLevel
original originalArrays lowerEvent (InitialParameters.useful δ) σ plan.density A
(FixedRows.repeats plan) cut v)
(prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut)
positive).symm
exact HierarchicalProjectedRawMeeting.meeting_probability_lower plan slots projected projection upper lower lowerLevel
original originalArrays lowerEvent σ A a cut v hδ externalLaw positive conditional
theorem expectedMeeting_ge_good_mass (hδ : 0 < δ)
(externalLaw : FiniteDistribution
(OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) :
factor plan upper *
(law plan projected upper lower A cut externalLaw).probability
(goodEvent plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut externalLaw) ≤
expectedMeeting plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut externalLaw := by
apply event_mass_le_expectation
· intro sample
exact HierarchicalProjectedRawMeeting.meetingProbability_nonneg plan slots projected projection upper lower lowerLevel
original originalArrays lowerEvent σ A a cut sample.2
· intro sample hs
exact meeting_lower_of_goodRecord plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ
A a cut hδ externalLaw sample.2 (Bool.and_eq_true_iff.mp hs).2
theorem J_and_prediction :
(fun sample => J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut sample &&
prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut sample) =
prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut := by
funext sample
unfold J prediction HierarchicalProjectedRawPrediction.rawJ HierarchicalProjectedRawPrediction.rawPrediction
HierarchicalPrediction.prediction
exact Bool.and_self_left _ _
theorem prediction_difference_le_good_mass (hδ : 0 < δ)
(externalLaw : FiniteDistribution
(OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) :
(law plan projected upper lower A cut externalLaw).probability
(prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut) -
rhoPred (InitialParameters.useful δ) plan.density *
(law plan projected upper lower A cut externalLaw).probability
(J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut) ≤
(law plan projected upper lower A cut externalLaw).probability
(goodEvent plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut externalLaw) := by
have h := PositiveFiberMass.difference_le_good_mass
(law plan projected upper lower A cut externalLaw)
(fun sample : Raw plan projected upper lower A cut => sample.2)
(J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut)
(prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut)
(rhoPred (InitialParameters.useful δ) plan.density)
(rhoPred_pos (InitialParameters.useful_pos hδ) plan.density_pos).le
rw [J_and_prediction] at h
exact h
theorem expectedMeeting_ge_prediction_difference (hδ : 0 < δ)
(externalLaw : FiniteDistribution
(OwnInputReference.Exterior projected (FixedRows.rows plan) upper lower)) :
factor plan upper *
((law plan projected upper lower A cut externalLaw).probability
(prediction plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut) -
rhoPred (InitialParameters.useful δ) plan.density *
(law plan projected upper lower A cut externalLaw).probability
(J plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A cut)) ≤
expectedMeeting plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ A a cut externalLaw :=
(mul_le_mul_of_nonneg_left
(prediction_difference_le_good_mass plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ
A a cut hδ externalLaw) (factor_nonneg plan upper)).trans
(expectedMeeting_ge_good_mass plan slots projected projection upper lower lowerLevel original originalArrays lowerEvent σ
A a cut hδ externalLaw)
end
end PerfectCompleteness.HierarchicalProjectedRawAverage
end
end OAI
|