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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.AllSamplingSemantics
import OAI.Combinatorics.ContingencyTables.Transport.ListedScheduledSampler
namespace OAI
/-! # Accuracy of the single computed sampler for arbitrary margins -/
namespace ContingencyTables.SmallGraphProfiles
open ProfilePrograms DensePrograms CompletionCounts FirstPaperPhysicalMarginal
open ResidualMixture FirstSuccess FairBits SmallContextCoordinates
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 "b" => smallBinaryLength r c
local notation "q" => largeTailCount r (d^20)*largeTailCount c (d^20)
/-- The mathematical draw whose code is the output of the single public program. -/
def scheduledTableDraw (htotal : (∑ i,r i)=∑ j,c j) (k : ℕ)
(bits : Fin (finiteSamplerBits r c k) → Bool) : Table r c :=
if h : hasLargeRectangle ((List.ofFn r,List.ofFn c),k)=true then
listedOuterDraw r c (computedCellEquiv r c)
((hasLargeRectangle_spec r c k).mp h).1 ((hasLargeRectangle_spec r c k).mp h).2
b (outerAccuracy d b k) (smallBinaryLength_parameters r c).1
(tableState r c (GreedyFeasibleTable.table r c htotal))
(outerSteps d b k) (outerTrials d k) (GreedyFeasibleTable.table r c htotal) bits
else
unitCodeOuterDraw r c (computedCellEquiv r c) (allSmallFibre r c (allSmall_of_no_large r c k h))
(tableState r c (GreedyFeasibleTable.table r c htotal)) (outerSteps d b k) (outerTrials d k)
(GreedyFeasibleTable.table r c htotal) (prefixWord (unitCodeBits_le r c k) bits)
theorem samplingMatrixAll_code (htotal : (∑ i,r i)=∑ j,c j) (k : ℕ)
(bits : Fin (finiteSamplerBits r c k) → Bool) :
samplingMatrixAll (((List.ofFn r,List.ofFn c),k),List.ofFn bits)=
matrixCode (scheduledTableDraw r c htotal k bits).val := by
unfold samplingMatrixAll scheduledTableDraw
by_cases h : hasLargeRectangle ((List.ofFn r,List.ofFn c),k)=true
· rw [ite_eq_left h,dite_eq_left h]
exact samplingMatrix_code r c ((hasLargeRectangle_spec r c k).mp h).1
((hasLargeRectangle_spec r c k).mp h).2 htotal k bits
· rw [ite_eq_right h,dite_eq_right h]
exact unitSamplingMatrix_code r c (allSmall_of_no_large r c k h) htotal k bits
theorem publicSamplingAll_spec (htotal : (∑ i,r i)=∑ j,c j) (k : ℕ)
(bits : Fin (finiteSamplerBits r c k) → Bool) :
publicSamplingAll (Algorithms.encodeSamplingInput r c k,List.ofFn bits)=
Algorithms.encodeMatrix (scheduledTableDraw r c htotal k bits).val := by
rw [publicSamplingAll,parsedSampling_spec,samplingMatrixAll_code r c htotal]
exact GreedyFeasibleTable.printMatrix_ofFn _
private lemma law_uniform_of_subsingleton {Y : Type*} [Fintype Y] [Subsingleton Y]
[Nonempty Y] (p : RationalLaw Y) : p=uniformLaw (α := Y) := by
apply RationalLaw.ext
funext y
let : Unique Y := ⟨⟨y⟩,fun z => Subsingleton.elim z y⟩
have ht : p.mass (default : Y)=1 := by simpa using p.total
have hy : p.mass y=1 := (congrArg p.mass (Subsingleton.elim y default)).trans ht
simp only [uniformLaw,uniformMass,Fintype.card_unique,Nat.cast_one,div_one,hy]
private lemma variation_uniform_of_subsingleton {Y : Type*} [Fintype Y] [Subsingleton Y]
[Nonempty Y] (p : RationalLaw Y) : variation p (uniformLaw (α := Y))=0 := by
rw [law_uniform_of_subsingleton p,variation_self]
/-- Only the two cited published inputs on the completion grid are used.
The small and zero-dimensional branches need neither input. -/
theorem scheduledTableDraw_accuracy
(hPL : PublishedInputs.FiniteBoxPrekopaLeindler q)
(hCheeger : PublishedInputs.FiniteCheeger (GridBoundary.Grid (I := Fin q) (4*d^8)))
(htotal : (∑ i,r i)=∑ j,c j) (k : ℕ) (hk : 1≤k) :
let _ : Nonempty (Table r c) := ⟨GreedyFeasibleTable.table r c htotal⟩
(variation (mapLaw (uniformLaw (α := Fin (finiteSamplerBits r c k) → Bool))
(scheduledTableDraw r c htotal k)) (uniformLaw (α := Table r c)):ℝ)≤1/(2:ℝ)^k := by
let : Nonempty (Table r c) := ⟨GreedyFeasibleTable.table r c htotal⟩
let : Nonempty (PaperStates r c) := ⟨tableState r c (GreedyFeasibleTable.table r c htotal)⟩
dsimp only
unfold scheduledTableDraw
split_ifs with h
· exact listedScheduledDraw_accuracy r c (computedCellEquiv r c)
((hasLargeRectangle_spec r c k).mp h).1 ((hasLargeRectangle_spec r c k).mp h).2
hPL hCheeger htotal (tableState r c (GreedyFeasibleTable.table r c htotal))
(GreedyFeasibleTable.table r c htotal) k hk
· have hs := allSmall_of_no_large r c k h
rcases isEmpty_or_nonempty (Fin m) with hm|hm
· let := hm
let : Subsingleton (Table r c) := ⟨fun X Y => Subtype.ext (by funext i; exact isEmptyElim i)⟩
rw [variation_uniform_of_subsingleton,Rat.cast_zero]
positivity
· rcases isEmpty_or_nonempty (Fin n) with hn|hn
· let := hn
let : Subsingleton (Table r c) := ⟨fun X Y => Subtype.ext (by funext i j; exact isEmptyElim j)⟩
rw [variation_uniform_of_subsingleton,Rat.cast_zero]
positivity
· let := hm
let := hn
rw [padded_draw_law,unitCodeOuterDraw_law r c (computedCellEquiv r c) htotal]
exact unitOuterDraw_accuracy r c hs htotal k hk (allSmallFibre r c hs)
(tableState r c (GreedyFeasibleTable.table r c htotal)) (GreedyFeasibleTable.table r c htotal)
end
end ContingencyTables.SmallGraphProfiles
end OAI
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