| import OAI.Probability.FreeIsing.Conditioning.Product |
| import OAI.Probability.FreeIsing.Spins.Kernel |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| namespace Problem367.BinaryRegression |
|
|
| variable {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {S : Ω → IsingSpin} |
|
|
| private theorem spin_event_measurable (hS : Measurable S) (s : IsingSpin) : |
| MeasurableSet {ω | S ω = s} := by |
| exact hS (by trivial : MeasurableSet ({s} : Set IsingSpin)) |
|
|
| theorem integral_spin_function (hS : Measurable S) |
| (hfair : ∀ s : IsingSpin, μ {ω | S ω = s} = ENNReal.ofReal (1/2 : ℝ)) |
| (φ : IsingSpin → ℝ) : |
| (∫ ω, φ (S ω) ∂μ) = (φ IsingSpin.neg + φ IsingSpin.pos) / 2 := by |
| classical |
| have hneg := spin_event_measurable hS IsingSpin.neg |
| have hpos := spin_event_measurable hS IsingSpin.pos |
| have hf : (fun ω => φ (S ω)) = |
| {ω | S ω = IsingSpin.neg}.indicator (fun _ => φ IsingSpin.neg) + |
| {ω | S ω = IsingSpin.pos}.indicator (fun _ => φ IsingSpin.pos) := by |
| funext ω |
| cases hs : S ω <;> simp [hs] |
| rw [hf] |
| change (∫ ω, ({ω | S ω = IsingSpin.neg}.indicator (fun _ => φ IsingSpin.neg)) ω + |
| ({ω | S ω = IsingSpin.pos}.indicator (fun _ => φ IsingSpin.pos)) ω ∂μ) = _ |
| rw [integral_add ((integrable_const _).indicator hneg) |
| ((integrable_const _).indicator hpos), integral_indicator hneg, |
| integral_indicator hpos, setIntegral_const, setIntegral_const] |
| simp only [smul_eq_mul, measureReal_def, hfair] |
| norm_num |
| ring |
|
|
| theorem exists_mul_spinSign_of_centered (hS : Measurable S) |
| (hfair : ∀ s : IsingSpin, μ {ω | S ω = s} = ENNReal.ofReal (1/2 : ℝ)) |
| {f : Ω → ℝ} (hf : Measurable[isingSpinMeasurable.comap S] f) |
| (hzero : ∫ ω, f ω ∂μ = 0) : |
| ∃ a : ℝ, ∀ ω, f ω = a * spinSign (S ω) := by |
| obtain ⟨φ, hφ, rfl⟩ := hf.exists_eq_measurable_comp |
| have hm := integral_spin_function hS hfair φ |
| have hs : φ IsingSpin.neg + φ IsingSpin.pos = 0 := by |
| simp only [Function.comp_apply] at hzero |
| rw [hzero] at hm |
| linarith |
| refine ⟨φ IsingSpin.pos, ?_⟩ |
| intro ω |
| cases he : S ω with |
| | neg => |
| simp [Function.comp_apply, he, spinSign] |
| linarith |
| | pos => simp [Function.comp_apply, he, spinSign] |
|
|
| theorem exists_condExp_mul_spinSign (hS : Measurable S) |
| (hfair : ∀ s : IsingSpin, μ {ω | S ω = s} = ENNReal.ofReal (1/2 : ℝ)) |
| {f : Ω → ℝ} (hzero : ∫ ω, f ω ∂μ = 0) : |
| ∃ a : ℝ, ∀ ω, μ[f | isingSpinMeasurable.comap S] ω = a * spinSign (S ω) := by |
| apply exists_mul_spinSign_of_centered hS hfair stronglyMeasurable_condExp.measurable |
| rw [integral_condExp hS.comap_le, hzero] |
|
|
| theorem exists_condExp_mul_spinSign_sq_le (hS : Measurable S) |
