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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactCoreProgram
namespace OAI
/-! # The actual finite prefixes of the fixed exact-sampling program -/
namespace ContingencyTables.ExactCoreProgram
open MatchingFPRAS.PhysicalStream ProfilePrograms
attribute [local irreducible] rareBranch exactOrdinary correctionPrefix exactPrecision
lemma bool_mean (p : Experiment Bool) (a b : ℚ) :
p.mean (fun x => if x then b else a)=
(1-p.mean (fun x => if x then 1 else 0))*a+
p.mean (fun x => if x then 1 else 0)*b := by
induction p with
| done x => cases x <;> simp [Experiment.mean]
| delay n p ih => exact ih
| read p ih =>
simp only [Experiment.mean,ih]
ring
lemma prefix_mean (z : DensePrograms.CompletionMargins) (N : ℕ)
(g : Option DensePrograms.MatrixCode → ℚ) :
(prefixExperiment z N).mean g=
(1-1/(2:ℚ)^(exactPrecision z))*((exactOrdinary z).map some).mean g+
(1/(2:ℚ)^(exactPrecision z))*(correctionPrefix z N).mean g := by
rw [prefixExperiment,Experiment.mean_bind]
have he (b : Bool) :
(if b then correctionPrefix z N else (exactOrdinary z).map some).mean g=
if b then (correctionPrefix z N).mean g else ((exactOrdinary z).map some).mean g := by
cases b <;> simp only [Bool.false_eq_true,ite_false,ite_true]
simp_rw [he]
rw [bool_mean,rareBranch_probability]
end ContingencyTables.ExactCoreProgram
namespace ContingencyTables.SmallGraphProfiles
open ProfilePrograms DensePrograms MatchingFPRAS MatchingFPRAS.PhysicalStream
open ResidualMixture FirstSuccess CompletionCounts
attribute [local irreducible] scheduledTableDraw exactOrdinary exactOrdinaryWord
exactSamplingParameters ExactCoreProgram.prefixExperiment correctionPrefix
open scoped BigOperators Classical
noncomputable section
variable {m n : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
local notation "d" => dimensionAllowance (I := Fin m) (J := Fin n)
local notation "D" => correctionPrecision r c
local notation "q" => largeTailCount r (d^20)*largeTailCount c (d^20)
lemma exactOrdinary_mean (htotal : (∑ i,r i)=∑ j,c j) (g : MatrixCode → ℚ) :
(exactOrdinary (List.ofFn r,List.ofFn c)).mean g=
average (fun bits : Bits (finiteSamplerBits r c (2*D)) =>
g (matrixCode (scheduledTableDraw r c htotal (2*D) bits).val)) := by
rw [exactOrdinary,FiniteWordExecution.experiment_mean,exactOrdinaryBits_ofFn]
simp_rw [exactOrdinaryWord_code r c htotal]
lemma table_event_average {L : ℕ} (draw : Bits L → Table r c) (X : Table r c) :
average (fun bits => if matrixCode (draw bits).val=matrixCode X.val then (1:ℚ) else 0)=
(mapLaw (uniformLaw (α := Bits L)) draw).mass X := by
have he (Y : Table r c) : matrixCode Y.val=matrixCode X.val ↔ Y=X := by
constructor
· intro h
apply Subtype.ext
change List.ofFn (fun i => List.ofFn (Y.val i))=List.ofFn (fun i => List.ofFn (X.val i)) at h
funext i
exact List.ofFn_injective (congrFun (List.ofFn_injective h) i)
· rintro rfl; rfl
simp_rw [he]
simp only [average,mapLaw,uniformLaw,uniformMass,Fintype.card_fun,Fintype.card_bool,
Fintype.card_fin,Finset.sum_div]
apply Finset.sum_congr rfl
intro bits _
split_ifs <;> simp
lemma exactOrdinary_mass (htotal : (∑ i,r i)=∑ j,c j) (X : Table r c) :
(exactOrdinary (List.ofFn r,List.ofFn c)).mean
(fun Y => if Y=matrixCode X.val then 1 else 0)=
(mapLaw (uniformLaw (α := Bits (finiteSamplerBits r c (2*D))))
(scheduledTableDraw r c htotal (2*D))).mass X :=
(exactOrdinary_mean r c htotal _).trans (table_event_average r c _ X)
lemma exactPrefix_failure
(hPL : PublishedInputs.FiniteBoxPrekopaLeindler q)
(hCheeger : PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*d^8)))
(htotal : (∑ i,r i)=∑ j,c j) (N : ℕ) :
(ExactCoreProgram.prefixExperiment (List.ofFn r,List.ofFn c) N).mean
(fun o => if o=none then 1 else 0)≤(1/(2:ℚ)^D)*(1/2:ℚ)^N := by
rw [ExactCoreProgram.prefix_mean,exactPrecision_ofFn,Experiment.mean_map]
have hz : (exactOrdinary (List.ofFn r,List.ofFn c)).mean
(fun a => if some a=none then (1:ℚ) else 0)=0 := by
rw [exactOrdinary_mean r c htotal]
simp only [reduceCtorEq,ite_false,average_const]
rw [hz,mul_zero,zero_add]
exact mul_le_mul_of_nonneg_left (correctionPrefix_failure r c hPL hCheeger htotal N)
(by positivity)
lemma exactPrefix_limit
(hPL : PublishedInputs.FiniteBoxPrekopaLeindler q)
(hCheeger : PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*d^8)))
(htotal : (∑ i,r i)=∑ j,c j) (X : Table r c) :
Filter.Tendsto (fun N => ((ExactCoreProgram.prefixExperiment (List.ofFn r,List.ofFn c) N).mean
(fun o => if o=some (matrixCode X.val) then 1 else 0):ℝ))
Filter.atTop (nhds ((Fintype.card (Table r c):ℝ)⁻¹)) := by
have ht := corrected_prefix_limit r c hPL hCheeger htotal X
convert ht using 1
funext N
rw [ExactCoreProgram.prefix_mean,exactPrecision_ofFn,Experiment.mean_map]
simp only [Option.some.injEq,exactOrdinary_mass r c htotal,Rat.cast_add,Rat.cast_mul,
Rat.cast_sub,Rat.cast_div,Rat.cast_pow,Rat.cast_one,Rat.cast_ofNat]
end
end ContingencyTables.SmallGraphProfiles
end OAI
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