File size: 2,266 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 | import OAI.Combinatorics.ContingencyTables.Transport.CompletionProposalLaw
namespace OAI
/-! # Identifying a complete transition law from its off-diagonal probabilities -/
namespace ContingencyTables.FirstSuccess
open scoped BigOperators
open ResidualMixture
variable {X : Type*} [Fintype X] [DecidableEq X]
/-- Normalization determines the remaining holding probability. -/
theorem law_ext_except (p q : RationalLaw X) (x : X)
(h : ∀ y,y≠x → p.mass y=q.mass y) : p=q := by
apply RationalLaw.ext
funext y
by_cases hy : y=x
· subst y
have hs : (∑ y,(p.mass y-q.mass y))=p.mass x-q.mass x := by
apply Finset.sum_eq_single x
· intro y _ hy
rw [h y hy,sub_self]
· simp
rw [Finset.sum_sub_distrib,p.total,q.total,sub_self] at hs
linarith
· exact h y hy
end ContingencyTables.FirstSuccess
namespace ContingencyTables.CompletionTranslation
open scoped BigOperators Classical
open ResidualMixture FirstSuccess
noncomputable section
variable {X I J : Type*} [Fintype X] [Fintype I] [Fintype J] [DecidableEq X]
variable (R : X → Option I → ℕ) (P : X → Option J → ℕ)
variable [∀ x,Nonempty (Table (R x) (P x))]
omit [∀ x,Nonempty (Table (R x) (P x))] in
/-- Only the proposal distribution at the current state is observed. -/
lemma completionStepLaw_proposal_at (p q : X → RationalLaw (Option X))
(draw : (z : X) → RationalLaw (Table (R z) (P z))) (x : X) (h : p x=q x) :
completionStepLaw R P p draw x=completionStepLaw R P q draw x := by
unfold completionStepLaw
rw [h]
/-- This identifies the actual draw-and-test transition, including holds;
no independent reversibility or stationarity assertion is imposed on it. -/
theorem uniformCompletionStep_eq (w : X → ℚ) (hw : ∀ y,0 ≤ w y)
(htotal : ∑ y,w y ≤ 1) (x : X) (K : RationalLaw X)
(hK : ∀ y,y≠x → K.mass y=w y*(acceptedCount R P x y:ℚ)/(blockCount R P x:ℚ)) :
completionStepLaw R P (fun _ => proposalLaw w hw htotal)
(fun z => uniformLaw (α := Table (R z) (P z))) x=K := by
apply law_ext_except _ _ x
intro y hy
rw [completionStepLaw_mass_ne R P w hw htotal _ x y hy.symm,uniformAcceptedMass,hK y hy]
ring
end
end ContingencyTables.CompletionTranslation
end OAI
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