openAI-math-dataset / lean /OAI /Probability /FreeIsing /Gaussian /BinaryPosteriorResponseBound.lean
| import OAI.Probability.FreeIsing.Energy.ProgressiveIsingCovariance | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter ProbabilityTheory | |
| open scoped NNReal ENNReal | |
| namespace Problem367 | |
| namespace BinaryPosteriorResponseBound | |
| lemma covariance_le_variance_of_pair_bounds {m n p : ℝ} | |
| (hm : |m| ≤ 1) (hp : |p + n| ≤ 1 + m) (hn : |n - p| ≤ 1 - m) : | |
| |p - m * n| ≤ 1 - m ^ 2 := by | |
| have hm0 : 0 ≤ 1 - m := by linarith [(abs_le.mp hm).2] | |
| have hm1 : 0 ≤ 1 + m := by linarith [(abs_le.mp hm).1] | |
| have hpU := mul_le_mul_of_nonneg_left (abs_le.mp hp).2 hm0 | |
| have hpL := mul_le_mul_of_nonneg_left (abs_le.mp hp).1 hm0 | |
| have hnU := mul_le_mul_of_nonneg_left (abs_le.mp hn).2 hm1 | |
| have hnL := mul_le_mul_of_nonneg_left (abs_le.mp hn).1 hm1 | |
| rw [abs_le] | |
| constructor <;> nlinarith | |
| lemma condExp_abs_le_of_abs_le | |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} | |
| {F : MeasurableSpace Ω} {f g : Ω → ℝ} | |
| (hf : Integrable f μ) (hg : Integrable g μ) | |
| (hfg : ∀ᵐ omega ∂μ, |f omega| ≤ g omega) : | |
| ∀ᵐ omega ∂μ, |μ[f | F] omega| ≤ μ[g | F] omega := by | |
| exact (abs_condExp_ae_le_condExp_abs f).trans (condExp_mono hf.abs hg hfg) | |
| theorem abs_posteriorCovariance_le_one_sub_sq | |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] | |
| {F : MeasurableSpace Ω} (hF : F ≤ mΩ) {f g : Ω → ℝ} | |
| (hf : AEStronglyMeasurable[mΩ] f μ) (hg : AEStronglyMeasurable[mΩ] g μ) | |
| (hfb : ∀ᵐ omega ∂μ, |f omega| ≤ 1) | |
| (hgb : ∀ᵐ omega ∂μ, |g omega| ≤ 1) : | |
| ∀ᵐ omega ∂μ, |posteriorCovariance (mΩ := mΩ) μ F f g omega| ≤ 1 - μ[f | F] omega ^ 2 := by | |
| have hfi : Integrable f μ := Integrable.of_bound hf 1 (by simpa using hfb) | |
| have hgi : Integrable g μ := Integrable.of_bound hg 1 (by simpa using hgb) | |
| have hfg : Integrable (fun omega => f omega * g omega) μ := by | |
| apply Integrable.of_bound (hf.mul hg) 1 | |
| filter_upwards [hfb, hgb] with omega hf hg | |
| rw [Real.norm_eq_abs, Pi.mul_apply, abs_mul] | |
| simpa using mul_le_mul hf hg (abs_nonneg _) (by norm_num : (0 : ℝ) ≤ 1) | |
| have hplus : ∀ᵐ omega ∂μ, |f omega * g omega + g omega| ≤ 1 + f omega := by | |
| filter_upwards [hfb, hgb] with omega hf hg | |
| have hnon : 0 ≤ 1 + f omega := by linarith [(abs_le.mp hf).1] | |
| calc | |
| |f omega * g omega + g omega| = |(1 + f omega) * g omega| := by congr 1; ring | |
| _ = (1 + f omega) * |g omega| := by rw [abs_mul, abs_of_nonneg hnon] | |
| _ ≤ 1 + f omega := by nlinarith | |
| have hminus : ∀ᵐ omega ∂μ, |g omega - f omega * g omega| ≤ 1 - f omega := by | |
| filter_upwards [hfb, hgb] with omega hf hg | |
| have hnon : 0 ≤ 1 - f omega := by linarith [(abs_le.mp hf).2] | |
| calc | |
| |g omega - f omega * g omega| = |(1 - f omega) * g omega| := by congr 1; ring | |
| _ = (1 - f omega) * |g omega| := by rw [abs_mul, abs_of_nonneg hnon] | |
| _ ≤ 1 - f omega := by nlinarith | |
| have hp := condExp_abs_le_of_abs_le (mΩ := mΩ) (μ := μ) (F := F) (hfg.add hgi) | |
| ((integrable_const 1).add hfi) hplus | |
| have hn := condExp_abs_le_of_abs_le (mΩ := mΩ) (μ := μ) (F := F) (hgi.sub hfg) | |
| ((integrable_const 1).sub hfi) hminus | |
| have hm := ae_bdd_norm_condExp_of_ae_bdd_norm (m := F) | |
| (show ∀ᵐ omega ∂μ, ‖f omega‖ ≤ 1 by simpa using hfb) | |
| filter_upwards [hp, hn, hm, condExp_add hfg hgi F, condExp_sub hgi hfg F, | |
