openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Construction /HiddenChildMixture.lean
| import OAI.Computability.PerfectCompleteness.Foundations.FiniteProduct | |
| import OAI.Computability.PerfectCompleteness.Sampling.DensityVariation | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.HiddenChildMixture | |
| open scoped BigOperators | |
| open UniqueGamesTheorem.Foundations.Games | |
| open PerfectCompleteness SmallBias DensityVariation | |
| noncomputable section | |
| variable {I : Type*} [Fintype I] [DecidableEq I] [Nonempty I] | |
| {Ω : I → Type*} [∀ i, Fintype (Ω i)] | |
| def averageDensity (q : (i : I) → Ω i → ℝ) (x : (i : I) → Ω i) : ℝ := | |
| (∑ i, q i (x i)) / Fintype.card I | |
| theorem averageDensity_mean (P : (i : I) → FiniteDistribution (Ω i)) | |
| (q : (i : I) → Ω i → ℝ) (mean : ∀ i, (P i).expectation (q i) = 1) : | |
| (FiniteProduct.law P).expectation (averageDensity q) = 1 := by | |
| change (FiniteProduct.law P).expectation | |
| (fun x => (∑ i, q i (x i)) / Fintype.card I) = 1 | |
| rw [expectation_div, expectation_sum] | |
| simp only [FiniteProduct.expectation_eval, mean, Finset.sum_const, Finset.card_univ, | |
| nsmul_eq_mul, mul_one] | |
| exact div_self (by exact_mod_cast Fintype.card_ne_zero (α := I)) | |
| def mixture (R : I → FiniteDistribution ((i : I) → Ω i)) : | |
| FiniteDistribution ((i : I) → Ω i) where | |
| weight x := (∑ i, (R i).weight x) / Fintype.card I | |
| nonnegative x := div_nonneg (Finset.sum_nonneg fun i _ => (R i).nonnegative x) | |
| (Nat.cast_nonneg _) | |
| normalized := by | |
| simp only [div_eq_mul_inv, ← Finset.sum_mul] | |
| rw [Finset.sum_comm] | |
| simp only [FiniteDistribution.normalized, Finset.sum_const, | |
| Finset.card_univ, nsmul_eq_mul, mul_one] | |
| exact mul_inv_cancel₀ (by exact_mod_cast Fintype.card_ne_zero (α := I)) | |
| theorem mixture_density (P : (i : I) → FiniteDistribution (Ω i)) | |
| (R : I → FiniteDistribution ((i : I) → Ω i)) | |
| (q : (i : I) → Ω i → ℝ) | |
| (weight : ∀ i x, (R i).weight x = (FiniteProduct.law P).weight x * q i (x i)) | |
| (x : (i : I) → Ω i) : | |
| (mixture R).weight x = (FiniteProduct.law P).weight x * averageDensity q x := by | |
| simp only [mixture, weight, ← Finset.mul_sum, averageDensity, mul_div_assoc] | |
| omit [Nonempty I] in | |
| theorem centered_cross (P : (i : I) → FiniteDistribution (Ω i)) | |
| (q : (i : I) → Ω i → ℝ) (mean : ∀ i, (P i).expectation (q i) = 1) | |
| (i j : I) (different : i ≠ j) : | |
| (FiniteProduct.law P).expectation (fun x => (q i (x i) - 1) * (q j (x j) - 1)) = 0 := by | |
| rw [FiniteProduct.expectation_eval_mul P i j different (fun a => q i a - 1) | |
| (fun a => q j a - 1)] | |
| simp only [expectation_sub, expectation_const, mean, sub_self, mul_zero] | |
| omit [Nonempty I] in | |
| theorem centered_sum_second_moment (P : (i : I) → FiniteDistribution (Ω i)) | |
| (q : (i : I) → Ω i → ℝ) (mean : ∀ i, (P i).expectation (q i) = 1) : | |
| (FiniteProduct.law P).expectation (fun x => (∑ i, (q i (x i) - 1)) ^ 2) = | |
| ∑ i, (P i).expectation (fun a => (q i a - 1) ^ 2) := by | |
| simp only [pow_two, Finset.sum_mul, Finset.mul_sum, expectation_sum] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [Finset.sum_eq_single i] | |
| · exact FiniteProduct.expectation_eval P i (fun a => (q i a - 1) * (q i a - 1)) | |
| · intro j _ different | |
| exact centered_cross P q mean j i different | |
| · simp | |
| omit [DecidableEq I] [∀ i, Fintype (Ω i)] in | |
| theorem average_centered (q : (i : I) → Ω i → ℝ) (x : (i : I) → Ω i) : | |
| averageDensity q x - 1 = (∑ i, (q i (x i) - 1)) / Fintype.card I := by | |
| have hn : (Fintype.card I : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero (α := I) | |
| simp only [averageDensity, Finset.sum_sub_distrib, Finset.sum_const, | |
| Finset.card_univ, nsmul_eq_mul, mul_one] | |
| field_simp | |
| theorem average_second_moment (P : (i : I) → FiniteDistribution (Ω i)) | |
| (q : (i : I) → Ω i → ℝ) (mean : ∀ i, (P i).expectation (q i) = 1) : | |
| (FiniteProduct.law P).expectation (fun x => (averageDensity q x - 1) ^ 2) = | |
| (∑ i, (P i).expectation (fun a => (q i a - 1) ^ 2)) / (Fintype.card I : ℝ) ^ 2 := by | |
| simp only [average_centered, div_pow, expectation_div, centered_sum_second_moment P q mean] | |
| theorem average_second_moment_le (P : (i : I) → FiniteDistribution (Ω i)) | |
| (q : (i : I) → Ω i → ℝ) (mean : ∀ i, (P i).expectation (q i) = 1) | |
| (v : ℝ) (bound : ∀ i, (P i).expectation (fun a => (q i a - 1) ^ 2) ≤ v) : | |
| (FiniteProduct.law P).expectation (fun x => (averageDensity q x - 1) ^ 2) ≤ | |
| v / Fintype.card I := by | |
| rw [average_second_moment P q mean] | |
| have hn : (Fintype.card I : ℝ) ≠ 0 := by exact_mod_cast Fintype.card_ne_zero (α := I) | |
| calc | |
| _ ≤ (∑ _i : I, v) / (Fintype.card I : ℝ) ^ 2 := | |
| div_le_div_of_nonneg_right (Finset.sum_le_sum fun i _ => bound i) (sq_nonneg _) | |
| _ = _ := by simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]; field_simp | |
| theorem observed_mixture_variation {Γ : Type*} [Fintype Γ] | |
| (P : (i : I) → FiniteDistribution (Ω i)) | |
| (R : I → FiniteDistribution ((i : I) → Ω i)) (q : (i : I) → Ω i → ℝ) | |
| (weight : ∀ i x, (R i).weight x = (FiniteProduct.law P).weight x * q i (x i)) | |
| (mean : ∀ i, (P i).expectation (q i) = 1) | |
| (v : ℝ) (bound : ∀ i, (P i).expectation (fun a => (q i a - 1) ^ 2) ≤ v) | |
| (observe : ((i : I) → Ω i) → Γ) : | |
| ((mixture R).pushforward observe).totalVariation | |
| ((FiniteProduct.law P).pushforward observe) ≤ Real.sqrt (v / Fintype.card I) / 2 := | |
| observed_variation_le_sqrt (FiniteProduct.law P) (mixture R) (averageDensity q) | |
| (mixture_density P R q weight) _ (average_second_moment_le P q mean v bound) observe | |
| end | |
| end PerfectCompleteness.HiddenChildMixture | |
| end | |
| end OAI | |