| import OAI.Probability.CriticalZ3.ConditionalQuality |
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| namespace OAI |
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| universe uI uOmega |
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| open MeasureTheory ProbabilityTheory Classical Filter |
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| open scoped ENNReal BigOperators Matrix |
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| open MeasureTheory ProbabilityTheory Classical |
| namespace ShellEstimates |
| variable {Ω : Type uOmega} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] |
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| / |
| theorem exists_layer {I : Type uI} [Fintype I] [Nonempty I] |
| (Q A : I → Set Ω) (hAm : ∀ i, MeasurableSet (A i)) |
| (hAd : Pairwise (fun i j => Disjoint (A i) (A j))) {α : ℝ} (hα : 0 < α) |
| (hblock : ∀ i, α*μ.real (Q i) ≤ μ.real (A i)) : |
| ∃ i, μ.real (Q i) ≤ 1/(α*Fintype.card I) := by |
| have hs : ∑ i, μ.real (A i) ≤ 1 := by |
| rw [←measureReal_iUnion_fintype hAd hAm] |
| exact measureReal_le_one |
| have hsQ : α*(∑ i, μ.real (Q i)) ≤ 1 := by |
| rw [Finset.mul_sum] |
| exact (Finset.sum_le_sum (fun i _ => hblock i)).trans hs |
| have hcard : (0 : ℝ) < Fintype.card I := by exact_mod_cast Fintype.card_pos |
| by_contra! hnone |
| have ht : (∑ _i : I, 1/(α*Fintype.card I)) < ∑ i, μ.real (Q i) := |
| Finset.sum_lt_sum_of_nonempty Finset.univ_nonempty (fun i _ => hnone i) |
| simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul] at ht |
| have he : (Fintype.card I : ℝ)*(1/(α*Fintype.card I)) = 1/α := by field_simp |
| rw [he] at ht |
| have hh := (div_lt_iff₀ hα).mp ht |
| nlinarith |
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| theorem success_count_bound (S : Ω → ℝ) (hS : MemLp S 2 μ) |
| {a : ℝ} (ha : 0 < a) (hmean : ∫ ω, S ω ∂μ = a) |
| (hvar : variance S μ ≤ a) : |
| μ.real {ω | S ω < a/2} ≤ 4/a := by |
| have hcheb := meas_ge_le_variance_div_sq hS (show 0 < a/2 by positivity) |
| rw [hmean] at hcheb |
| have hs : {ω | S ω < a/2} ⊆ {ω | a/2 ≤ |S ω-a|} := by |
| intro ω hω |
| have hh : a/2 ≤ a-S ω := by dsimp at hω ⊢; linarith |
| exact hh.trans (le_abs_self _ |>.trans_eq (abs_sub_comm a (S ω))) |
| have hle : μ {ω | S ω < a/2} ≤ ENNReal.ofReal (variance S μ/(a/2)^2) := |
| (measure_mono hs).trans hcheb |
| have hv0 : 0 ≤ variance S μ/(a/2)^2 := div_nonneg (variance_nonneg _ _) (sq_nonneg _) |
| have hr : μ.real {ω | S ω < a/2} ≤ variance S μ/(a/2)^2 := by |
| exact (ENNReal.toReal_le_toReal (measure_ne_top _ _) ENNReal.ofReal_ne_top).mpr hle |>.trans_eq |
| (ENNReal.toReal_ofReal hv0) |
| apply hr.trans |
| apply (div_le_iff₀ (sq_pos_of_pos (show 0 < a/2 by positivity))).mpr |
| have he : (4/a)*(a/2)^2 = a := by field_simp; ring |
| rwa [he] |
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| /- Bad successes only need a first moment; no mutual independence is used. -/ |
| omit [IsProbabilityMeasure μ] in |
| theorem bad_count_bound (Z : Ω → ℝ) (hZ : Integrable Z μ) (hZ0 : ∀ ω, 0 ≤ Z ω) |
