import OAI.Probability.GaussianReplacement.Refinement namespace OAI open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal NNReal RealInnerProductSpace Topology noncomputable section namespace CurrentProjection open InnerProductSpace open InformationTheory def criticalSignal {d : ℕ} {W : Type*} [MeasurableSpace W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsFiniteMeasure P] (ℓ : ℕ) : Measure (Sphere d×(W×(ℕ×ℤ))) := refineSignal P (criticalRefinement P ℓ) instance criticalSignal_probability {d : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] (ℓ : ℕ) : IsProbabilityMeasure (criticalSignal P ℓ) := refineSignal_probability P _ (measurable_criticalRefinement P ℓ) def criticalB {d : ℕ} {W : Type*} [MeasurableSpace W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsFiniteMeasure P] (ℓ : ℕ) (v : W×(ℕ×ℤ)) : ℝ := Real.exp ((v.2.2:ℝ)+(ℓ:ℝ)*Real.log 4+refinementLoss P (criticalRefinement P ℓ) v) def criticalD {d : ℕ} {W : Type*} [MeasurableSpace W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsFiniteMeasure P] (ℓ : ℕ) (v : W×(ℕ×ℤ)) : ℝ := Real.exp (-Real.log ((criticalSignal P ℓ).snd {v}).toReal)*(3*4^d) lemma criticalSignal_fst {d : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] (ℓ : ℕ) : (criticalSignal P ℓ).fst = P.fst := refineSignal_fst P _ (measurable_criticalRefinement P ℓ) lemma criticalSignal_posterior {d ℓ : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] {v : W×(ℕ×ℤ)} (hv : (criticalSignal P ℓ).snd {v} ≠ 0) : messagePosterior (criticalSignal P ℓ) v = criticalPosterior (messagePosterior P v.1) ℓ v.2.1 v.2.2 ∧ 0 < messagePosterior P v.1 (criticalClass (messagePosterior P v.1) ℓ v.2.1 v.2.2) ∧ P.snd {v.1} ≠ 0 := by have he := measurable_criticalRefinement P ℓ have hw := refineSignal_positive_original P _ he hv have hc : {s : Sphere d | criticalRefinement P ℓ (s,v.1) = v.2} = criticalClass (messagePosterior P v.1) ℓ v.2.1 v.2.2 := by ext s simp only [criticalRefinement,criticalClass,mem_ofPred_eq,Prod.ext_iff] have hmass := refineSignal_mass P _ he hw v.2 rw [hc] at hmass change (criticalSignal P ℓ).snd {v} = _ at hmass have hη : messagePosterior P v.1 (criticalClass (messagePosterior P v.1) ℓ v.2.1 v.2.2) ≠ 0 := by intro hz rw [hmass,hz,mul_zero] at hv exact hv rfl refine ⟨?_,hη.bot_lt,hw⟩ have hp := messagePosterior_refinement P _ he hv rw [hc] at hp exact hp lemma criticalB_eq {d ℓ : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] {v : W×(ℕ×ℤ)} (hv : (criticalSignal P ℓ).snd {v} ≠ 0) : criticalB P ℓ v = (4:ℝ)^ℓ * Real.exp (v.2.2:ℝ) / (messagePosterior P v.1 (criticalClass (messagePosterior P v.1) ℓ v.2.1 v.2.2)).toReal := by have he := measurable_criticalRefinement P ℓ have hc : {s : Sphere d | criticalRefinement P ℓ (s,v.1) = v.2} = criticalClass (messagePosterior P v.1) ℓ v.2.1 v.2.2 := by ext s simp only [criticalRefinement,criticalClass,mem_ofPred_eq,Prod.ext_iff] have hδ := refinementLoss_eq P _ he hv rw [hc] at hδ have hη := (criticalSignal_posterior P hv).2.1 have hηr := ENNReal.toReal_pos hη.ne' (measure_ne_top _ _) unfold criticalB rw [hδ,Real.exp_add,Real.exp_add,Real.exp_neg,Real.exp_log hηr, ←Real.log_pow,Real.exp_log (by positivity : (0:ℝ)<4^ℓ)] ring lemma criticalD_eq {d ℓ : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] {v : W×(ℕ×ℤ)} (hv : (criticalSignal P ℓ).snd {v} ≠ 0) : criticalD P ℓ v = ((criticalSignal P ℓ).snd {v}).toReal⁻¹*(3*4^d) := by rw [criticalD,Real.exp_neg,Real.exp_log (ENNReal.toReal_pos hv (measure_ne_top _ _))] lemma criticalFamily_growth {d ℓ : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] (hσ : P.fst = uniformSphere d) (hℓ : ℓ < d-1) {v : W×(ℕ×ℤ)} (hv : (criticalSignal P ℓ).snd {v} ≠ 0) : LocalGrowth (messagePosterior (criticalSignal P ℓ) v) ℓ (criticalB P ℓ v) ∧ LocalGrowth (messagePosterior (criticalSignal P ℓ) v) (d-1) (criticalD P ℓ v) ∧ 0 ≤ (v.2.2:ℝ)+(ℓ:ℝ)*Real.log 4+refinementLoss P (criticalRefinement P ℓ) v ∧ messagePosterior (criticalSignal P ℓ) v ≪ uniformSphere d := by obtain ⟨hpost,hη,hw⟩ := criticalSignal_posterior P hv have hd : 1 ≤ d := by omega have hg := posterior_sphere_growth P hσ hd hw let D := (P.snd {v.1}).toReal⁻¹*(3*4^d) have hD : 1 ≤ D := by have ha := ENNReal.toReal_pos hw (measure_ne_top _ _) have ha1 : (P.snd {v.1}).toReal ≤ 1 := by exact ENNReal.toReal_le_of_le_ofReal (by norm_num) (by simpa using (prob_le_one : P.snd {v.1} ≤ 1)) have hinv : 1 ≤ (P.snd {v.1}).toReal⁻¹ := (one_le_inv₀ ha).mpr ha1 have hp : (1:ℝ) ≤ 4^d := one_le_pow₀ (by norm_num) dsimp only [D] nlinarith have hsigma : (criticalSignal P ℓ).fst = uniformSphere d := (criticalSignal_fst P ℓ).trans hσ refine ⟨?_,?_,?_,?_⟩ · rw [hpost,criticalB_eq P hv] exact criticalPosterior_growth (messagePosterior P v.1) D (by linarith) hg hℓ v.2.2 hη · rw [criticalD_eq P hv] exact posterior_sphere_growth (criticalSignal P ℓ) hsigma hd hv · obtain ⟨s,hs⟩ := nonempty_of_measure_ne_zero hη.ne' have hq := (criticalLevel_bounds (messagePosterior P v.1) D hD hg hℓ s).1 rw [hs.2] at hq have hl4 : 1 ≤ Real.log 4 := by have hh : Real.log 4 = 2*Real.log 2 := by have hh := (Real.log_pow (2:ℝ) 2) norm_num at hh ⊢ exact hh linarith [Real.log_two_gt_d9] have ht := refinementLoss_nonneg P _ (measurable_criticalRefinement P ℓ) hv nlinarith [Nat.cast_nonneg (α:=ℝ) ℓ] · have hh := messagePosterior_le (criticalSignal P ℓ) hv rw [hsigma] at hh exact hh.absolutelyContinuous.trans Measure.smul_absolutelyContinuous lemma criticalSignal_entropy {d ℓ : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] (hσ : P.fst = uniformSphere d) (hℓ : ℓ < d-1) (hH : shannonEntropy P.snd ≠ ∞) : shannonEntropy (P.map (criticalRefinement P ℓ)) ≠ ∞ ∧ (shannonEntropy (P.map (criticalRefinement P ℓ))).toReal ≤ 1+2*Real.log (20*((d:ℝ)+1+(shannonEntropy P.snd).toReal)+1) ∧ shannonEntropy (criticalSignal P ℓ).snd ≠ ∞ ∧ (shannonEntropy (criticalSignal P ℓ).snd).toReal ≤ (shannonEntropy P.snd).toReal+ (shannonEntropy (P.map (criticalRefinement P ℓ))).toReal := by obtain ⟨he,hb⟩ := criticalRefinement_entropy P hσ hℓ hH obtain ⟨hp,hpb,_,_⟩ := refinement_entropy_and_loss P _ (measurable_criticalRefinement P ℓ) hH he exact ⟨he,hb,hp,hpb⟩ lemma criticalLevel_integrable {d ℓ : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] (hσ : P.fst = uniformSphere d) (hℓ : ℓ < d-1) (hH : shannonEntropy P.snd ≠ ∞) : Integrable (fun v : W×(ℕ×ℤ) => (v.2.2:ℝ)) (criticalSignal P ℓ).snd := by let e := criticalRefinement P ℓ have he := measurable_criticalRefinement P ℓ have hw : Integrable (fun p : Sphere d×W => -Real.log (P.snd {p.2}).toReal) P := (integrable_map_measure (measurable_of_countable _).aestronglyMeasurable measurable_snd.aemeasurable).mp (entropy_finite_integrable P.snd hH) have hbi : Integrable (fun p : Sphere d×W => 20*((d:ℝ)+1-Real.log (P.snd {p.2}).toReal)) P := by convert ((integrable_const ((d:ℝ)+1)).add hw).const_mul 20 using 1 have hqi : Integrable (fun p => ((e p).2:ℝ)) P := by apply hbi.mono' ((measurable_of_countable (fun x : ℕ×ℤ => (x.2:ℝ))).comp he).aestronglyMeasurable have hp := (ae_map_iff measurable_snd.aemeasurable ((Set.to_countable _).measurableSet)).mp (ae_positive_atom P.snd) filter_upwards [hp] with p hp have hb := criticalRefinement_size P hσ hℓ hp change ‖((e p).2:ℝ)‖ ≤ _ rw [Real.norm_eq_abs] have habs : |((e p).2:ℝ)| = ((e p).2.natAbs:ℝ) := by simp rw [habs] change ((e p).1:ℝ)+2*((e p).2.natAbs:ℝ) ≤ _ at hb nlinarith [Nat.cast_nonneg (α:=ℝ) (e p).1,Nat.cast_nonneg (α:=ℝ) (e p).2.natAbs] change Integrable _ (refineSignal P e).snd rw [refineSignal_snd P e he] exact (integrable_map_measure (measurable_of_countable _).aestronglyMeasurable (measurable_snd.prodMk he).aemeasurable).mpr hqi lemma criticalFamily_moment_mean {d ℓ : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] (hσ : P.fst = uniformSphere d) (hℓ : ℓ < d-1) (hH : shannonEntropy P.snd ≠ ∞) : Integrable (fun v => Real.log (1+momentBound 9 d (criticalB P ℓ v) (criticalD P ℓ v))) (criticalSignal P ℓ).snd ∧ (∫ v, Real.log (1+momentBound 9 d (criticalB P ℓ v) (criticalD P ℓ v)) ∂(criticalSignal P ℓ).snd) ≤ (∫ v : W×(ℕ×ℤ), (v.2.2:ℝ) ∂(criticalSignal P ℓ).snd) + (shannonEntropy (P.map (criticalRefinement P ℓ))).toReal + 18*d + Real.log (1+(shannonEntropy (criticalSignal P ℓ).snd).toReal) := by let π := (criticalSignal P ℓ).snd let e := criticalRefinement P ℓ let s : W×(ℕ×ℤ) → ℝ := fun v => -Real.log (π {v}).toReal have hs : Integrable s π := entropy_finite_integrable π (criticalSignal_entropy P hσ hℓ hH).2.2.1 have hsm : Measurable s := measurable_of_countable _ have hs0 : ∀ v, 0 ≤ s v := atom_surprisal_nonneg π have hlog := integrable_log_one_add_real π hs hsm (Eventually.of_forall hs0) have hlogmean := integral_log_one_add_real_le π hs hsm (Eventually.of_forall hs0) have hsmean : (∫ v, s v ∂π) = (shannonEntropy π).toReal := integral_surprisal π rw [hsmean] at hlogmean obtain ⟨_,_,hδ,hδmean⟩ := refinement_entropy_and_loss P e (measurable_criticalRefinement P ℓ) hH (criticalSignal_entropy P hσ hℓ hH).1 have hqi := criticalLevel_integrable P hσ hℓ hH have hupper : Integrable (fun v : W×(ℕ×ℤ) => ((v.2.2:ℝ)+refinementLoss P e v)+18*d+Real.log (1+s v)) π := ((hqi.add hδ).add (integrable_const _)).add hlog have hle : ∀ᵐ v ∂π, Real.log (1+momentBound 9 d (criticalB P ℓ v) (criticalD P ℓ v)) ≤ ((v.2.2:ℝ)+refinementLoss P e v)+18*d+Real.log (1+s v) := by filter_upwards [ae_positive_atom π] with v hv have hb := (criticalFamily_growth P hσ hℓ hv).2.2.1 exact critical_moment_log_bound (by omega) (by omega) _ _ _ (hs0 v) hb have hL : Integrable (fun v => Real.log (1+momentBound 9 d (criticalB P ℓ v) (criticalD P ℓ v))) π := by apply hupper.mono' (measurable_of_countable _).aestronglyMeasurable filter_upwards [hle] with v hv have hB : 0 ≤ criticalB P ℓ v := (Real.exp_pos _).le rw [Real.norm_eq_abs,abs_of_nonneg (Real.log_nonneg (le_add_of_nonneg_right (momentBound_nonneg 9 d hB)))] exact hv refine ⟨hL,?_⟩ have hh := integral_mono_ae hL hupper hle have hqδ : Integrable (fun v : W×(ℕ×ℤ) => (v.2.2:ℝ)+refinementLoss P e v) π := hqi.add hδ have hu : Integrable (fun v : W×(ℕ×ℤ) => (v.2.2:ℝ)+refinementLoss P e v+18*(d:ℝ)) π := hqδ.add (integrable_const _) rw [integral_add hu hlog,integral_add hqδ (integrable_const (18*(d:ℝ))),integral_add hqi hδ, integral_const,probReal_univ,one_smul] at hh exact hh.trans (add_le_add (add_le_add (add_le_add le_rfl hδmean) le_rfl) hlogmean) lemma completeFamily_restrict_apply {W T : Type*} [MeasurableSpace W] (π : Measure W) (f : W → T) (t₀ : T) {w : W} (hw : π {w} ≠ 0) : completeFamily π (fun v => f v.1) t₀ w = f w := by have hp : 0 < π {w} := pos_iff_ne_zero.mpr hw rw [completeFamily, dite_eq_left hp] lemma completeFamily_restrict_ae {W T : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] (π : Measure W) (f : W → T) (t₀ : T) : completeFamily π (fun v => f v.1) t₀ =ᵐ[π] f := by filter_upwards [ae_positive_atom π] with