File size: 3,011 Bytes
2f887db | 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 | import OAI.Computability.PerfectCompleteness.Sampling.BucketSampler
import OAI.Computability.PerfectCompleteness.Sampling.UniformDifference
namespace OAI
section
namespace PerfectCompleteness.BucketResamplingLaw
noncomputable section
open scoped Classical
open UniqueGamesTheorem.Foundations.Games
open BucketSampler (F2 Direction Tape)
variable {H : Type*} [AddCommGroup H] [Module F2 H] [Fintype H]
private theorem pushforward_id {A : Type*} [Fintype A] (μ : FiniteDistribution A) :
μ.pushforward (fun a => a) = μ := by
apply FiniteDistribution.eq_of_weight_eq
intro a
simp [FiniteDistribution.pushforward]
theorem assembled_difference_law (ℓ : Nat) (μ : FiniteDistribution (Tape ℓ H))
(a : Direction ℓ) :
(μ.product (FiniteDistribution.uniform H)).pushforward
(fun p => (BucketSampler.evaluate ℓ id p.1, p.2 - p.1 a)) =
(μ.pushforward (BucketSampler.evaluate ℓ id)).product (FiniteDistribution.uniform H) := by
rw [UniformDifference.observed_difference_law μ (fun t => t a)
(BucketSampler.evaluate ℓ id)]
have hp := FiniteDistribution.product_pushforward μ (FiniteDistribution.uniform H)
(BucketSampler.evaluate ℓ (id : H → H)) (id : H → H)
have hi : (FiniteDistribution.uniform H).pushforward (id : H → H) =
FiniteDistribution.uniform H := pushforward_id (FiniteDistribution.uniform H)
rw [hi] at hp
simpa only [id_eq] using hp
theorem resampled_array_pair_law (ℓ : Nat) (μ : FiniteDistribution (Tape ℓ H))
(a : Direction ℓ) :
(μ.product (FiniteDistribution.uniform H)).pushforward
(fun p => (BucketSampler.evaluate ℓ id p.1,
BucketSampler.evaluate ℓ id (Function.update p.1 a p.2))) =
((μ.pushforward (BucketSampler.evaluate ℓ id)).product
(FiniteDistribution.uniform H)).pushforward
(fun p => (p.1, p.1 + BucketSampler.rankOne a.val p.2)) := by
have h := congrArg
(fun ν : FiniteDistribution ((Fin ℓ → H) × H) => ν.pushforward
(fun p => (p.1, p.1 + BucketSampler.rankOne a.val p.2)))
(assembled_difference_law ℓ μ a)
rw [FiniteDistribution.pushforward_comp] at h
simpa only [BucketSampler.evaluate_update, id_eq] using h
theorem resampled_event_probability (ℓ : Nat) (μ : FiniteDistribution (Tape ℓ H))
(a : Direction ℓ) (event : (Fin ℓ → H) × (Fin ℓ → H) → Bool) :
(μ.product (FiniteDistribution.uniform H)).probability
(fun p => event (BucketSampler.evaluate ℓ id p.1,
BucketSampler.evaluate ℓ id (Function.update p.1 a p.2))) =
((μ.pushforward (BucketSampler.evaluate ℓ id)).product
(FiniteDistribution.uniform H)).probability
(fun p => event (p.1, p.1 + BucketSampler.rankOne a.val p.2)) := by
have h := congrArg (fun ν => FiniteDistribution.probability ν event)
(resampled_array_pair_law ℓ μ a)
simpa only [FiniteDistribution.probability_pushforward] using h
end
end PerfectCompleteness.BucketResamplingLaw
end
end OAI
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