| import Mathlib.Algebra.BigOperators.Expect |
| import Mathlib.Algebra.BigOperators.Field |
| import Mathlib.Tactic.FieldSimp |
| import Mathlib.Tactic.Ring |
| import OAI.Computability.UniqueGames.Foundations.Conditioning |
|
|
| namespace OAI |
|
|
| section |
|
|
| /-! A finite experiment whose sample weights are constant on the selected |
| event becomes exactly uniform after conditioning on that event. The event is |
| allowed to have arbitrary positive mass, so this applies after the complete |
| visible datum and the chosen column constraints have both been fixed. -/ |
|
|
| namespace UniqueGamesTheorem.Decoder.UniformConditioning |
|
|
| open Foundations.Games |
| open scoped BigOperators Classical |
|
|
| noncomputable section |
|
|
| variable {Ω : Type*} [Fintype Ω] |
|
|
| theorem sum_over_event (given : Ω → Bool) (f : Ω → ℝ) : |
| (∑ x, if given x then f x else 0) = ∑ x : {x : Ω // given x = true}, f x.val := by |
| rw [← Finset.sum_filter] |
| exact Finset.sum_subtype _ (fun x => by simp) f |
|
|
| theorem probability_as_subtype_sum (μ : FiniteDistribution Ω) (given : Ω → Bool) : |
| μ.probability given = ∑ x : {x : Ω // given x = true}, μ.weight x.val := |
| sum_over_event given μ.weight |
|
|
| theorem probability_and_as_subtype_sum (μ : FiniteDistribution Ω) |
| (given event : Ω → Bool) : |
| μ.probability (fun x => given x && event x) = |
| ∑ x : {x : Ω // given x = true}, if event x.val then μ.weight x.val else 0 := by |
| calc |
| _ = ∑ x, if given x then (if event x then μ.weight x else 0) else 0 := by |
| apply Finset.sum_congr rfl |
| intro x _ |
| change (if given x && event x then μ.weight x else 0) = _ |
| cases given x <;> cases event x <;> rfl |
| _ = _ := sum_over_event given _ |
|
|
| theorem probability_of_constant_weight (μ : FiniteDistribution Ω) |
| (given : Ω → Bool) (c : ℝ) (hconstant : ∀ x, given x = true → μ.weight x = c) : |
| μ.probability given = (Fintype.card {x : Ω // given x = true} : ℝ) * c := by |
| rw [probability_as_subtype_sum] |
| calc |
| _ = ∑ _x : {x : Ω // given x = true}, c := by |
| apply Finset.sum_congr rfl |
| intro x _ |
| exact hconstant x.val x.property |
| _ = _ := by simp [nsmul_eq_mul] |
|
|
| / |
| the conditioning event justifies cancellation; no unconditional uniform law |
| on the original ambient experiment is required. -/ |
| theorem condition_probability_uniform (μ : FiniteDistribution Ω) |
| (given event : Ω → Bool) (positive : 0 < μ.probability given) |
| (c : ℝ) (hconstant : ∀ x, given x = true → μ.weight x = c) : |
| (μ.condition given positive).probability event = |
| 𝔼 x : {x : Ω // given x = true}, if event x.val then (1 : ℝ) else 0 := by |
| have hp := probability_of_constant_weight μ given c hconstant |
| have hc : c ≠ 0 := by |
| intro hz |
| rw [hp, hz, mul_zero] at positive |
| exact (lt_irrefl 0) positive |
| have hn : (Fintype.card {x : Ω // given x = true} : ℝ) ≠ 0 := by |
| intro hz |
| rw [hp, hz, zero_mul] at positive |
| exact (lt_irrefl 0) positive |
| rw [FiniteDistribution.probability_condition, probability_and_as_subtype_sum, hp, |
| Fintype.expect_eq_sum_div_card] |
| have hnum : (∑ x : {x : Ω // given x = true}, if event x.val then μ.weight x.val else 0) = |
| c * ∑ x : {x : Ω // given x = true}, if event x.val then (1 : ℝ) else 0 := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro x _ |
| rw [hconstant x.val x.property] |
| cases event x.val <;> simp |
| rw [hnum] |
| field_simp [hc, hn] |
|
|
| / |
| conditional target probability with that slice's complete uniform average. -/ |
| theorem condition_probability_equiv {Y : Type*} [Fintype Y] |
| (μ : FiniteDistribution Ω) (given event : Ω → Bool) |
| (positive : 0 < μ.probability given) (c : ℝ) |
| (hconstant : ∀ x, given x = true → μ.weight x = c) |
| (e : {x : Ω // given x = true} ≃ Y) (target : Y → Bool) |
| (hevent : ∀ x, event x.val = target (e x)) : |
| (μ.condition given positive).probability event = |
| 𝔼 y : Y, if target y then (1 : ℝ) else 0 := by |
| rw [condition_probability_uniform μ given event positive c hconstant] |
| apply Fintype.expect_equiv e |
| intro x |
| rw [hevent x] |
|
|
| end |
| end UniqueGamesTheorem.Decoder.UniformConditioning |
|
|
| end |
|
|
| end OAI |
|
|
