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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 | import OAI.Combinatorics.ContingencyTables.Transport.TranslatedCompletionCounts
import OAI.Combinatorics.ContingencyTables.Transport.RationalWalkApproximation
namespace OAI
/-! # Completion replacement in the actual proposal-and-translation step -/
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]
/-- A proposal is either a hold or a target state. The sampled completion
is discarded after its literal translated-nonnegativity test. -/
def completionDecision (R : X → Option I → ℕ) (P : X → Option J → ℕ)
(x : X) (proposal : Option X) (a : Table (R x) (P x)) : X :=
match proposal with
| none => x
| some y => if ∀ i j,0 ≤ (a.val i j:ℤ)+between (R x) (R y) (P x) (P y) i j then y else x
def completionStepLaw (R : X → Option I → ℕ) (P : X → Option J → ℕ)
(proposal : X → RationalLaw (Option X))
(draw : (x : X) → RationalLaw (Table (R x) (P x))) (x : X) : RationalLaw X :=
bindLaw (proposal x) (fun y => mapLaw (draw x) (completionDecision R P x y))
/-- From a common state only the completion draw changes. Mapping to the
accept/reject decision and averaging over the same proposal cannot enlarge TV. -/
theorem completionStep_variation (R : X → Option I → ℕ) (P : X → Option J → ℕ)
(proposal : X → RationalLaw (Option X))
(μ ν : (x : X) → RationalLaw (Table (R x) (P x))) (η : ℚ)
(h : ∀ x,variation (μ x) (ν x) ≤ η) (x : X) :
variation (completionStepLaw R P proposal μ x) (completionStepLaw R P proposal ν x) ≤ η := by
have hv := variation_bind_uniform (proposal x) (proposal x)
(fun y => mapLaw (μ x) (completionDecision R P x y))
(fun y => mapLaw (ν x) (completionDecision R P x y)) η
(fun y => (variation_map _ _ _).trans (h x))
simpa only [completionStepLaw,variation,sub_self,abs_zero,Finset.sum_const_zero,zero_div,zero_add] using hv
/-- The transition draws and the final completion all use fresh bits. The
bound is uniform even though the state supplied to each draw is adaptive. -/
theorem completionWalk_terminal_variation (R : X → Option I → ℕ) (P : X → Option J → ℕ)
{Y : Type*} [Fintype Y] [DecidableEq Y]
(proposal : X → RationalLaw (Option X))
(μ ν : (x : X) → RationalLaw (Table (R x) (P x)))
(output : (x : X) → Table (R x) (P x) → Y) (η : ℚ)
(h : ∀ x,variation (μ x) (ν x) ≤ η) (x : X) (n : ℕ) :
variation
(bindLaw (walkLaw (completionStepLaw R P proposal μ) x n) (fun z => mapLaw (μ z) (output z)))
(bindLaw (walkLaw (completionStepLaw R P proposal ν) x n) (fun z => mapLaw (ν z) (output z))) ≤
(n+1)*η :=
walkLaw_terminal_variation _ _ _ _ η
(completionStep_variation R P proposal μ ν η h)
(fun z => (variation_map _ _ _).trans (h z)) x n
end
end ContingencyTables.CompletionTranslation
end OAI
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