w hw exact completeFamily_restrict_apply π f t₀ hw lemma actual_posterior_completed {d m r q : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (hdim : AdmissibleDimensions d m r q) (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] (ν : Measure (Sphere d)) [IsProbabilityMeasure ν] (hInd : (P.map swapLast).fst = ν.prod (gaussianRows m d)) (H : ℝ) (hν : ν ≤ ENNReal.ofReal H • uniformSphere d) (hH : shannonEntropy P.fst.snd ≠ ∞) (B D : W → ℝ) (hB : ∀ w, P.fst.snd {w} ≠ 0 → 0 < B w) (hD : ∀ w, P.fst.snd {w} ≠ 0 → 0 < D w) (hAC : ∀ w, P.fst.snd {w} ≠ 0 → messagePosterior P.fst w ≪ uniformSphere d) (hgB : ∀ w, P.fst.snd {w} ≠ 0 → LocalGrowth (messagePosterior P.fst w) (m+r) (B w)) (hgD : ∀ w, P.fst.snd {w} ≠ 0 → LocalGrowth (messagePosterior P.fst w) (d-1) (D w)) (hCI : Integrable (fun w => Real.log (1+momentBound 9 d (B w) (D w))) P.fst.snd) : ActualRowConclusion (r:=r) (q:=q) P (completeFamily (P.map swapLast).snd (fun w => messagePosterior P.fst w.1) 0) (completeFamily (P.map swapLast).snd (fun w => B w.1) 1) (completeFamily (P.map swapLast).snd (fun w => D w.1) 1) := by let π := (P.map swapLast).snd have hπ : π = P.fst.snd := swapLast_snd P let ρ : PositiveMessage π → Measure (Sphere d) := fun w => messagePosterior P.fst w.1 let b : PositiveMessage π → ℝ := fun w => B w.1 let c : PositiveMessage π → ℝ := fun w => D w.1 have hw (w : PositiveMessage π) : P.fst.snd {w.1} ≠ 0 := by rw [←hπ]; exact w.2.ne' let f : W → ℝ := fun w => Real.log (1+momentBound 9 d (B w) (D w)) let f' : W → ℝ := fun w => Real.log (1+momentBound 9 d (completeFamily π b 1 w) (completeFamily π c 1 w)) have hf : f' =ᵐ[π] f := by filter_upwards [ae_positive_atom π] with w hw dsimp only [f',f,b,c] rw [completeFamily_restrict_apply π B 1 hw,completeFamily_restrict_apply π D 1 hw] have hfbase : (fun sw : Sphere d×W => f' sw.2) =ᵐ[P.fst] (fun sw => f sw.2) := by apply (ae_map_iff measurable_snd.aemeasurable (measurableSet_eq_fun (measurable_of_countable f') (measurable_of_countable f))).mp change f' =ᵐ[P.fst.snd] f rw [←hπ] exact hf have hCI' : Integrable (fun sw : Sphere d×W => f' sw.2) P.fst := by apply Integrable.congr _ hfbase.symm exact (integrable_map_measure (measurable_of_countable f).aestronglyMeasurable measurable_snd.aemeasurable).mp hCI exact actualRowMain hdim P ν hInd H hν (by rwa [swapLast_snd]) ρ (fun w => hAC w.1 (hw w)) b c (fun w => hB w.1 (hw w)) (fun w => hD w.1 (hw w)) (fun w => hgB w.1 (hw w)) (fun w => hgD w.1 (hw w)) hCI' lemma actualRowConclusion_observed {d m r q : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] (ρ : W → Measure (Sphere d)) [∀ w, IsFiniteMeasure (ρ w)] (B D : W → ℝ) (hmain : ActualRowConclusion (r:=r) (q:=q) P ρ B D) : Integrable (fun z => Real.log (1+(exactDensity (ρ z.2) z.1.1 z.1.2).toReal)) (observedMessage (alignedExperiment (r:=r) P)) ∧ Integrable (fun z => logPositive (exactDensity (ρ z.2) z.1.1 z.1.2)) (observedMessage (alignedExperiment (r:=r) P)) ∧ (∫ z, Real.log (1+(exactDensity (ρ z.2) z.1.1 z.1.2).toReal) ∂observedMessage (alignedExperiment (r:=r) P)) ≤ (rowInformation P (gaussianRows m d)).toReal/q + ∫ w, Real.log (1+momentBound 9 d (B w) (D w)) ∂P.fst.snd := by let O := observedMessage (alignedExperiment (r:=r) P) let p : (Rows (m+r) d×(Fin (m+r)→ℝ))×W → ℝ≥0∞ := fun z => exactDensity (ρ z.2) z.1.1 z.1.2 have hp : Measurable p := measurable_from_prod_countable_left fun w => measurable_exactDensity (k:=m+r) (ρ w) let T : ((Sphere d×W)×Rows m d)×Rows r d → (Rows (m+r) d×(Fin (m+r)→ℝ))×W := observedMessageMap ∘ (fun p => (p.1.1,Fin.append p.1.2 p.2)) have hT : Measurable T := measurable_observedMessageMap.comp measurable_alignedMap have hO : O = (actualExperiment (r:=r) P).map T := by dsimp only [O] rw [observedMessage,alignedExperiment,Measure.map_map measurable_observedMessageMap measurable_alignedMap] have hLi : Integrable (fun z => Real.log (1+(p z).toReal)) O := by rw [hO] apply (integrable_map_measure (measurable_log_one_add hp).aestronglyMeasurable hT.aemeasurable).mpr exact hmain.exact_integrable have hPi : Integrable (fun z => logPositive (p z)) O := by rw [hO] apply (integrable_map_measure ((hp.ennreal_toReal.log.max measurable_const).aestronglyMeasurable) hT.aemeasurable).mpr exact hmain.positive_log_integrable refine ⟨hLi,hPi,?_⟩ change (∫ z, Real.log (1+(p z).toReal) ∂O) ≤ _ rw [hO,integral_map hT.aemeasurable (measurable_log_one_add hp).aestronglyMeasurable] have hb := hmain.exact_bound rw [←integral_map measurable_snd.aemeasurable (measurable_of_countable (fun w => Real.log (1+momentBound 9 d (B w) (D w)))).aestronglyMeasurable] at hb exact hb lemma observedMessage_aligned_positive {d m r : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] : ∀ᵐ z ∂observedMessage (alignedExperiment (r:=r) P), (P.map swapLast).snd {z.2} ≠ 0 := by have hpos : ∀ᵐ w ∂(observedMessage (alignedExperiment (r:=r) P)).snd, (P.map swapLast).snd {w} ≠ 0 := by rw [observedMessage_snd,alignedExperiment_fst,swapLast_snd] exact ae_positive_atom P.fst.snd exact (ae_map_iff measurable_snd.aemeasurable (Set.to_countable {w : W | (P.map swapLast).snd {w} ≠ 0}).measurableSet).mp hpos lemma actual_posterior_log_bound {d m r q : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (hdim : AdmissibleDimensions d m r q) (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] (ν : Measure (Sphere d)) [IsProbabilityMeasure ν] (hInd : (P.map swapLast).fst = ν.prod (gaussianRows m d)) (H : ℝ) (hν : ν ≤ ENNReal.ofReal H • uniformSphere d) (hH : shannonEntropy P.fst.snd ≠ ∞) (B D : W → ℝ) (hB : ∀ w, P.fst.snd {w} ≠ 0 → 0 < B w) (hD : ∀ w, P.fst.snd {w} ≠ 0 → 0 < D w) (hAC : ∀ w, P.fst.snd {w} ≠ 0 → messagePosterior P.fst w ≪ uniformSphere d) (hgB : ∀ w, P.fst.snd {w} ≠ 0 → LocalGrowth (messagePosterior P.fst w) (m+r) (B w)) (hgD : ∀ w, P.fst.snd {w} ≠ 0 → LocalGrowth (messagePosterior P.fst w) (d-1) (D w)) (hCI : Integrable (fun w => Real.log (1+momentBound 9 d (B w) (D w))) P.fst.snd) : Integrable (fun z => Real.log (1+(exactDensity (messagePosterior P.fst z.2) z.1.1 z.1.2).toReal)) (observedMessage (alignedExperiment (r:=r) P)) ∧ Integrable (fun z => logPositive (exactDensity (messagePosterior P.fst z.2) z.1.1 z.1.2)) (observedMessage (alignedExperiment (r:=r) P)) ∧ (∫ z, Real.log (1+(exactDensity (messagePosterior P.fst z.2) z.1.1 z.1.2).toReal) ∂observedMessage (alignedExperiment (r:=r) P)) ≤ (rowInformation P (gaussianRows m d)).toReal/q + ∫ w, Real.log (1+momentBound 9 d (B w) (D w)) ∂P.fst.snd := by let π := (P.map swapLast).snd let ρ' := completeFamily π (fun w => messagePosterior P.fst w.1) 0 let B' := completeFamily π (fun w => B w.1) 1 let D' := completeFamily π (fun w => D w.1) 1 have hmain := actual_posterior_completed hdim P ν hInd H hν hH B D hB hD hAC hgB hgD hCI obtain ⟨hLi,hPi,hBound⟩ := actualRowConclusion_observed P ρ' B' D' hmain have heqp : (fun z => exactDensity (ρ' z.2) z.1.1 z.1.2) =ᵐ[observedMessage (alignedExperiment (r:=r) P)] (fun z => exactDensity (messagePosterior P.fst z.2) z.1.1 z.1.2) := by filter_upwards [observedMessage_aligned_positive (r:=r) P] with z hz dsimp only [ρ'] rw [completeFamily_restrict_apply π (messagePosterior P.fst) 0 hz] have heqlog := heqp.mono (fun z hz => congrArg (fun t : ℝ≥0∞ => Real.log (1+t.toReal)) hz) have heqpos := heqp.mono (fun z hz => congrArg logPositive hz) refine ⟨hLi.congr heqlog,hPi.congr heqpos,?