openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /FiniteConditionalAveraging.lean
| import Mathlib.Probability.ProbabilityMassFunction.Constructions | |
| import OAI.Combinatorics.BalancedRyser.Probability.PatternMoments | |
| namespace OAI | |
| namespace Balanced.FiniteConditionalAveraging | |
| open scoped BigOperators | |
| open PatternMoments | |
| noncomputable section | |
| attribute [local instance] Classical.propDecidable | |
| variable {Ω Y : Type*} [Fintype Ω] [Fintype Y] | |
| def mean (p : PMF Ω) (f : Ω → ℝ) : ℝ := | |
| expectation (fun ω => (p ω).toReal) f | |
| theorem probability_eq_mean_indicator (p : PMF Ω) (s : Set Ω) : | |
| (p.toOuterMeasure s).toReal = mean p (s.indicator (fun _ => 1)) := by | |
| rw [PMF.toOuterMeasure_apply_fintype, ENNReal.toReal_sum] | |
| · apply Finset.sum_congr rfl | |
| intro ω _ | |
| by_cases hω : ω ∈ s <;> simp [hω] | |
| · intro ω _ | |
| by_cases hω : ω ∈ s <;> simp [hω, p.apply_ne_top ω] | |
| theorem mean_sum {A : Type*} [Fintype A] (p : PMF Ω) (f : A → Ω → ℝ) : | |
| mean p (fun ω => ∑ a, f a ω) = ∑ a, mean p (f a) := | |
| expectation_sum _ _ | |
| theorem mean_const_mul (p : PMF Ω) (c : ℝ) (f : Ω → ℝ) : | |
| mean p (fun ω => c * f ω) = c * mean p f := | |
| expectation_const_mul _ _ _ | |
| theorem mean_mul_const (p : PMF Ω) (f : Ω → ℝ) (c : ℝ) : | |
| mean p (fun ω => f ω * c) = mean p f * c := | |
| expectation_mul_const _ _ _ | |
| theorem mean_nonneg (p : PMF Ω) (f : Ω → ℝ) (hf : ∀ ω, 0 ≤ f ω) : | |
| 0 ≤ mean p f := by | |
| exact Finset.sum_nonneg (fun ω _ => mul_nonneg ENNReal.toReal_nonneg (hf ω)) | |
| theorem mean_filter (p : PMF Ω) (s : Set Ω) | |
| (hs : ∃ ω ∈ s, ω ∈ p.support) (f : Ω → ℝ) : | |
| mean (p.filter s hs) f = mean p (s.indicator f) / (p.toOuterMeasure s).toReal := by | |
| unfold mean expectation | |
| simp only [PMF.filter_apply, ← PMF.toOuterMeasure_apply, ENNReal.toReal_mul, | |
| ENNReal.toReal_inv, div_eq_mul_inv, Finset.sum_mul] | |
| apply Finset.sum_congr rfl | |
| intro ω _ | |
| by_cases hω : ω ∈ s | |
| · simp [Set.indicator_apply, hω, mul_assoc, mul_comm] | |
| · simp [Set.indicator_apply, hω] | |
| omit [Fintype Ω] in | |
| theorem probability_pos (p : PMF Ω) (s : Set Ω) | |
| (hs : ∃ ω ∈ s, ω ∈ p.support) : 0 < (p.toOuterMeasure s).toReal := by | |
| apply ENNReal.toReal_pos | |
| · intro hz | |
| have hd := (PMF.toOuterMeasure_apply_eq_zero_iff p s).mp hz | |
| obtain ⟨ω, hωs, hωp⟩ := hs | |
| exact Set.disjoint_left.mp hd hωp hωs | |
| · rw [PMF.toOuterMeasure_apply] | |
| exact p.tsum_coe_indicator_ne_top s | |
| theorem weighted_mean_le (p : PMF Ω) (hfull : ∀ ω, ω ∈ p.support) | |
| (expose : Ω → Y) (f Z : Ω → ℝ) (B : ℝ) | |
| (hf : ∀ ω, 0 ≤ f ω) | |
| (hdepends : ∀ ω ω', expose ω = expose ω' → f ω = f ω') | |
| (hbound : ∀ a, mean (p.filter {ω | expose ω = expose a} | |
| ⟨a, rfl, hfull a⟩) Z ≤ B) : | |
| mean p (fun ω => f ω * Z ω) ≤ B * mean p f := by | |
| classical | |
| apply weighted_fiber_bound (fun ω => (p ω).toReal) Z f expose B hf hdepends | |
| intro y | |
| cases isEmpty_or_nonempty {ω // expose ω = y} with | |
| | inl h => | |
| let := h | |
| simp | |
| | inr h => | |
| obtain ⟨a⟩ := h | |
| have ha := hbound a | |
| rw [mean_filter] at ha | |
| have hp := probability_pos p {ω | expose ω = expose a} | |
| ⟨a, rfl, hfull a⟩ | |
| have hmul := (div_le_iff₀ hp).mp ha | |
| rw [probability_eq_mean_indicator] at hmul | |
| have heq : ({ω | expose ω = expose a} : Set Ω) = {ω | expose ω = y} := by | |
| simp only [a.property] | |
| rw [heq] at hmul | |
| have hsum (g : Ω → ℝ) : (∑ ω, if expose ω = y then g ω else 0) = | |
| ∑ ω : {ω // expose ω = y}, g ω := by | |
| rw [← Finset.sum_filter] | |
| exact Finset.sum_subtype _ (by simp) _ | |
| simp only [mean, expectation, Set.indicator_apply, Set.mem_ofPred_eq, | |
| mul_ite, mul_zero, mul_one] at hmul | |
| rw [hsum, hsum] at hmul | |
| exact hmul | |
| end | |
| end Balanced.FiniteConditionalAveraging | |
| end OAI | |