| import OAI.Probability.FreeIsing.Conditioning.Characteristic |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal RealInnerProductSpace |
|
|
| namespace Problem367.ConditionalGaussianVector |
|
|
| variable {Ω : Type*} {m : MeasurableSpace Ω} [mΩ : MeasurableSpace Ω] |
|
|
| section Hilbert |
|
|
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℝ H] |
| [CompleteSpace H] [MeasurableSpace H] [BorelSpace H] [SecondCountableTopology H] |
|
|
| theorem map_restrict_eq_smul_of_condExp_inner_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) {X : Ω → H} (hX : Measurable[mΩ] X) |
| (ν : Measure H) [IsProbabilityMeasure ν] |
| (hchar : ∀ t : H, |
| P[(fun ω => Complex.exp ((⟪X ω, t⟫ : ℝ) * Complex.I)) | m] =ᵐ[P] |
| fun _ => charFun ν t) |
| {s : Set Ω} (hs : MeasurableSet[m] s) : |
| Measure.map X (P.restrict s) = P s • ν := by |
| let : IsFiniteMeasure (P s • ν) := ν.smul_finite (measure_ne_top P s) |
| apply Measure.ext_of_charFun |
| funext t |
| have hi : Integrable (fun ω => Complex.exp ((⟪X ω, t⟫ : ℝ) * Complex.I)) P := by |
| have hi := ConditionalCharacteristic.integrable_character (P := P) |
| (show Measurable (fun ω => ⟪X ω, t⟫) by fun_prop) 1 |
| convert hi using 1 |
| funext ω |
| simp [ConditionalCharacteristic.character] |
| rw [charFun_apply, integral_map hX.aemeasurable (by fun_prop), |
| charFun_apply, integral_smul_measure] |
| rw [← setIntegral_condExp hm hi hs] |
| calc |
| (∫ ω in s, P[(fun ω => Complex.exp ((⟪X ω, t⟫ : ℝ) * Complex.I)) | m] ω ∂P) = |
| ∫ _ in s, charFun ν t ∂P := |
| integral_congr_ae (ae_restrict_of_ae (hchar t)) |
| _ = _ := by simp [charFun_apply, measureReal_def] |
|
|
| theorem law_and_indep_of_condExp_inner_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) {X : Ω → H} (hX : Measurable[mΩ] X) |
| (ν : Measure H) [IsProbabilityMeasure ν] |
| (hchar : ∀ t : H, |
| P[(fun ω => Complex.exp ((⟪X ω, t⟫ : ℝ) * Complex.I)) | m] =ᵐ[P] |
| fun _ => charFun ν t) |
| {E : Type*} [MeasurableSpace E] {Y : Ω → E} (hY : Measurable[m] Y) : |
| HasLaw X ν P ∧ IndepFun X Y P := by |
| have hlaw : Measure.map X P = ν := by |
| simpa using map_restrict_eq_smul_of_condExp_inner_character hm hX ν hchar |
| (MeasurableSet.univ : MeasurableSet[m] Set.univ) |
| refine ⟨⟨hX.aemeasurable, hlaw⟩, ?_⟩ |
| rw [indepFun_iff_measure_inter_preimage_eq_mul] |
| intro a b ha hb |
| have hrestrict := congrArg (fun μ : Measure H => μ a) |
| (map_restrict_eq_smul_of_condExp_inner_character hm hX ν hchar (hY hb)) |
| rw [Measure.map_apply hX ha, Measure.restrict_apply (hX ha), |
| Measure.smul_apply, smul_eq_mul, ← hlaw, Measure.map_apply hX ha] at hrestrict |
| exact hrestrict.trans (mul_comm _ _) |
|
|
| end Hilbert |
|
|
| section FiniteVector |
|
|
| variable {ι : Type*} [Fintype ι] |
|
|
| theorem charFun_product_gaussian (v : ι → ℝ≥0) (t : EuclideanSpace ℝ ι) : |
| charFun ((Measure.pi (fun i => gaussianReal 0 (v i))).map (WithLp.toLp 2)) t = |
| Complex.exp (-((∑ i, (v i : ℝ) * (t i) ^ 2 : ℝ) : ℂ) / 2) := by |
| simp_rw [charFun_pi, charFun_gaussianReal, ← Complex.exp_sum] |
| simp [Finset.sum_div, neg_div, Complex.ofReal_sum, Complex.ofReal_mul, |
| Complex.ofReal_pow] |
|
|
| theorem product_gaussian_law_and_indep_of_condExp_linear_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) {X : Ω → ι → ℝ} (hX : Measurable[mΩ] X) |
