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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 | import OAI.Combinatorics.ContingencyTables.Rooted.RootedOracleAcceptance
import OAI.Combinatorics.ContingencyTables.Counting.CompanionBooleanPhase
namespace OAI
/-! # Actual finite-word Metropolis steps using the selected-root oracle -/
namespace ContingencyTables.RootedBooleanPhase
open CompanionFactorialApplication CompanionOracleDensity CompanionWeightOracle CompanionRationalUpdates
open CommonBasesFPRAS FirstSuccess FairBits TwoLiftStates GridBoundary ResidualMixture
open CompanionBooleanPhase (State proposal proposal_mass proposalBits proposal_codes stepBits stepWordEquiv)
open scoped BigOperators Classical
noncomputable section
variable {I J : Type*} [Fintype I] [Fintype J]
/-- The realized step evaluates the current and proposed weights from
separate words, then uses a third independent word for its rational coin. -/
def step (A : Instance I J) (R : RootedCompletion.Selection A.T) (C K j k : ℕ)
(hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : Pair A → Pair A → ℚ)
(S : State A) (bits : Fin (stepBits A C K j k xi) → Bool) : State A :=
let b := stepWordEquiv A C K j k xi bits
if b.1 0 then S else
match proposal A S b.2.1 with
| none => S
| some T =>
if rationalThresholdAccept k
(RootedOracleAcceptance.acceptance A R C K j hK hW xi v w S.val T.val b.2.2.1) b.2.2.2
then T else S
/-- A nontrivial move uses exactly its proposal code and the fresh
acceptance experiment. Failed codes and the independent holding bit
contribute only to the diagonal. -/
theorem step_eq_iff (A : Instance I J) (R : RootedCompletion.Selection A.T) (C K j k : ℕ)
(hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : Pair A → Pair A → ℚ)
(S T : State A) (hST : S ≠ T) (bits : Fin (stepBits A C K j k xi) → Bool) :
step A R C K j k hK hW xi v w S bits=T ↔
(stepWordEquiv A C K j k xi bits).1 0=false ∧
proposal A S (stepWordEquiv A C K j k xi bits).2.1=some T ∧
rationalThresholdAccept k
(RootedOracleAcceptance.acceptance A R C K j hK hW xi v w S.val T.val
(stepWordEquiv A C K j k xi bits).2.2.1)
(stepWordEquiv A C K j k xi bits).2.2.2 := by
dsimp only [step]
generalize stepWordEquiv A C K j k xi bits = b
cases b.1 0 with
| true => simp [hST]
| false =>
cases proposal A S b.2.1 with
| none => simp [hST]
| some U =>
by_cases hUT : U=T
· subst U
simp [hST]
· by_cases ha : rationalThresholdAccept k
(RootedOracleAcceptance.acceptance A R C K j hK hW xi v w S.val U.val b.2.2.1) b.2.2.2
· simp [hUT,ha]
· simp [hST,hUT,ha]
def transitionLaw (A : Instance I J) (R : RootedCompletion.Selection A.T) (C K j k : ℕ)
(hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : Pair A → Pair A → ℚ) (S : State A) :
RationalLaw (State A) :=
mapLaw (uniformLaw (α := Fin (stepBits A C K j k xi) → Bool))
(step A R C K j k hK hW xi v w S)
def walk (A : Instance I J) (R : RootedCompletion.Selection A.T) (C K j k n : ℕ)
(hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : Pair A → Pair A → ℚ) (S : State A)
(bits : Fin (n*stepBits A C K j k xi) → Bool) : State A :=
wordWalk (step A R C K j k hK hW xi v w) S n
(oracleWalkWordEquiv n (stepBits A C K j k xi) bits)
/-- The actual flattened word gives independent repeated transitions;
its law follows from the accepted finite-word walk construction. -/
theorem walk_law (A : Instance I J) (R : RootedCompletion.Selection A.T) (C K j k n : ℕ)
(hK : 1 ≤ K) (hW : 100*A.d*K ≤ A.W) (xi : ℚ)
(v : Grid (I := Fin (dimension A)) K) (w : Pair A → Pair A → ℚ) (S : State A) :
mapLaw (uniformLaw (α := Fin (n*stepBits A C K j k xi) → Bool))
(walk A R C K j k n hK hW xi v w S)=
walkLaw (transitionLaw A R C K j k hK hW xi v w) S n := by
unfold walk
rw [←map_equiv (oracleWalkWordEquiv n (stepBits A C K j k xi)),uniform_equiv]
exact wordWalkLaw_eq _ _ _
end
end ContingencyTables.RootedBooleanPhase
end OAI
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