| (hfair : ∀ s : IsingSpin, μ {ω | S ω = s} = ENNReal.ofReal (1/2 : ℝ)) |
| {f : Ω → ℝ} (hzero : ∫ ω, f ω ∂μ = 0) |
| (hf2 : Integrable (fun ω => (f ω)^2) μ) : |
| ∃ a : ℝ, (∀ ω, μ[f | isingSpinMeasurable.comap S] ω = a * spinSign (S ω)) ∧ |
| a^2 ≤ ∫ ω, (f ω)^2 ∂μ := by |
| obtain ⟨a, ha⟩ := exists_condExp_mul_spinSign hS hfair hzero |
| refine ⟨a, ha, ?_⟩ |
| have hint : Integrable (fun ω => ‖f ω‖ ^ (2 : ℝ)) μ := by |
| simpa only [Real.rpow_two, Real.norm_eq_abs, sq_abs] using hf2 |
| have hb := integral_norm_condExp_rpow_le (μ := μ) (m := isingSpinMeasurable.comap S) |
| (by norm_num : (1 : ℝ) ≤ 2) hint |
| simp only [Real.rpow_two, Real.norm_eq_abs, sq_abs, ha, mul_pow, spinSign_sq, mul_one] at hb |
| simpa using hb |
|
|
| theorem abs_correlation_le_of_binary_condExp {T : Ω → IsingSpin} |
| (hS : Measurable S) (hT : Measurable T) |
| (hSfair : ∀ s : IsingSpin, μ {ω | S ω = s} = ENNReal.ofReal (1/2 : ℝ)) |
| (hTfair : ∀ s : IsingSpin, μ {ω | T ω = s} = ENNReal.ofReal (1/2 : ℝ)) |
| {f g : Ω → ℝ} (hfzero : ∫ ω, f ω ∂μ = 0) (hgzero : ∫ ω, g ω ∂μ = 0) |
| (hf2 : Integrable (fun ω => (f ω)^2) μ) (hg2 : Integrable (fun ω => (g ω)^2) μ) |
| (hfactor : (∫ ω, f ω * g ω ∂μ) = ∫ ω, |
| μ[f | isingSpinMeasurable.comap S] ω * μ[g | isingSpinMeasurable.comap T] ω ∂μ) : |
| |∫ ω, f ω * g ω ∂μ| ≤ |
| Real.sqrt (∫ ω, (f ω)^2 ∂μ) * Real.sqrt (∫ ω, (g ω)^2 ∂μ) * |
| |∫ ω, spinSign (S ω) * spinSign (T ω) ∂μ| := by |
| obtain ⟨a, ha, hsa⟩ := exists_condExp_mul_spinSign_sq_le hS hSfair hfzero hf2 |
| obtain ⟨b, hb, hsb⟩ := exists_condExp_mul_spinSign_sq_le hT hTfair hgzero hg2 |
| have hfa : 0 ≤ ∫ ω, (f ω)^2 ∂μ := integral_nonneg (fun _ => sq_nonneg _) |
| have hgb : 0 ≤ ∫ ω, (g ω)^2 ∂μ := integral_nonneg (fun _ => sq_nonneg _) |
| have ha' : |a| ≤ Real.sqrt (∫ ω, (f ω)^2 ∂μ) := by |
| apply (Real.le_sqrt (abs_nonneg a) hfa).2 |
| simpa only [sq_abs] using hsa |
| have hb' : |b| ≤ Real.sqrt (∫ ω, (g ω)^2 ∂μ) := by |
| apply (Real.le_sqrt (abs_nonneg b) hgb).2 |
| simpa only [sq_abs] using hsb |
| have heq : (∫ ω, f ω * g ω ∂μ) = |
| (a*b) * ∫ ω, spinSign (S ω) * spinSign (T ω) ∂μ := by |
| rw [hfactor, ← integral_const_mul] |
| apply integral_congr_ae |
| filter_upwards with ω |
| rw [ha, hb] |
| ring |
| rw [heq, abs_mul, abs_mul] |
| exact mul_le_mul_of_nonneg_right |
| (mul_le_mul ha' hb' (abs_nonneg b) (Real.sqrt_nonneg _)) (abs_nonneg _) |
|
|
| theorem abs_correlation_le_of_endpoint_condIndep [StandardBorelSpace Ω] |
| {T : Ω → IsingSpin} {m : MeasurableSpace Ω} (hm : m ≤ mΩ) |
| (hS : Measurable[mΩ] S) (hT : Measurable[mΩ] T) |
| (hSfair : ∀ s : IsingSpin, μ {ω | S ω = s} = ENNReal.ofReal (1/2 : ℝ)) |