| condExp_add (integrable_const (1 : ℝ)) hfi F, | |
| condExp_sub (integrable_const (1 : ℝ)) hfi F] | |
| with omega hp hn hm hpa hna hpc hnc | |
| simp only [Pi.add_apply, Pi.sub_apply, condExp_const hF] at hpa hna hpc hnc | |
| rw [hpa, hpc] at hp | |
| rw [hna, hnc] at hn | |
| exact covariance_le_variance_of_pair_bounds hm hp hn | |
| theorem abs_normalized_posteriorCovariance_le_one | |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {μ : Measure Ω} [IsFiniteMeasure μ] | |
| {F : MeasurableSpace Ω} (hF : F ≤ mΩ) {f g : Ω → ℝ} | |
| (hf : AEStronglyMeasurable[mΩ] f μ) (hg : AEStronglyMeasurable[mΩ] g μ) | |
| (hfb : ∀ᵐ omega ∂μ, |f omega| ≤ 1) | |
| (hgb : ∀ᵐ omega ∂μ, |g omega| ≤ 1) : | |
| ∀ᵐ omega ∂μ, | |
| |posteriorCovariance (mΩ := mΩ) μ F f g omega / (1 - μ[f | F] omega ^ 2)| ≤ 1 := by | |
| filter_upwards [abs_posteriorCovariance_le_one_sub_sq (mΩ := mΩ) hF hf hg hfb hgb] | |
| with omega hbound | |
| have hnon : 0 ≤ 1 - μ[f | F] omega ^ 2 := (abs_nonneg _).trans hbound | |
| by_cases hzero : 1 - μ[f | F] omega ^ 2 = 0 | |
| · simp [hzero] | |
| · rw [abs_div, abs_of_nonneg hnon] | |
| exact (div_le_one (lt_of_le_of_ne hnon (Ne.symm hzero))).mpr hbound | |
| end BinaryPosteriorResponseBound | |
| open PlantedSymmetry | |
| theorem HasFreeIsingLaw.infiniteGibbsResponse_abs_le_one_gaussian | |
| {d : ℕ} {beta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (t : ℝ≥0) (v w : TreeVertex d) : | |
| ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), | |
| |infiniteGibbsResponse beta v w (observation t p)| ≤ 1 := by | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| have hs : Measurable spinSign := by intro s hs; trivial | |
| have hm (u : TreeVertex d) : Measurable | |
| (fun p : (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) => spinSign (p.1 u)) := | |
| hs.comp ((measurable_pi_apply u).comp measurable_fst) | |
| have hb (u : TreeVertex d) : ∀ᵐ p ∂μ.prod (gaussianNoiseLaw (TreeVertex d) t), | |
| |spinSign (p.1 u)| ≤ 1 := by | |
| apply Eventually.of_forall | |
| intro p | |
| cases h : p.1 u <;> norm_num [spinSign, h] | |
| have hc := BinaryPosteriorResponseBound.abs_normalized_posteriorCovariance_le_one | |
| (observationSigma_le (TreeVertex d) t) (hm v).aestronglyMeasurable | |
| (hm w).aestronglyMeasurable (hb v) (hb w) | |
| filter_upwards [hc, hμ.infiniteGibbsCovariance_eq_posterior t v w, | |
| hμ.infiniteGibbsSpinMean_eq_posterior t v] with p hc hcov hmean | |
| unfold infiniteGibbsResponse | |
| rw [hcov, hmean] | |
| exact hc | |
| theorem HasFreeIsingLaw.infiniteGibbsResponse_abs_le_one_brownian | |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) | |
| (ν : TreeVertex d → Measure Ω) | |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} | |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) | |
| (hmB : ∀ i t, Measurable (B i t)) (t : ℝ≥0) (v w : TreeVertex d) : | |
| ∀ᵐ p ∂μ.prod (Measure.infinitePi ν), | |
| |infiniteGibbsResponse beta v w | |
| (fun i => (t : ℝ) * spinSign (p.1 i) + B i t (p.2 i))| ≤ 1 := by | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := | |
| (hB i).isGaussianProcess.isProbabilityMeasure | |
| let T : ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) → | |
| (TreeVertex d → IsingSpin) × (TreeVertex d → ℝ) := | |
| Prod.map id (fun omega i => B i t (omega i)) | |
| have hT : MeasurePreserving T (μ.prod (Measure.infinitePi ν)) | |
| (μ.prod (gaussianNoiseLaw (TreeVertex d) t)) := | |
| (MeasurePreserving.id μ).prod (brownian_array_eval_measurePreserving ν hB hmB t) | |
