File size: 3,632 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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactMatrixLimits
import OAI.Combinatorics.ContingencyTables.Sampling.ExactMatrixLookup
import OAI.Combinatorics.ContingencyTables.Transport.IntegerLawEmbedding
import OAI.Combinatorics.ContingencyTables.Machines.MatrixCodeEquality
namespace OAI
/-! # Exact weighted matrix selection realizes a table law -/
namespace ContingencyTables.ExactMatrixTrial
open MatchingFPRAS MatchingFPRAS.PhysicalStream DensePrograms ProfilePrograms OriginalTableBox
open ExactIntegerChoice ResidualMixture FirstSuccess
open scoped BigOperators Classical
noncomputable section
attribute [local irreducible] lookup originalCandidates
lemma normalized_counts {K Q : ℕ} (p : IntegerLaw (Fin K) Q)
(hQ : 0<Q) (a : Fin K) : (p.count a:ℚ)/totalWeight p.count=(p.toRational hQ).mass a := by
unfold totalWeight
rw [p.total]
rfl
lemma count_weight_pos {K Q : ℕ} (p : IntegerLaw (Fin K) Q)
(hQ : 0<Q) : 0<totalWeight p.count := by
unfold totalWeight
rwa [p.total]
/-- An integer law's full original-box vector pushes back to precisely
that table law under the executed matrix lookup. -/
lemma table_normalized_event {m n Q : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
(p : IntegerLaw (Table r c) Q) (hQ : 0<Q) (fallback X : Table r c) :
(∑ i : Index m n (∑ j,r j),
if lookup ((originalCandidates (List.ofFn r,List.ofFn c),matrixCode fallback.val),i.val)=
matrixCode X.val then normalized (p.push (index r c le_rfl)).count i else 0)=
(p.toRational hQ).mass X := by
let test : Index m n (∑ j,r j) → Bool := fun i => decide
(lookup ((originalCandidates (List.ofFn r,List.ofFn c),matrixCode fallback.val),i.val)=matrixCode X.val)
have hn (i : Index m n (∑ j,r j)) :
normalized (p.push (index r c le_rfl)).count i=
((p.push (index r c le_rfl)).toRational hQ).mass i :=
normalized_counts _ hQ i
simp_rw [hn]
have hm : (mapLaw ((p.push (index r c le_rfl)).toRational hQ) test).mass true=
(mapLaw (p.toRational hQ) (fun T => test (index r c le_rfl T))).mass true := by
rw [IntegerLaw.toRational_push,mapLaw_comp]
have he (T : Table r c) : test (index r c le_rfl T)=true ↔ T=X := by
simp only [test,decide_eq_true_eq,lookup_index]
exact ⟨fun h => Subtype.ext (matrixCode_injective h),fun h => by rw [h]⟩
simp only [mapLaw] at hm
simp only [test,decide_eq_true_eq] at hm
rw [hm]
have htest (T : Table r c) :
(lookup ((originalCandidates (List.ofFn r,List.ofFn c),matrixCode fallback.val),
(index r c le_rfl T).val)=matrixCode X.val) ↔ T=X := by
simpa only [test,decide_eq_true_eq] using he T
simp only [htest,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
/-- The finite prefixes of the actual exact matrix loop converge to the
input table law. -/
theorem table_prefix_limit {m n Q : ℕ} (r : Fin m → ℕ) (c : Fin n → ℕ)
(p : IntegerLaw (Table r c) Q) (hQ : 0<Q) (fallback X : Table r c) :
Filter.Tendsto (fun N =>
((OnlineIntegerRetry.prefixExperiment (f := trial)
((originalCandidates (List.ofFn r,List.ofFn c),matrixCode fallback.val),
List.ofFn (p.push (index r c le_rfl)).count) N).mean
(fun o => if o=some (matrixCode X.val) then 1 else 0):ℝ))
Filter.atTop (nhds ((p.toRational hQ).mass X:ℝ)) := by
have h := prefix_limit (originalCandidates (List.ofFn r,List.ofFn c),matrixCode fallback.val)
(p.push (index r c le_rfl)).count (count_weight_pos _ hQ) (matrixCode X.val)
rw [table_normalized_event r c p hQ fallback X] at h
exact h
end
end ContingencyTables.ExactMatrixTrial
end OAI
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