| 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 _ |
|
|
| / |
| 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) |
|
|
| / |
| 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 |
|
|
