openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Sampling /DensityVariation.lean
| import Mathlib.Analysis.Real.Sqrt | |
| import OAI.Computability.PerfectCompleteness.Foundations.RecursiveSpaceEquivLemmas | |
| import OAI.Computability.UniqueGames.Foundations.MixtureLemmas | |
| import OAI.Computability.UniqueGames.Foundations.ValueLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.DensityVariation | |
| open scoped BigOperators | |
| open UniqueGamesTheorem.Foundations.Games | |
| open PerfectCompleteness.SmallBias | |
| noncomputable section | |
| variable {Ω Γ : Type*} [Fintype Ω] [Fintype Γ] | |
| theorem expectation_sub (μ : FiniteDistribution Ω) (f g : Ω → ℝ) : | |
| μ.expectation (fun x => f x - g x) = μ.expectation f - μ.expectation g := by | |
| simp only [FiniteDistribution.expectation, mul_sub, Finset.sum_sub_distrib] | |
| theorem expectation_mul_const (μ : FiniteDistribution Ω) (f : Ω → ℝ) (c : ℝ) : | |
| μ.expectation (fun x => f x * c) = μ.expectation f * c := by | |
| simp only [FiniteDistribution.expectation, ← mul_assoc, Finset.sum_mul] | |
| theorem expectation_div (μ : FiniteDistribution Ω) (f : Ω → ℝ) (c : ℝ) : | |
| μ.expectation (fun x => f x / c) = μ.expectation f / c := by | |
| simpa only [div_eq_mul_inv] using expectation_mul_const μ f c⁻¹ | |
| def reweight (μ : FiniteDistribution Ω) (density : Ω → ℝ) | |
| (nonnegative : ∀ x, 0 ≤ density x) (mean_one : μ.expectation density = 1) : | |
| FiniteDistribution Ω where | |
| weight x := μ.weight x * density x | |
| nonnegative x := mul_nonneg (μ.nonnegative x) (nonnegative x) | |
| normalized := mean_one | |
| theorem variation_eq_density (μ ν : FiniteDistribution Ω) (density : Ω → ℝ) | |
| (h : ∀ x, ν.weight x = μ.weight x * density x) : | |
| ν.totalVariation μ = μ.expectation (fun x => |density x - 1|) / 2 := by | |
| unfold FiniteDistribution.totalVariation FiniteDistribution.expectation | |
| congr 1 | |
| apply Finset.sum_congr rfl | |
| intro x _ | |
| rw [h x, show μ.weight x * density x - μ.weight x = | |
| μ.weight x * (density x - 1) by ring, abs_mul, | |
| abs_of_nonneg (μ.nonnegative x)] | |
| theorem variation_le_sqrt (μ ν : FiniteDistribution Ω) (density : Ω → ℝ) | |
| (h : ∀ x, ν.weight x = μ.weight x * density x) (v : ℝ) | |
| (hv : μ.expectation (fun x => (density x - 1) ^ 2) ≤ v) : | |
| ν.totalVariation μ ≤ Real.sqrt v / 2 := by | |
| rw [variation_eq_density μ ν density h] | |
| apply div_le_div_of_nonneg_right _ (by norm_num : (0 : ℝ) ≤ 2) | |
| apply Real.le_sqrt_of_sq_le | |
| have hcs := UniqueGamesTheorem.Foundations.Information.weighted_sum_sq_le μ.weight | |
| (fun x => |density x - 1|) ⟨μ.nonnegative, μ.normalized⟩ | |
| change μ.expectation (fun x => |density x - 1|) ^ 2 ≤ | |
| μ.expectation (fun x => |density x - 1| ^ 2) at hcs | |
| simpa only [sq_abs] using hcs.trans (by simpa only [sq_abs] using hv) | |
| theorem variation_pushforward_le (μ ν : FiniteDistribution Ω) (observe : Ω → Γ) : | |
| (μ.pushforward observe).totalVariation (ν.pushforward observe) ≤ | |
| μ.totalVariation ν := by | |
| classical | |
| unfold FiniteDistribution.totalVariation | |
| apply div_le_div_of_nonneg_right _ (by norm_num : (0 : ℝ) ≤ 2) | |
| have hdiff (y : Γ) : | |
| (μ.pushforward observe).weight y - (ν.pushforward observe).weight y = | |
| ∑ x, if observe x = y then μ.weight x - ν.weight x else 0 := by | |
| simp only [FiniteDistribution.pushforward, ← Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro x _ | |
| by_cases hx : observe x = y <;> simp [hx] | |
| calc | |
| _ ≤ ∑ y, ∑ x, if observe x = y then |μ.weight x - ν.weight x| else 0 := by | |
| apply Finset.sum_le_sum | |
| intro y _ | |
| rw [hdiff] | |
| calc | |
| _ ≤ ∑ x, |if observe x = y then μ.weight x - ν.weight x else 0| := | |
| Finset.abs_sum_le_sum_abs _ _ | |
| _ = _ := by | |
| apply Finset.sum_congr rfl | |
| intro x _ | |
| split <;> simp | |
| _ = _ := by rw [Finset.sum_comm]; simp | |
| theorem observed_variation_le_sqrt (μ ν : FiniteDistribution Ω) (density : Ω → ℝ) | |
| (h : ∀ x, ν.weight x = μ.weight x * density x) (v : ℝ) | |
| (hv : μ.expectation (fun x => (density x - 1) ^ 2) ≤ v) | |
| (observe : Ω → Γ) : | |
| (ν.pushforward observe).totalVariation (μ.pushforward observe) ≤ Real.sqrt v / 2 := | |
| (variation_pushforward_le ν μ observe).trans (variation_le_sqrt μ ν density h v hv) | |
| end | |
| end PerfectCompleteness.DensityVariation | |
| end | |
| end OAI | |