| have hb := hμ.infiniteGibbsResponse_abs_le_one_gaussian t v w | |
| rw [← hT.map_eq] at hb | |
| have hp := ae_of_ae_map hT.measurable.aemeasurable hb | |
| exact hp | |
| theorem HasFreeIsingLaw.memLp_infiniteGibbsResponse_brownian | |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) | |
| (ν : TreeVertex d → Measure Ω) | |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} | |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) | |
| (hmB : ∀ i t, Measurable (B i t)) (t : ℝ≥0) (v w : TreeVertex d) (p : ℝ≥0∞) : | |
| MemLp (fun q : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω) => | |
| infiniteGibbsResponse beta v w | |
| (fun i => (t : ℝ) * spinSign (q.1 i) + B i t (q.2 i))) p | |
| (μ.prod (Measure.infinitePi ν)) := by | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := | |
| (hB i).isGaussianProcess.isProbabilityMeasure | |
| have hs : Measurable spinSign := by intro s hs; trivial | |
| have hobs : Measurable (fun q : (TreeVertex d → IsingSpin) × (TreeVertex d → Ω) => | |
| fun i => (t : ℝ) * spinSign (q.1 i) + B i t (q.2 i)) := | |
| Measurable.of_eval fun i => | |
| (measurable_const.mul (hs.comp ((measurable_pi_apply i).comp measurable_fst))).add | |
| ((hmB i t).comp ((measurable_pi_apply i).comp measurable_snd)) | |
| apply MemLp.of_bound ((measurable_infiniteGibbsResponse beta v w).comp hobs).aestronglyMeasurable 1 | |
| simpa only [Real.norm_eq_abs, Function.comp_def] using | |
| hμ.infiniteGibbsResponse_abs_le_one_brownian ν hB hmB t v w | |
| theorem HasFreeIsingLaw.memLp_infiniteGibbsResponse_time_brownian | |
| {d : ℕ} {beta : ℝ} {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure (TreeVertex d → IsingSpin)} | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) | |
| (ν : TreeVertex d → Measure Ω) | |
| {B : TreeVertex d → ℝ≥0 → Ω → ℝ} | |
| (hB : ∀ i, IsPreBrownianReal (B i) (ν i)) | |
| (hmB : ∀ i t, Measurable (B i t)) | |
| (hjoint : ∀ i, Measurable (fun z : ℝ≥0 × Ω => B i z.1 z.2)) | |
| (η : Measure ℝ≥0) [IsFiniteMeasure η] (v w : TreeVertex d) (p : ℝ≥0∞) : | |
| MemLp (fun q : ℝ≥0 × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) => | |
| infiniteGibbsResponse beta v w | |
| (fun i => (q.1 : ℝ) * spinSign (q.2.1 i) + B i q.1 (q.2.2 i))) p | |
| (η.prod (μ.prod (Measure.infinitePi ν))) := by | |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure | |
| have (i : TreeVertex d) : IsProbabilityMeasure (ν i) := | |
| (hB i).isGaussianProcess.isProbabilityMeasure | |
| have hs : Measurable spinSign := by intro s hs; trivial | |
| have hobs : Measurable | |
| (fun q : ℝ≥0 × ((TreeVertex d → IsingSpin) × (TreeVertex d → Ω)) => | |
| fun i => (q.1 : ℝ) * spinSign (q.2.1 i) + B i q.1 (q.2.2 i)) := by | |
| apply Measurable.of_eval | |
| intro i | |
| exact ((measurable_subtype_coe.comp measurable_fst).mul | |
| (hs.comp ((measurable_pi_apply i).comp (measurable_fst.comp measurable_snd)))).add | |
| ((hjoint i).comp (measurable_fst.prodMk | |
| ((measurable_pi_apply i).comp (measurable_snd.comp measurable_snd)))) | |
| have hresponse := (measurable_infiniteGibbsResponse beta v w).comp hobs | |
| apply MemLp.of_bound hresponse.aestronglyMeasurable 1 | |
| apply (Measure.ae_prod_iff_ae_ae | |
| (measurableSet_le hresponse.norm measurable_const)).mpr | |
| apply Eventually.of_forall | |
| intro t | |
| simpa only [Real.norm_eq_abs, Function.comp_def] using | |
| hμ.infiniteGibbsResponse_abs_le_one_brownian ν hB hmB t v w | |
| end Problem367 | |
| end | |
| end OAI | |