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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 | import OAI.Computability.PerfectCompleteness.Construction.SourcePhysicalTaggedChildren
import OAI.Computability.PerfectCompleteness.Construction.SourceQuestionRawSwap
import OAI.Computability.PerfectCompleteness.Sampling.CandidateCoupling
namespace OAI
section
namespace PerfectCompleteness.SourcePhysicalChildExpectation
noncomputable section
open scoped Classical
open RecursiveSpaces
open UniqueGamesTheorem.Foundations.Games
variable {branch : Nat → Nat} {n t v m : Nat} {C : Type*} [Fintype C]
(rows : Nat → Nat) (clauses : Fin m → SourceClause.NormalizedClause v)
(designated : Fin (branch n) → Slots branch n)
def observe (q : PreliminarySampler.Questions branch (n + 1) t m)
(choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t))
(raw : SourceChildKernelJoint.RawTuple (C := C) (t := t) rows clauses designated) :
Σ flags : SourceProjectedTag.Flags (branch := branch) (n := n),
CutChildGrouping.Raw (C := C)
(SourceProjectedTag.mixedInside clauses designated q choices flags) rows :=
SourcePhysicalTaggedChildren.observe rows clauses designated
(SourceQuestionKernelJoint.sources designated q choices) raw
@[simp] theorem observe_flags
(q : PreliminarySampler.Questions branch (n + 1) t m)
(choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t))
(raw : SourceChildKernelJoint.RawTuple (C := C) (t := t) rows clauses designated) :
(observe rows clauses designated q choices raw).1 =
fun child => SourceChildKernel.rawProjected rows clauses designated child (raw child) := rfl
theorem observe_sample_heq
(sample : SourceChildKernel.Sample (C := C) (t := t) rows clauses designated) :
HEq (observe rows clauses designated
(SourceQuestionRawSwap.insideQuestions rows clauses designated sample)
(SourceQuestionRawSwap.positions rows clauses designated sample)
(SourceQuestionRawSwap.raw rows clauses designated sample))
(SourcePhysicalTaggedChildren.observe rows clauses designated
(fun child => ((sample child).1, (sample child).2.1))
(fun child => (sample child).2.2)) := by
have congrObserve (sources other : SourceChildKernel.Sources (m := m) (t := t) designated)
(hs : sources = other)
(raw : SourceChildKernelJoint.RawTuple (C := C) (t := t) rows clauses designated) :
HEq (SourcePhysicalTaggedChildren.observe rows clauses designated sources raw)
(SourcePhysicalTaggedChildren.observe rows clauses designated other raw) := by
cases hs
rfl
exact congrObserve _ _
(SourceQuestionRawSwap.sources_insideQuestions_positions rows clauses designated sample)
(SourceQuestionRawSwap.raw rows clauses designated sample)
theorem kernel_expectation
(flag : Fin (branch n) → FiniteDistribution Bool)
(q : PreliminarySampler.Questions branch (n + 1) t m)
(choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t))
(F : (flags : SourceProjectedTag.Flags (branch := branch) (n := n)) →
CutChildGrouping.Raw (C := C)
(SourceProjectedTag.mixedInside clauses designated q choices flags) rows → ℝ) :
(SourceQuestionRawSwap.kernelLaw (C := C) rows clauses designated flag q choices).expectation
(fun raw =>
let observed := observe rows clauses designated q choices raw
F observed.1 observed.2) =
(SourceProjectedTag.flagLaw flag).expectation (fun flags =>
(CutChildGrouping.rawLaw (C := C)
(SourceProjectedTag.mixedInside clauses designated q choices flags) rows).expectation
(F flags)) := by
let statistic :
(Σ flags : SourceProjectedTag.Flags (branch := branch) (n := n),
CutChildGrouping.Raw (C := C)
(SourceProjectedTag.mixedInside clauses designated q choices flags) rows) → ℝ :=
fun observed => F observed.1 observed.2
let physical := fun flags : SourceProjectedTag.Flags (branch := branch) (n := n) =>
CutChildGrouping.rawLaw (C := C)
(SourceProjectedTag.mixedInside clauses designated q choices flags) rows
have hlaw :
(SourceQuestionRawSwap.kernelLaw (C := C) rows clauses designated flag q choices).pushforward
(observe rows clauses designated q choices) =
CompletionSoundness.sigmaLaw (SourceProjectedTag.flagLaw flag) physical :=
SourcePhysicalTaggedChildren.kernels_tagged (C := C) rows clauses designated flag
(SourceQuestionKernelJoint.sources designated q choices)
calc
_ = ((SourceQuestionRawSwap.kernelLaw (C := C) rows clauses designated flag q choices).pushforward
(observe rows clauses designated q choices)).expectation statistic :=
(FiniteDistribution.expectation_pushforward
(SourceQuestionRawSwap.kernelLaw (C := C) rows clauses designated flag q choices)
(observe rows clauses designated q choices) statistic).symm
_ = (CompletionSoundness.sigmaLaw (SourceProjectedTag.flagLaw flag) physical).expectation
statistic := congrArg (fun μ => FiniteDistribution.expectation μ statistic) hlaw
_ = _ := CandidateCoupling.expectation_sigmaLaw
(SourceProjectedTag.flagLaw flag) physical statistic
theorem originalLaw_expectation [NeZero m]
(flag : Fin (branch n) → FiniteDistribution Bool)
(F : (q : PreliminarySampler.Questions branch (n + 1) t m) →
(choices : SourceQuestionKernelJoint.ChoiceTuple (branch := branch) (n := n) (t := t)) →
(flags : SourceProjectedTag.Flags (branch := branch) (n := n)) →
CutChildGrouping.Raw (C := C)
(SourceProjectedTag.mixedInside clauses designated q choices flags) rows → ℝ) :
(SourceChildKernel.originalLaw (C := C) (t := t) rows clauses designated flag).expectation
(fun sample =>
let q := SourceQuestionRawSwap.insideQuestions rows clauses designated sample
let choices := SourceQuestionRawSwap.positions rows clauses designated sample
let observed := observe rows clauses designated q choices
(SourceQuestionRawSwap.raw rows clauses designated sample)
F q choices observed.1 observed.2) =
(PreliminarySampler.questionsLaw
(branch := branch) (n := n + 1) (t := t) (m := m)).expectation
(fun q => (SourceProjectedTag.positionLaw
(branch := branch) (n := n) (t := t)).expectation
(fun choices => (SourceProjectedTag.flagLaw flag).expectation
(fun flags => (CutChildGrouping.rawLaw (C := C)
(SourceProjectedTag.mixedInside clauses designated q choices flags) rows).expectation
(F q choices flags)))) := by
rw [SourceQuestionRawSwap.originalLaw_expectation rows clauses designated flag
(fun q choices raw =>
let observed := observe rows clauses designated q choices raw
F q choices observed.1 observed.2)]
apply FiniteDistribution.expectation_congr
intro q
apply FiniteDistribution.expectation_congr
intro choices
exact kernel_expectation rows clauses designated flag q choices (F q choices)
end
end PerfectCompleteness.SourcePhysicalChildExpectation
end
end OAI
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