File size: 4,804 Bytes
d5b7741 | 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 | import OAI.Combinatorics.ContingencyTables.Sampling.AllSmallRational
import OAI.Combinatorics.ContingencyTables.Sampling.FirstPaperBooleanProposal
import OAI.Combinatorics.ContingencyTables.Machines.FiniteWordWalk
import OAI.Combinatorics.ContingencyTables.Dense.DenseBooleanWalk
import OAI.Combinatorics.ContingencyTables.Transport.FiniteDeterministicJoint
import OAI.Combinatorics.ContingencyTables.Sampling.FirstPaperIdealRejection
namespace OAI
/-! # Boolean implementation when all cells are retained -/
namespace ContingencyTables.SmallGraphProfiles
open scoped BigOperators Classical
open ResidualMixture FirstSuccess FairBits CompletionTranslation CompletionCounts
noncomputable section
universe u
variable {I J : Type u} [Fintype I] [Fintype J] [LinearOrder I] [LinearOrder J]
variable (r : I → ℕ) (c : J → ℕ)
local notation "s" => smallProposalBits (I := I) (J := J)
local instance : DecidableEq (PaperStates r c) := Classical.decEq _
def unitBitStep (x : PaperStates r c) (bits : Fin s → Bool) : PaperStates r c :=
(smallBitProposal r c x bits).getD x
lemma unitBitStep_law (x : PaperStates r c) :
mapLaw (uniformLaw (α := Fin s → Bool)) (unitBitStep r c x)=unitSmallStep r c x := by
unfold unitBitStep
calc
_ = mapLaw (mapLaw (uniformLaw (α := Fin s → Bool)) (smallBitProposal r c x))
(fun z : Option (PaperStates r c) => z.getD x) :=
(mapLaw_comp _ (smallBitProposal r c x)
(fun z : Option (PaperStates r c) => z.getD x)).symm
_ = _ := by rw [smallBitProposal_law_decidableEq r c _ x]; rfl
def flatUnitWalk (x : PaperStates r c) (n : ℕ) (bits : Fin (n*s) → Bool) : PaperStates r c :=
wordWalk (unitBitStep r c) x n (GridBoundary.gridWordEquiv n s bits)
lemma flatUnitWalk_law (x : PaperStates r c) (n : ℕ) :
mapLaw (uniformLaw (α := Fin (n*s) → Bool)) (flatUnitWalk r c x n)=
walkLaw (unitSmallStep r c) x n := by
unfold flatUnitWalk
rw [←map_equiv (GridBoundary.gridWordEquiv n s),uniform_equiv]
change wordWalkLaw _ _ _=_
rw [wordWalkLaw_eq]
exact congrArg (fun K => walkLaw K x n) (funext (unitBitStep_law r c))
lemma uniform_singleton_joint [Nonempty (PaperStates r c)]
(hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true)
(fallback : (z : PaperStates r c) → PaperFibre r c z) :
mapLaw (uniformLaw (α := PaperStates r c))
(fun z => (⟨z,fallback z⟩ : PaperStateJoint r c))=
uniformLaw (α := PaperStateJoint r c) := by
rw [mapLaw_deterministicJoint]
have he (z : PaperStates r c) : pointLaw (fallback z)=uniformLaw (α := PaperFibre r c z) := by
let : Subsingleton (PaperFibre r c z) :=
PhysicalCompletionFibres.completionFibre_subsingleton _ _ _ _ _ _ _ hsmall
exact pointLaw_eq_uniform _
simp_rw [he]
apply dependentJointLaw_uniform
intro z
simp only [uniformLaw,uniformMass,paperFibre_card_allSmall r c hsmall,
Nat.cast_one,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one]
def unitJointDraw (fallback : (z : PaperStates r c) → PaperFibre r c z)
(x : PaperStates r c) (n : ℕ) (bits : Fin (n*s) → Bool) : PaperStateJoint r c :=
let z := flatUnitWalk r c x n bits
⟨z,fallback z⟩
lemma unitJointDraw_law (fallback : (z : PaperStates r c) → PaperFibre r c z)
(x : PaperStates r c) (n : ℕ) :
mapLaw (uniformLaw (α := Fin (n*s) → Bool)) (unitJointDraw r c fallback x n)=
mapLaw (walkLaw (unitSmallStep r c) x n)
(fun z => (⟨z,fallback z⟩ : PaperStateJoint r c)) := by
unfold unitJointDraw
calc
_ = mapLaw (mapLaw (uniformLaw (α := Fin (n*s) → Bool)) (flatUnitWalk r c x n))
(fun z => (⟨z,fallback z⟩ : PaperStateJoint r c)) :=
(mapLaw_comp _ (flatUnitWalk r c x n)
(fun z => (⟨z,fallback z⟩ : PaperStateJoint r c))).symm
_ = _ := by rw [flatUnitWalk_law]
theorem unitJointDraw_variation [Nonempty I] [Nonempty J] [Nonempty (PaperStates r c)]
(hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true)
(fallback : (z : PaperStates r c) → PaperFibre r c z)
(x : PaperStates r c) (n : ℕ) :
(variation (mapLaw (uniformLaw (α := Fin (n*s) → Bool))
(unitJointDraw r c fallback x n)) (uniformLaw (α := PaperStateJoint r c)):ℝ) ≤
((stateBound r c+1:ℕ):ℝ)^(2*dimensionAllowance (I := I) (J := J))*
Real.exp (-(n:ℝ)/(dimensionAllowance (I := I) (J := J):ℝ)^160) := by
rw [unitJointDraw_law]
have hv := variation_map (walkLaw (unitSmallStep r c) x n)
(uniformLaw (α := PaperStates r c)) (fun z => (⟨z,fallback z⟩ : PaperStateJoint r c))
rw [uniform_singleton_joint r c hsmall fallback] at hv
exact ((Rat.cast_le (K := ℝ)).mpr hv).trans (unitSmallWalk_variation r c hsmall x n)
end
end ContingencyTables.SmallGraphProfiles
end OAI
|