| (v : ι → ℝ≥0) |
| (hchar : ∀ a : ι → ℝ, |
| P[(fun ω => Complex.exp (((∑ i, a i * X ω i : ℝ) : ℂ) * Complex.I)) | m] =ᵐ[P] |
| fun _ => Complex.exp (-((∑ i, (v i : ℝ) * (a i) ^ 2 : ℝ) : ℂ) / 2)) |
| {E : Type*} [MeasurableSpace E] {Y : Ω → E} (hY : Measurable[m] Y) : |
| HasLaw X (Measure.pi (fun i => gaussianReal 0 (v i))) P ∧ IndepFun X Y P := by |
| let Z : Ω → EuclideanSpace ℝ ι := fun ω => WithLp.toLp 2 (X ω) |
| let ν : Measure (EuclideanSpace ℝ ι) := |
| (Measure.pi (fun i => gaussianReal 0 (v i))).map (WithLp.toLp 2) |
| have hZ : Measurable Z := (WithLp.measurable_toLp 2 _).comp hX |
| have hν : IsProbabilityMeasure ν := inferInstance |
| have hc : ∀ t : EuclideanSpace ℝ ι, |
| P[(fun ω => Complex.exp ((⟪Z ω, t⟫ : ℝ) * Complex.I)) | m] =ᵐ[P] |
| fun _ => charFun ν t := by |
| intro t |
| simpa only [Z, ν, PiLp.inner_apply, WithLp.ofLp_toLp, Real.inner_apply, mul_comm, |
| charFun_product_gaussian] using hchar t.ofLp |
| obtain ⟨hlaw, hind⟩ := law_and_indep_of_condExp_inner_character hm hZ ν hc hY |
| constructor |
| · refine ⟨hX.aemeasurable, ?_⟩ |
| apply (MeasurableEquiv.toLp 2 (ι → ℝ)).map_measurableEquiv_injective |
| change (P.map X).map (WithLp.toLp 2) = ν |
| rw [Measure.map_map (WithLp.measurable_toLp 2 _) hX] |
| exact hlaw.map_eq |
| · simpa only [Function.comp_def, Z, WithLp.ofLp_toLp, id_eq] using |
| hind.comp (WithLp.measurable_ofLp 2 (ι → ℝ)) measurable_id |
|
|
| theorem condExp_linear_character_of_unit_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) {X : Ω → ι → ℝ} (v : ℝ≥0) |
| (hchar : ∀ a : ι → ℝ, (∑ i, (a i) ^ 2) = 1 → ∀ u : ℝ, |
| P[ConditionalCharacteristic.character (fun ω => ∑ i, a i * X ω i) u | m] =ᵐ[P] |
| fun _ => Complex.exp (-((v : ℝ) : ℂ) * (u : ℂ) ^ 2 / 2)) : |
| ∀ a : ι → ℝ, |
| P[(fun ω => Complex.exp (((∑ i, a i * X ω i : ℝ) : ℂ) * Complex.I)) | m] =ᵐ[P] |
| fun _ => Complex.exp (-((∑ i, (v : ℝ) * (a i) ^ 2 : ℝ) : ℂ) / 2) := by |
| classical |
| intro a |
| by_cases hz : (∑ i, (a i) ^ 2) = 0 |
| · have ha : ∀ i, a i = 0 := by |
| intro i |
| have hs := (Finset.sum_eq_zero_iff_of_nonneg (fun j _ => sq_nonneg (a j))).mp hz |
| exact sq_eq_zero_iff.mp (hs i (Finset.mem_univ i)) |
| simpa [ha] using |
| Filter.EventuallyEq.of_eq (condExp_const (μ := P) hm (1 : ℂ)) |
| let q : ℝ := Real.sqrt (∑ i, (a i) ^ 2) |
| have hsq : 0 < ∑ i, (a i) ^ 2 := |
| lt_of_le_of_ne (Finset.sum_nonneg (fun i _ => sq_nonneg (a i))) (Ne.symm hz) |
| have hq : 0 < q := Real.sqrt_pos.mpr hsq |
| have hq2 : q ^ 2 = ∑ i, (a i) ^ 2 := Real.sq_sqrt hsq.le |
| let b : ι → ℝ := fun i => a i / q |
| have hb : (∑ i, (b i) ^ 2) = 1 := by |
| simp only [b, div_pow, ← Finset.sum_div, hq2] |
| exact div_self hz |
| have heq : ConditionalCharacteristic.character (fun ω => ∑ i, b i * X ω i) q = |
| (fun ω => Complex.exp (((∑ i, a i * X ω i : ℝ) : ℂ) * Complex.I)) := by |
| funext ω |
| have hs : q * (∑ i, b i * X ω i) = ∑ i, a i * X ω i := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| dsimp [b] |
| field_simp |
| simp only [ConditionalCharacteristic.character, ← Complex.ofReal_mul, hs] |
| have h := hchar b hb q |
| rw [heq] at h |
| convert h using 1 |
| funext ω |
| congr 1 |
| rw [← Complex.ofReal_pow, hq2, ← Finset.mul_sum] |
| push_cast |
| ring |
|
|