_⟩ have hf : (fun w => Real.log (1+momentBound 9 d (B' w) (D' w))) =ᵐ[P.fst.snd] (fun w => Real.log (1+momentBound 9 d (B w) (D w))) := by rw [←swapLast_snd P] filter_upwards [ae_positive_atom π] with w hw dsimp only [B',D'] rw [completeFamily_restrict_apply π B 1 hw,completeFamily_restrict_apply π D 1 hw] rw [integral_congr_ae heqlog,integral_congr_ae hf] at hBound exact hBound lemma criticalFamily_cover_fibre {d ℓ : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] (hσ : P.fst = uniformSphere d) (hℓ : ℓ < d-1) {v : W×(ℕ×ℤ)} (hv : (criticalSignal P ℓ).snd {v} ≠ 0) (hac : ∀ᵐ G ∂gaussianRows ℓ d, labelLaw (messagePosterior (criticalSignal P ℓ) v) G ≪ volume) (hpi : Integrable (fun z : Rows ℓ d×(Fin ℓ→ℝ) => logPositive (exactDensity (messagePosterior (criticalSignal P ℓ) v) z.1 z.2)) (projectionJoint (messagePosterior (criticalSignal P ℓ) v) (gaussianRows ℓ d))) : -(∫ z, Real.log (exactDensity (messagePosterior (criticalSignal P ℓ) v) z.1 z.2).toReal ∂projectionJoint (messagePosterior (criticalSignal P ℓ) v) (gaussianRows ℓ d)) ≤ 1+5*(ℓ:ℝ)-(v.2.2:ℝ) := by let R := criticalSignal P ℓ obtain ⟨hpost,hη,hw⟩ := criticalSignal_posterior P hv have hpi' : Integrable (fun z : Rows ℓ d×Sphere d => logPositive (exactDensity (messagePosterior R v) z.1 (label z.1 (z.2:Vector d)))) ((gaussianRows ℓ d).prod (messagePosterior R v)) := by rw [projectionJoint] at hpi exact (integrable_map_measure ((measurable_exactDensity _).ennreal_toReal.log.max measurable_const).aestronglyMeasurable (measurable_fst.prodMk measurable_own_label).aemeasurable).mp hpi change ∀ᵐ G ∂gaussianRows ℓ d, labelLaw (messagePosterior R v) G ≪ volume at hac rw [show messagePosterior R v = criticalPosterior (messagePosterior P v.1) ℓ v.2.1 v.2.2 from hpost] at hpi' hac have hbound := (criticalPosterior_cover_entropy (messagePosterior P v.1) ((P.snd {v.1}).toReal⁻¹*(3*4^d)) (by positivity) (posterior_sphere_growth P hσ (by omega) hw) hℓ v.2.2 hη hac hpi').2 rw [projectionJoint,integral_map (measurable_fst.prodMk measurable_own_label).aemeasurable (measurable_exactDensity _).ennreal_toReal.log.aestronglyMeasurable] simpa only [hpost] using hbound lemma criticalFamily_cover_mean {d ℓ : ℕ} {W : Type*} [MeasurableSpace W] [Countable W] [MeasurableSingletonClass W] [Nonempty (Sphere d)] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] (hσ : P.fst = uniformSphere d) (hℓ : ℓ < d-1) (hH : shannonEntropy P.snd ≠ ∞) (hAC : ∀ᵐ v ∂(criticalSignal P ℓ).snd, ∀ᵐ G ∂gaussianRows ℓ d, labelLaw (messagePosterior (criticalSignal P ℓ) v) G ≪ volume) (hpos : Integrable (fun z => logPositive (exactDensity (messagePosterior (criticalSignal P ℓ) z.2) z.1.1 z.1.2)) (observedMessage ((criticalSignal P ℓ).prod (gaussianRows ℓ d)))) (hlog : Integrable (fun z => Real.log (exactDensity (messagePosterior (criticalSignal P ℓ) z.2) z.1.1 z.1.2).toReal) (observedMessage ((criticalSignal P ℓ).prod (gaussianRows ℓ d)))) : -(∫ z, Real.log (exactDensity (messagePosterior (criticalSignal P ℓ) z.2) z.1.1 z.1.2).toReal ∂observedMessage ((criticalSignal P ℓ).prod (gaussianRows ℓ d))) ≤ 1+5*(ℓ:ℝ)-(∫ v : W×(ℕ×ℤ), (v.2.2:ℝ) ∂(criticalSignal P ℓ).snd) := by let R := criticalSignal P ℓ let π := R.snd let κ := posteriorProjectionKernel R (gaussianRows ℓ d) let f : (W×(ℕ×ℤ))×(Rows ℓ d×(Fin ℓ→ℝ)) → ℝ := fun z => Real.log (exactDensity (messagePosterior R z.1) z.2.1 z.2.2).toReal let fp : (W×(ℕ×ℤ))×(Rows ℓ d×(Fin ℓ→ℝ)) → ℝ := fun z => logPositive (exactDensity (messagePosterior R z.1) z.2.1 z.2.2) have hm : Measurable (fun z : (Rows ℓ d×(Fin ℓ→ℝ))×(W×(ℕ×ℤ)) => exactDensity (messagePosterior R z.2) z.1.1 z.1.2) := measurable_from_prod_countable_left fun v => measurable_exactDensity (k:=ℓ) (messagePosterior R v) have hfi : Integrable f (π ⊗ₘ κ) := by rw [observedMessage_independent_disintegrate] at hlog exact (integrable_map_measure hm.ennreal_toReal.log.aestronglyMeasurable measurable_swap.aemeasurable).mp hlog have hfpi : Integrable fp (π ⊗ₘ κ) := by rw [observedMessage_independent_disintegrate] at hpos exact (integrable_map_measure (hm.ennreal_toReal.log.max measurable_const).aestronglyMeasurable measurable_swap.aemeasurable).mp hpos have hji : Integrable (fun v => ∫ z, f (v,z) ∂κ v) π := by simpa only [Kernel.prodMkLeft_apply,Kernel.const_apply] using hfi.integral_compProd have hposae : ∀ᵐ v ∂π, Integrable (fun z => fp (v,z)) (κ v) := (Measure.integrable_compProd_iff hfpi.aestronglyMeasurable).mp hfpi |>.1 have hb : ∀ᵐ v ∂π, -(∫ z, f (v,z) ∂κ v) ≤ 1+5*(ℓ:ℝ)-(v.2.2:ℝ) := by filter_upwards [ae_positive_atom π,hposae,hAC] with v hv hpi hac have hk : κ v = projectionJoint (messagePosterior R v) (gaussianRows ℓ d) := posteriorProjectionKernel_apply R _ _ dsimp only [fp] at hpi rw [hk] at hpi dsimp only [f] rw [hk] exact criticalFamily_cover_fibre P hσ hℓ hv hac hpi have hqi := criticalLevel_integrable P hσ hℓ hH have hupper : Integrable (fun v : W×(ℕ×ℤ) => 1+5*(ℓ:ℝ)-(v.2.2:ℝ)) π := (integrable_const _).sub hqi have he := integral_mono_ae hji.neg hupper hb simp only [Pi.neg_apply] at he rw [integral_neg,integral_sub (integrable_const (1+5*(ℓ:ℝ))) hqi, integral_const,probReal_univ,one_smul] at he rw [observedMessage_independent_disintegrate, integral_map measurable_swap.aemeasurable hm.ennreal_toReal.log.aestronglyMeasurable] change -(∫ z, f z ∂π ⊗ₘ κ) ≤ _ rw [Measure.integral_compProd hfi] exact he def refineRows {d m : ℕ} {W E : Type*} [MeasurableSpace W] [MeasurableSpace E] (P : Measure ((Sphere d×W)×Rows m d)) (e : Sphere d×W → E) : Measure ((Sphere d×(W×E))×Rows m d) := P.map (fun p => ((p.1.1,(p.1.2,e p.1)),p.2)) lemma measurable_refineRowsMap {d m : ℕ} {W E : Type*} [MeasurableSpace W] [MeasurableSpace E] (e : Sphere d×W → E) (he : Measurable e) : Measurable (fun p : (Sphere d×W)×Rows m d => ((p.1.1,(p.1.2,e p.1)),p.2)) := (measurable_fst.fst.prodMk (measurable_fst.snd.prodMk (he.comp measurable_fst))).prodMk measurable_snd lemma refineRows_probability {d m : ℕ} {W E : Type*} [MeasurableSpace W] [MeasurableSpace E] (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] (e : Sphere d×W → E) (he : Measurable e) : IsProbabilityMeasure (refineRows P e) := by unfold refineRows exact (Measure.isProbabilityMeasure_map_iff (measurable_refineRowsMap e he).aemeasurable).mpr inferInstance lemma refineRows_fst {d m : ℕ} {W E : Type*} [MeasurableSpace W] [MeasurableSpace E] (P : Measure ((Sphere d×W)×Rows m d)) (e : Sphere d×W → E) (he : Measurable e) : (refineRows P e).fst = refineSignal P.fst e := by rw [refineRows,Measure.fst,refineSignal,Measure.fst, Measure.map_map measurable_fst (measurable_refineRowsMap e he), Measure.map_map (measurable_fst.prodMk (measurable_snd.prodMk he)) measurable_fst] rfl lemma refineRows_signal_rows {d m : ℕ} {W E : Type*} [MeasurableSpace W] [MeasurableSpace E] (P : Measure ((Sphere d×W)×Rows m d)) (e : Sphere d×W → E) (he : Measurable e) : ((refineRows P e).map swapLast).fst = (P.map swapLast).fst := by rw [Measure.fst,Measure.fst,Measure.map_map measurable_fst swapLast.measurable, Measure.map_map measurable_fst swapLast.measurable,refineRows, Measure.map_map (measurable_fst.comp swapLast.measurable) (measurable_refineRowsMap e he)] rfl lemma refineRows_drop {d m : ℕ} {W E : Type*} [MeasurableSpace W] [MeasurableSpace E] (P : Measure ((Sphere d×W)×Rows m d)) (e : Sphere d×W → E) (he : Measurable e) : (refineRows P e).map (fun p => ((p.1.1,p.1.2.1),p.2)) = P := by rw [refineRows,Measure.map_map ((measurable_fst.fst.prodMk measurable_fst.snd.fst).prodMk measurable_snd) (measurable_refineRowsMap e he)] exact Measure.map_id lemma alignedExperiment_drop {d m r : ℕ} {W E : Type*} [MeasurableSpace W] [MeasurableSpace E] (P : Measure ((Sphere d×(W×E))×Rows m d)) [IsProbabilityMeasure P] : (alignedExperiment (r:=r) P).map (fun p => ((p.1.1,p.1.2.1),p.2)) = alignedExperiment (r:=r) (P.map (fun p => ((p.1.1,p.1.2.1),p.2))) := by let T : (Sphere d×(W×E))×Rows m d → (Sphere d×W)×Rows m d := fun p => ((p.1.1,p.1.2.1),p.2) have hT : Measurable T := (measurable_fst.fst.prodMk measurable_fst.snd.fst).prodMk measurable_snd have hprod : (P.map T).prod (gaussianRows r d) = (P.prod (gaussianRows r d)).map (Prod.map T id) := by simpa only [Measure.map_id] using (Measure.map_prod_map P (gaussianRows r d) hT measurable_id) rw [alignedExperiment,Measure.map_map ((measurable_fst.fst.prodMk measurable_fst.snd.fst).prodMk measurable_snd) measurable_alignedMap] change _ = ((P.map T).prod (gaussianRows r d)).map _ rw [hprod,Measure.map_map measurable_alignedMap (hT.prodMap measurable_id)] rfl lemma independent_refineSignal_drop {d k : ℕ} {W E : Type*} [MeasurableSpace W] [MeasurableSpace E] (P : Measure (Sphere d×W)) [IsFiniteMeasure P] (e : Sphere d×W → E) (he : Measurable e) : ((refineSignal P e).prod (gaussianRows k d)).map (fun p => ((p.1.1,p.1.2.1),p.2)) = P.prod (gaussianRows k d) := by let : IsFiniteMeasure (refineSignal P e) := Measure.isFiniteMeasure_map _ _ have hb : (refineSignal P e).map (fun p => (p.1,p.2.1)) = P := by rw [refineSignal,Measure.map_map (measurable_fst.prodMk measurable_snd.fst) (measurable_fst.prodMk (measurable_snd.prodMk he))] exact Measure.map_id have hh := Measure.map_prod_map (refineSignal P e) (gaussianRows k d) (measurable_fst.prodMk measurable_snd.fst) measurable_id rw [Measure.map_id,hb] at hh exact hh.symm lemma alignedExperiment_independent {d m r : ℕ} {W : Type*} [MeasurableSpace W] (P : Measure (Sphere d×W)) [IsProbabilityMeasure P] : alignedExperiment (r:=r) (P.prod (gaussianRows m d)) = P.prod (gaussianRows (m+r) d) := by change (((P.prod (gaussianRows m d)).prod (gaussianRows r d)).map ((Prod.map id (fun p : Rows m d×Rows r d => Fin.append p.1 p.2)) ∘ MeasurableEquiv.prodAssoc)) = _ rw [←Measure.map_map (measurable_id.prodMap (measurable_append (E:=Vector d) m r)) MeasurableEquiv.prodAssoc.measurable,Measure.prodAssoc_prod, ←Measure.map_prod_map _ _ measurable_id (measurable_append (E:=Vector d) m r),Measure.map_id] rw [show ((gaussianRows m d).prod (gaussianRows r d)).map (fun p => Fin.append p.1 p.2) = gaussianRows (m+r) d from map_append_pi (stdGaussian (Vector d)) m r] lemma signal_marginal_of_independence {S W G : Type*} [MeasurableSpace S] [MeasurableSpace W] [MeasurableSpace G] (P : Measure ((S×W)×G)) (ν : Measure S) (γ : Measure G) [IsProbabilityMeasure γ] [SFinite ν] (hInd : (P.map swapLast).fst = ν.prod γ) : P.fst.fst = ν := by have hh := congrArg Measure.fst hInd rw [Measure.fst_prod] at hh convert hh using 1 simp only [Measure.fst,Measure.map_map measurable_fst measurable_fst, Measure.map_map measurable_fst swapLast.measurable, Measure.map_map measurable_fst (measurable_fst.comp swapLast.measurable)] rfl end CurrentProjection end end OAI