| {a γ τ : ℝ} (ha : 0 < a) (_hτ : 0 < τ) |
| (hmean : ∫ ω, Z ω ∂μ ≤ a*γ/τ) : |
| μ.real {ω | a/2 ≤ Z ω} ≤ 2*γ/τ := by |
| have h := mul_meas_ge_le_integral_of_nonneg (Filter.Eventually.of_forall hZ0) hZ (a/2) |
| have hm : (a/2)*μ.real {ω | a/2 ≤ Z ω} ≤ a*γ/τ := h.trans hmean |
| have he : (a/2)*(2*γ/τ) = a*γ/τ := by ring |
| rw [←he] at hm |
| exact (mul_le_mul_iff_right₀ (show 0 < a/2 by positivity)).mp hm |
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| /- If there is no good success, either successes fall below half their mean |
| or at least half their mean are bad. This is purely deterministic. -/ |
| omit [MeasurableSpace Ω] in |
| theorem no_good_subset (S Z : Ω → ℝ) (G : Set Ω) (a : ℝ) |
| (h : ∀ ω, ω ∉ G → S ω ≤ Z ω) : |
| Gᶜ ⊆ {ω | S ω < a/2} ∪ {ω | a/2 ≤ Z ω} := by |
| intro ω hω |
| by_cases hs : S ω < a/2 |
| · exact Or.inl hs |
| · exact Or.inr (le_trans (le_of_not_gt hs) (h ω hω)) |
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| end ShellEstimates |
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| namespace TrialCounts |
| open MeasureTheory ProbabilityTheory Classical |
| variable {Ω : Type uOmega} {I : Type uI} [MeasurableSpace Ω] [Fintype I] |
| variable (μ : Measure Ω) [IsProbabilityMeasure μ] |
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| noncomputable def count (A : I → Set Ω) : Ω → ℝ := |
| fun ω => ∑ i, (A i).indicator (fun _ => (1 : ℝ)) ω |
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| theorem count_memLp (A : I → Set Ω) (hA : ∀ i, MeasurableSet (A i)) : |
| MemLp (count A) 2 μ := by |
| apply memLp_finsetSum |
| intro i _ |
| exact (memLp_const (1 : ℝ)).indicator (hA i) |
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| omit [MeasurableSpace Ω] in |
| theorem count_nonneg (A : I → Set Ω) (ω : Ω) : 0 ≤ count A ω := by |
| apply Finset.sum_nonneg |
| intro i _ |
| by_cases h : ω ∈ A i <;> simp [Set.indicator,h] |
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| omit [IsProbabilityMeasure μ] in |
| theorem indicator_mean (A : Set Ω) (hA : MeasurableSet A) : |
| (∫ ω, A.indicator (fun _ => (1 : ℝ)) ω ∂μ) = μ.real A := |
| integral_indicator_one hA |
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| theorem count_mean (A : I → Set Ω) (hA : ∀ i, MeasurableSet (A i)) : |
| (∫ ω, count A ω ∂μ) = ∑ i, μ.real (A i) := by |
| unfold count |
| rw [integral_finsetSum] |
| · apply Finset.sum_congr rfl |
| intro i _ |
| exact integral_indicator_one (hA i) |
| · intro i _ |
| exact ((memLp_const (1 : ℝ)).indicator (hA i)).integrable (by norm_num : (1 : ENNReal) ≤ 2) |
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| omit [MeasurableSpace Ω] in |
| theorem indicator_product (A B : Set Ω) : |
| A.indicator (fun _ => (1 : ℝ)) * B.indicator (fun _ => (1 : ℝ)) = |
| (A ∩ B).indicator (fun _ => (1 : ℝ)) := by |
| funext ω |
| by_cases hA : ω ∈ A <;> by_cases hB : ω ∈ B <;> simp [Set.indicator,hA,hB] |
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| / |
| theorem count_variance_le (A : I → Set Ω) (hA : ∀ i, MeasurableSet (A i)) |
