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760f90e | 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 | import OAI.Computability.UniqueGames.Foundations.Conditioning
import OAI.Computability.UniqueGames.Foundations.ValueLemmas
namespace OAI
section
/-!
Weighted total probability for the public clean-coordinate observation.
The observation may have zero-probability fibres. Only its positive fibres
require a conditional bound; the identity itself is proved from finite sums.
-/
namespace UniqueGamesTheorem.Clean.WeightedPartition
open scoped BigOperators
open Foundations.Games
noncomputable section
variable {Ω H : Type*} [Fintype Ω] [Fintype H] [DecidableEq H]
def fibre (observe : Ω → H) (h : H) : Ω → Bool :=
fun x => decide (observe x = h)
theorem sum_fibre_inter (μ : FiniteDistribution Ω) (observe : Ω → H)
(event : Ω → Bool) :
(∑ h, μ.probability (fun x => fibre observe h x && event x)) =
μ.probability event := by
classical
unfold FiniteDistribution.probability fibre
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro x _
by_cases hx : event x = true <;> simp [hx]
theorem sum_fibre_bound (μ : FiniteDistribution Ω) (observe : Ω → H)
(bound : H → ℝ) :
(∑ h, μ.probability (fibre observe h) * bound h) =
μ.expectation (fun x => bound (observe x)) := by
classical
unfold FiniteDistribution.probability FiniteDistribution.expectation fibre
simp_rw [Finset.sum_mul]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro x _
simp only [decide_eq_true_eq, ite_mul, zero_mul]
simp
/-- The averaging step for a genuine weighted observation. No conditional
independence or law-of-total-probability equality is supplied as a premise. -/
theorem probability_le_conditional_bounds (μ : FiniteDistribution Ω)
(observe : Ω → H) (event : Ω → Bool) (bound : H → ℝ)
(conditional : ∀ h (positive : 0 < μ.probability (fibre observe h)),
(μ.condition (fibre observe h) positive).probability event ≤ bound h) :
μ.probability event ≤ μ.expectation (fun x => bound (observe x)) := by
rw [← sum_fibre_inter μ observe event, ← sum_fibre_bound μ observe bound]
apply Finset.sum_le_sum
intro h _
by_cases positive : 0 < μ.probability (fibre observe h)
· rw [μ.probability_inter_eq_mul_conditional (fibre observe h) event positive]
exact mul_le_mul_of_nonneg_left (conditional h positive) positive.le
· have zero : μ.probability (fibre observe h) = 0 :=
le_antisymm (le_of_not_gt positive) (μ.probability_nonnegative _)
rw [μ.probability_and_eq_zero_of_probability_eq_zero (fibre observe h) event zero,
zero, zero_mul]
/-- A constant bound survives arbitrary public outside observations. -/
theorem probability_le_of_conditional_le (μ : FiniteDistribution Ω)
(observe : Ω → H) (event : Ω → Bool) (bound : ℝ)
(conditional : ∀ h (positive : 0 < μ.probability (fibre observe h)),
(μ.condition (fibre observe h) positive).probability event ≤ bound) :
μ.probability event ≤ bound := by
have estimate := probability_le_conditional_bounds μ observe event
(fun _ => bound) conditional
have constant : μ.expectation (fun _ => bound) = bound := by
rw [FiniteDistribution.expectation, ← Finset.sum_mul, μ.normalized, one_mul]
simpa only [constant] using estimate
end
end UniqueGamesTheorem.Clean.WeightedPartition
end
end OAI
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