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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 | import OAI.Combinatorics.ContingencyTables.Sampling.ExactMatrixTrial
namespace OAI
/-! # Limiting output law of the executed matrix rejection loop -/
namespace ContingencyTables.ExactMatrixTrial
open MatchingFPRAS MatchingFPRAS.PhysicalStream DensePrograms
open ExactIntegerChoice ResidualMixture FirstSuccess
open scoped BigOperators Classical
noncomputable section
lemma prefix_mass {K : ℕ} (candidates : Lookup) (w : Fin K → ℕ)
(hw : 0<totalWeight w) (N : ℕ) (X : MatrixCode) :
(OnlineIntegerRetry.prefixExperiment (f := trial) (candidates,List.ofFn w) N).mean
(fun o => if o=some X then 1 else 0)=
∑ i : Fin K,if lookup (candidates,i.val)=X then
(retryIntegerLaw w hw (Nat.le_pow_clog (by decide : 1<2) (totalWeight w)) N).mass (some i)
else 0 := by
rw [prefix_mean candidates w hw,Fintype.sum_option]
simp only [Option.map_none,reduceCtorEq,ite_false,mul_zero,zero_add,Option.map_some,
Option.some.injEq,mul_ite,mul_one,mul_zero]
/-- The integer rejection loop tends to the exact weighted pushforward,
including repeated matrix codes in a candidate list. -/
theorem prefix_limit {K : ℕ} (candidates : Lookup) (w : Fin K → ℕ)
(hw : 0<totalWeight w) (X : MatrixCode) :
Filter.Tendsto (fun N =>
((OnlineIntegerRetry.prefixExperiment (f := trial) (candidates,List.ofFn w) N).mean
(fun o => if o=some X then 1 else 0):ℝ))
Filter.atTop (nhds ((∑ i : Fin K,if lookup (candidates,i.val)=X then normalized w i else 0:ℚ):ℝ)) := by
simp_rw [prefix_mass candidates w hw,Rat.cast_sum]
apply tendsto_finsetSum
intro i _
by_cases hi : lookup (candidates,i.val)=X
· simp only [hi,ite_true]
exact retryIntegerLaw_limit w hw _ (clog_power_lt_twice (totalWeight w) hw) i
· simp only [hi,ite_false,Rat.cast_zero]
exact tendsto_const_nhds
end
end ContingencyTables.ExactMatrixTrial
end OAI
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