| (hTfair : ∀ s : IsingSpin, μ {ω | T ω = s} = ENNReal.ofReal (1/2 : ℝ)) |
| {f g : Ω → ℝ} (hf : Measurable[mΩ] f) (hg : Measurable[mΩ] g) |
| (hfzero : ∫ ω, f ω ∂μ = 0) (hgzero : ∫ ω, g ω ∂μ = 0) |
| (hf2 : MemLp f 2 μ) (hg2 : MemLp g 2 μ) |
| (hind : ProbabilityTheory.CondIndepFun m hm f g μ) |
| (hleft : μ[f | m] =ᵐ[μ] μ[f | isingSpinMeasurable.comap S]) |
| (hright : μ[g | m] =ᵐ[μ] μ[g | isingSpinMeasurable.comap T]) : |
| |∫ ω, f ω * g ω ∂μ| ≤ |
| Real.sqrt (∫ ω, (f ω)^2 ∂μ) * Real.sqrt (∫ ω, (g ω)^2 ∂μ) * |
| |∫ ω, spinSign (S ω) * spinSign (T ω) ∂μ| := by |
| let : MeasurableSpace Ω := mΩ |
| apply abs_correlation_le_of_binary_condExp hS hT hSfair hTfair hfzero hgzero |
| hf2.integrable_sq hg2.integrable_sq |
| calc |
| (∫ ω, f ω * g ω ∂μ) = ∫ ω, μ[f | m] ω * μ[g | m] ω ∂μ := |
| integral_mul_eq_integral_condExp_mul_of_condIndepFun hm hind hf hg |
| (hf2.integrable (by norm_num)) (hg2.integrable (by norm_num)) |
| ((hf2.integrable_mul hg2).congr (Filter.Eventually.of_forall fun _ => rfl)) |
| _ = ∫ ω, μ[f | isingSpinMeasurable.comap S] ω * |
| μ[g | isingSpinMeasurable.comap T] ω ∂μ := |
| integral_congr_ae (hleft.mul hright) |
|
|
| theorem abs_correlation_le_of_spatial_condIndep [StandardBorelSpace Ω] |
| {T : Ω → IsingSpin} (hS : Measurable S) (hT : Measurable T) |
| (hSfair : ∀ s : IsingSpin, μ {ω | S ω = s} = ENNReal.ofReal (1/2 : ℝ)) |
| (hTfair : ∀ s : IsingSpin, μ {ω | T ω = s} = ENNReal.ofReal (1/2 : ℝ)) |
| {f g : Ω → ℝ} (hf : Measurable f) (hg : Measurable g) |
| (hfzero : ∫ ω, f ω ∂μ = 0) (hgzero : ∫ ω, g ω ∂μ = 0) |
| (hf2 : MemLp f 2 μ) (hg2 : MemLp g 2 μ) |
| (hind : ProbabilityTheory.CondIndepFun |
| (isingSpinMeasurable.comap S ⊔ isingSpinMeasurable.comap T) |
| (sup_le hS.comap_le hT.comap_le) f g μ) |
| (hleft : ProbabilityTheory.CondIndepFun (isingSpinMeasurable.comap S) hS.comap_le T f μ) |
| (hright : ProbabilityTheory.CondIndepFun (isingSpinMeasurable.comap T) hT.comap_le S g μ) : |
| |∫ ω, f ω * g ω ∂μ| ≤ |
| Real.sqrt (∫ ω, (f ω)^2 ∂μ) * Real.sqrt (∫ ω, (g ω)^2 ∂μ) * |
| |∫ ω, spinSign (S ω) * spinSign (T ω) ∂μ| := by |
| let : Nonempty IsingSpin := ⟨IsingSpin.neg⟩ |
| apply abs_correlation_le_of_endpoint_condIndep (sup_le hS.comap_le hT.comap_le) |
| hS hT hSfair hTfair hf hg hfzero hgzero hf2 hg2 hind |
| · exact condExp_sup_eq_of_condIndepFun hS hT hf (hf2.integrable (by norm_num)) hleft |
| · simpa only [sup_comm] using |
| condExp_sup_eq_of_condIndepFun hT hS hg (hg2.integrable (by norm_num)) hright |
|
|
| end Problem367.BinaryRegression |
|
|
| end |
|
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| end OAI |
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