| 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 |
| |
