File size: 3,381 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 67 68 69 70 71 72 73 | import OAI.Combinatorics.ContingencyTables.Sampling.AllSmallChain
import OAI.Combinatorics.ContingencyTables.Sampling.FirstPaperProposal
import OAI.Combinatorics.ContingencyTables.Transport.FiniteProposalStay
import OAI.Combinatorics.ContingencyTables.Transport.RationalChainMixing
namespace OAI
/-! # The actual rational walk on an empty large rectangle -/
namespace ContingencyTables.SmallGraphProfiles
open scoped BigOperators Classical
open ResidualMixture FirstSuccess CompletionTranslation CompletionCounts CommonBasesFPRAS
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 instance : DecidableEq (PaperStates r c) := Classical.decEq _
def unitSmallStep (x : PaperStates r c) : RationalLaw (PaperStates r c) :=
proposalStayLaw (proposalWeights r c x) (proposalWeights_nonnegative r c x)
(proposalWeights_total r c x) x
lemma unitSmallStep_mass (x y : PaperStates r c) :
(unitSmallStep r c x).mass y=proposalWeights r c x y+
if x=y then 1-∑ z,proposalWeights r c x z else 0 :=
proposalStayLaw_mass _ _ _ x y
variable [Nonempty (PaperStates r c)]
lemma unitSmallStep_cast (x y : PaperStates r c) :
((unitSmallStep r c x).mass y:ℝ)=(unitSmallChain r c).P x y := by
rw [unitSmallStep_mass]
simp only [Rat.cast_add,apply_ite,Rat.cast_sub,Rat.cast_one,Rat.cast_zero,Rat.cast_sum]
simp only [proposalWeights,apply_ite,smallProposalRat_cast,Rat.cast_zero]
simp only [unitSmallChain,MetropolisChain.chain,FiniteChain.ofMoves,MetropolisChain.move,
div_self (one_ne_zero : (1:ℝ)≠0),min_self,mul_one]
by_cases hxy : x=y <;> simp only [hxy,ite_true,ite_false]
lemma unitSmallChain_pi_lower
(hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true)
(z : PaperStates r c) :
1/(((stateBound r c+1:ℕ):ℝ)^(2*d)) ≤ (unitSmallChain r c).π z := by
have hs : 0 < ∑ y,stateWeight r c y :=
Finset.sum_pos (fun y _ => stateWeight_positive r c y) Finset.univ_nonempty
rw [unitSmallChain_pi_weight r c hsmall]
exact (one_div_le_one_div_of_le hs (state_total_mass_le r c)).trans
(div_le_div_of_nonneg_right (stateWeight_ge_one r c z) hs.le)
theorem unitSmallWalk_variation [Nonempty I] [Nonempty J]
(hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true)
(x : PaperStates r c) (n : ℕ) :
(variation (walkLaw (unitSmallStep r c) x n)
(uniformLaw (α := PaperStates r c)):ℝ) ≤
((stateBound r c+1:ℕ):ℝ)^(2*d)*Real.exp (-(n:ℝ)/(d:ℝ)^160) := by
have hd : (1:ℝ) ≤ d := by exact_mod_cast (show 1 ≤ d by unfold dimensionAllowance; omega)
have hm := (unitSmallChain r c).point_mixing_exp ((d:ℝ)^160)
(1/(((stateBound r c+1:ℕ):ℝ)^(2*d))) (one_le_pow₀ hd) (by positivity)
(unitSmallChain_pi_lower r c hsmall) (unitSmallChain_poincare r c hsmall) n x
apply rationalWalk_variation (unitSmallStep r c) (unitSmallChain r c)
(unitSmallStep_cast r c) (uniformLaw (α := PaperStates r c))
(fun y => ?_) x n _ (by positivity)
· simpa only [one_div,div_eq_mul_inv,one_mul,inv_inv,mul_comm] using hm
· simp only [uniformLaw,uniformMass,Rat.cast_div,Rat.cast_one,Rat.cast_natCast,
unitSmallChain_pi]
end
end ContingencyTables.SmallGraphProfiles
end OAI
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