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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 99 100 101 102 103 104 | import OAI.Combinatorics.ContingencyTables.Sampling.AllSmallBoolean
import OAI.Combinatorics.ContingencyTables.Sampling.AllSmallInitialization
import OAI.Combinatorics.ContingencyTables.Sampling.FirstPaperInitialization
import OAI.Combinatorics.ContingencyTables.Sampling.FirstPaperBinaryParameters
namespace OAI
/-! # Complete finite rejection sampling when the large rectangle is empty -/
namespace ContingencyTables.SmallGraphProfiles
open scoped BigOperators Classical
open ResidualMixture FirstSuccess FairBits 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 "d" => dimensionAllowance (I := I) (J := J)
local notation "s" => smallProposalBits (I := I) (J := J)
local instance : DecidableEq (PaperStates r c) := Classical.decEq _
/-- Each trial reads a disjoint word; failed trials use the feasible fallback. -/
def unitOuterDraw (fallback : (z : PaperStates r c) → PaperFibre r c z)
(x : PaperStates r c) (n M : ℕ) (z₀ : Table r c)
(bits : Fin (M*(n*s)) → Bool) : Table r c :=
wordRetry (fun w => stateTrial r c (unitJointDraw r c fallback x n w)) z₀ M
(GridBoundary.gridWordEquiv M (n*s) bits)
lemma unitOuterDraw_law (fallback : (z : PaperStates r c) → PaperFibre r c z)
(x : PaperStates r c) (n M : ℕ) (z₀ : Table r c) :
mapLaw (uniformLaw (α := Fin (M*(n*s)) → Bool))
(unitOuterDraw r c fallback x n M z₀)=
retryLaw (mapLaw (uniformLaw (α := Fin (n*s) → Bool))
(unitJointDraw r c fallback x n)) (stateTrial r c) z₀ M := by
unfold unitOuterDraw
rw [←map_equiv (GridBoundary.gridWordEquiv M (n*s)),uniform_equiv]
change wordRetryLaw _ _ _=_
rw [wordRetryLaw_eq,←retryLaw_map]
lemma unitOuterDraw_error [Nonempty I] [Nonempty J] [Nonempty (PaperStates r c)]
[Nonempty (Table r c)] (hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true)
(htotal : (∑ i,r i)=∑ j,c j)
(fallback : (z : PaperStates r c) → PaperFibre r c z)
(x : PaperStates r c) (n M : ℕ) (z₀ : Table r c) :
(variation (mapLaw (uniformLaw (α := Fin (M*(n*s)) → Bool))
(unitOuterDraw r c fallback x n M z₀)) (uniformLaw (α := Table r c)):ℝ) ≤
Real.exp (-(M:ℝ)/(2*(1+(d:ℝ)^2)))+
M*(((stateBound r c+1:ℕ):ℝ)^(2*d)*Real.exp (-(n:ℝ)/(d:ℝ)^160)) := by
have hd : 14 ≤ d := by
have hi := Fintype.card_pos (α := I)
have hj := Fintype.card_pos (α := J)
unfold dimensionAllowance
nlinarith
let p := mapLaw (uniformLaw (α := Fin (n*s) → Bool)) (unitJointDraw r c fallback x n)
let q := uniformLaw (α := PaperStateJoint r c)
have hj := unitJointDraw_variation r c hsmall fallback x n
have hr := retryLaw_variation p q (stateTrial r c) z₀ M
have hi : (variation (retryLaw q (stateTrial r c) z₀ M)
(uniformLaw (α := Table r c)):ℝ) ≤ Real.exp (-(M:ℝ)/(2*(1+(d:ℝ)^2))) := by
apply uniform_partial_retry_exp (StateJointSuccess r c) (originalStateSuccessEquiv r c)
z₀ M (2*(1+(d:ℝ)^2))
rw [paperStateJoint_card]
exact stationary_success_ratio r c htotal hd
have ht := variation_triangle (retryLaw p (stateTrial r c) z₀ M)
(retryLaw q (stateTrial r c) z₀ M) (uniformLaw (α := Table r c))
rw [unitOuterDraw_law]
have hrR : (variation (retryLaw p (stateTrial r c) z₀ M)
(retryLaw q (stateTrial r c) z₀ M):ℝ) ≤ M*(variation p q:ℝ) := by
exact_mod_cast hr
have htR := (Rat.cast_le (K := ℝ)).mpr ht
simp only [Rat.cast_add] at htR
exact htR.trans (by nlinarith)
/-- The all-small branch reaches the same requested error as the branch
with a nonempty large rectangle, using no completion-sampling oracle. -/
theorem unitOuterDraw_accuracy [Nonempty I] [Nonempty J] [Nonempty (PaperStates r c)]
[Nonempty (Table r c)] (hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true)
(htotal : (∑ i,r i)=∑ j,c j) (k : ℕ) (hk : 1 ≤ k)
(fallback : (z : PaperStates r c) → PaperFibre r c z)
(x : PaperStates r c) (z₀ : Table r c) :
(variation (mapLaw (uniformLaw (α := Fin (outerTrials d k*(outerSteps d (smallBinaryLength r c) k*s)) → Bool))
(unitOuterDraw r c fallback x (outerSteps d (smallBinaryLength r c) k) (outerTrials d k) z₀))
(uniformLaw (α := Table r c)):ℝ) ≤ 1/(2:ℝ)^k := by
have hd : 14 ≤ d := by
have hi := Fintype.card_pos (α := I)
have hj := Fintype.card_pos (α := J)
unfold dimensionAllowance
nlinarith
have he := outer_total_error d (smallBinaryLength r c) k (stateBound r c) hd
(smallBinaryLength_parameters r c).2.1 hk (smallBinaryLength_parameters r c).2.2
have hv := unitOuterDraw_error r c hsmall htotal fallback x
(outerSteps d (smallBinaryLength r c) k) (outerTrials d k) z₀
apply hv.trans
have hn : 0 ≤ (outerTrials d k:ℝ)*(outerSteps d (smallBinaryLength r c) k+1)/
(2:ℝ)^(outerAccuracy d (smallBinaryLength r c) k) :=
div_nonneg (mul_nonneg (Nat.cast_nonneg _) (by
exact add_nonneg (Nat.cast_nonneg _) zero_le_one)) (le_of_lt (pow_pos (by norm_num) _))
nlinarith
end
end ContingencyTables.SmallGraphProfiles
end OAI
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