| theorem condExp_weighted_character_of_unit_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) {X : Ω → ι → ℝ} (v : ℝ≥0) |
| (hchar : ∀ a : ι → ℝ, (∑ i, (a i) ^ 2) = 1 → ∀ u : ℝ, |
| P[ConditionalCharacteristic.character (fun ω => ∑ i, a i * X ω i) u | m] =ᵐ[P] |
| fun _ => Complex.exp (-((v : ℝ) : ℂ) * (u : ℂ) ^ 2 / 2)) : |
| ∀ a : ι → ℝ, |
| P[ConditionalCharacteristic.character (fun ω => ∑ i, a i * X ω i) 1 | m] =ᵐ[P] |
| fun _ => Complex.exp (-(∑ i, ((v : ℝ) : ℂ) * (a i : ℂ)^2) / 2) := by |
| intro a |
| have he : ConditionalCharacteristic.character (fun ω => ∑ i, a i * X ω i) 1 = |
| (fun ω => Complex.exp (((∑ i, a i * X ω i : ℝ) : ℂ) * Complex.I)) := by |
| funext ω |
| simp [ConditionalCharacteristic.character] |
| rw [he] |
| simpa only [Complex.ofReal_sum, Complex.ofReal_mul, Complex.ofReal_pow] using |
| condExp_linear_character_of_unit_character hm v hchar a |
|
|
| theorem product_gaussian_law_and_indep_of_condExp_unit_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) {X : Ω → ι → ℝ} (hX : Measurable[mΩ] X) |
| (v : ℝ≥0) |
| (hchar : ∀ a : ι → ℝ, (∑ i, (a i) ^ 2) = 1 → ∀ u : ℝ, |
| P[ConditionalCharacteristic.character (fun ω => ∑ i, a i * X ω i) u | m] =ᵐ[P] |
| fun _ => Complex.exp (-((v : ℝ) : ℂ) * (u : ℂ) ^ 2 / 2)) |
| {E : Type*} [MeasurableSpace E] {Y : Ω → E} (hY : Measurable[m] Y) : |
| HasLaw X (Measure.pi (fun _ : ι => gaussianReal 0 v)) P ∧ IndepFun X Y P := by |
| exact product_gaussian_law_and_indep_of_condExp_linear_character hm hX |
| (fun _ => v) (condExp_linear_character_of_unit_character hm v hchar) hY |
|
|
| end FiniteVector |
|
|
| section ArbitraryArray |
|
|
| variable {ι : Type*} |
|
|
| theorem infinite_product_gaussian_law_and_indep_of_condExp_linear_character |
| {P : Measure Ω} [IsProbabilityMeasure P] |
| (hm : m ≤ mΩ) {X : Ω → ι → ℝ} (hX : Measurable[mΩ] X) |
| (v : ι → ℝ≥0) |
| (hchar : ∀ (I : Finset ι) (a : I → ℝ), |
| P[(fun ω => Complex.exp (((∑ i : I, a i * X ω i : ℝ) : ℂ) * Complex.I)) | m] =ᵐ[P] |
| fun _ => Complex.exp (-((∑ i : I, (v i : ℝ) * (a i) ^ 2 : ℝ) : ℂ) / 2)) |
| {E : Type*} [MeasurableSpace E] {Y : Ω → E} (hY : Measurable[m] Y) : |
| HasLaw X (Measure.infinitePi (fun i => gaussianReal 0 (v i))) P ∧ |
| iIndepFun (fun i ω => X ω i) P ∧ IndepFun X Y P := by |
| classical |
| have hfinite (I : Finset ι) : |
| HasLaw (fun ω (i : I) => X ω i) (Measure.pi (fun i : I => gaussianReal 0 (v i))) P ∧ |
| IndepFun (fun ω (i : I) => X ω i) Y P := |
| product_gaussian_law_and_indep_of_condExp_linear_character hm |
| (by fun_prop) (fun i : I => v i) (hchar I) hY |
| have hcoordinate (i : ι) : HasLaw (fun ω => X ω i) (gaussianReal 0 (v i)) P := by |
| let j : ({i} : Finset ι) := ⟨i, Finset.mem_singleton_self i⟩ |
| exact (measurePreserving_eval (fun k : ({i} : Finset ι) => gaussianReal 0 (v k)) j).fun_comp_hasLaw (hfinite {i}).1 |
| have hind : iIndepFun (fun i ω => X ω i) P := by |
| apply iIndepFun_iff_finset.mpr |
| intro I |
| exact (iIndepFun_iff_hasLaw_pi_pi (fun i : I => hcoordinate i)).mpr (hfinite I).1 |
| refine ⟨hind.hasLaw_infinitePi hcoordinate hX.aemeasurable, hind, ?_⟩ |
| exact IndepFun.process_indepFun (fun i => (measurable_pi_apply i).comp hX) |
| (hY.mono hm le_rfl) (fun I => (hfinite I).2) |
|
|
| end ArbitraryArray |
|
|
| end Problem367.ConditionalGaussianVector |
|
|
| end |
|
|
| end OAI |
|
|