| {c : ℝ} (hc : ∀ i, μ.real (A i) = c) |
| (hpair : ∀ i j, i ≠ j → μ.real (A i ∩ A j) ≤ c*c) : |
| variance (count A) μ ≤ Fintype.card I*c := by |
| let X (i : I) : Ω → ℝ := (A i).indicator (fun _ => (1 : ℝ)) |
| have hX (i : I) : MemLp (X i) 2 μ := (memLp_const (1 : ℝ)).indicator (hA i) |
| have hcov (i j : I) : covariance (X i) (X j) μ ≤ if i = j then c else 0 := by |
| have hprod : X i * X j = (A i ∩ A j).indicator (fun _ => (1 : ℝ)) := |
| indicator_product (A i) (A j) |
| rw [covariance_eq_sub (hX i) (hX j),hprod, |
| indicator_mean μ _ ((hA i).inter (hA j))] |
| change μ.real (A i ∩ A j) - |
| (∫ ω, (A i).indicator (fun _ => (1 : ℝ)) ω ∂μ)* |
| (∫ ω, (A j).indicator (fun _ => (1 : ℝ)) ω ∂μ) ≤ _ |
| rw [indicator_mean μ _ (hA i),indicator_mean μ _ (hA j),hc i,hc j] |
| by_cases hij : i = j |
| · subst j |
| simp only [ite_true,Set.inter_self,hc] |
| nlinarith [sq_nonneg c] |
| · simp only [hij,ite_false] |
| linarith [hpair i j hij] |
| have he : count A = ∑ i, X i := by ext ω; simp [count,X] |
| rw [he,variance_sum hX] |
| calc |
| _ ≤ ∑ i : I, ∑ j : I, if i = j then c else 0 := |
| Finset.sum_le_sum (fun i _ => Finset.sum_le_sum (fun j _ => hcov i j)) |
| _ = Fintype.card I*c := by simp |
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| / |
| theorem no_good_probability (A B : I → Set Ω) |
| (hA : ∀ i, MeasurableSet (A i)) (hB : ∀ i, MeasurableSet (B i)) |
| {c γ τ : ℝ} (hcpos : 0 < Fintype.card I*c) (hτ : 0 < τ) |
| (hc : ∀ i, μ.real (A i) = c) |
| (hpair : ∀ i j, i ≠ j → μ.real (A i ∩ A j) ≤ c*c) |
| (hbad : ∀ i, μ.real (B i) ≤ c*γ/τ) : |
| μ.real {ω | ∀ i, ω ∈ A i → ω ∈ B i} ≤ 4/(Fintype.card I*c)+2*γ/τ := by |
| let a : ℝ := Fintype.card I*c |
| have hm : (∫ ω, count A ω ∂μ) = a := by |
| rw [count_mean μ A hA] |
| simp only [hc,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,a] |
| have hs := ShellEstimates.success_count_bound μ (count A) (count_memLp μ A hA) |
| hcpos hm (count_variance_le μ A hA hc hpair) |
| have hbmean : (∫ ω, count B ω ∂μ) ≤ a*γ/τ := by |
| rw [count_mean μ B hB] |
| calc |
| _ ≤ ∑ _i : I, c*γ/τ := Finset.sum_le_sum (fun i _ => hbad i) |
| _ = a*γ/τ := by simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,a]; ring |
| have hb := ShellEstimates.bad_count_bound μ (count B) |
| ((count_memLp μ B hB).integrable (by norm_num)) (count_nonneg B) hcpos hτ hbmean |
| have hsub : {ω | ∀ i, ω ∈ A i → ω ∈ B i} ⊆ |
| {ω | count A ω < a/2} ∪ {ω | a/2 ≤ count B ω} := by |
| intro ω hω |
| by_cases hs : count A ω < a/2 |
| · exact Or.inl hs |
| · apply Or.inr |
| apply (le_of_not_gt hs).trans |
| apply Finset.sum_le_sum |
| intro i _ |
| by_cases hi : ω ∈ A i |
| · simp [Set.indicator,hi,hω i hi] |
| · simp only [Set.indicator,hi,ite_false] |
| split_ifs <;> norm_num |
| exact (measureReal_mono hsub).trans ((measureReal_union_le _ _).trans (add_le_add hs hb)) |
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| end TrialCounts |
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| end OAI |
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