| import OAI.Probability.GaussianReplacement.Entropy |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
| noncomputable section |
| namespace CurrentProjection |
| open InnerProductSpace |
| open InformationTheory |
|
|
|
|
| def actualExperiment {d m r : ℕ} {W : Type*} [MeasurableSpace W] |
| (P : Measure ((Sphere d × W) × Rows m d)) : |
| Measure (((Sphere d × W) × Rows m d) × Rows r d) := P.prod (gaussianRows r d) |
| |
| instance actualExperiment_isProbabilityMeasure {d m r : ℕ} {W : Type*} [MeasurableSpace W] |
| (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] : |
| IsProbabilityMeasure (actualExperiment (r:=r) P) := |
| inferInstanceAs (IsProbabilityMeasure (P.prod (gaussianRows r d))) |
| |
| def actualRowsSignal {d m r : ℕ} {W : Type*} |
| (p : ((Sphere d × W) × Rows m d) × Rows r d) : Rows (m+r) d × Sphere d := |
| (Fin.append p.1.2 p.2,p.1.1.1) |
| |
| def actualObservation {d m r : ℕ} {W : Type*} |
| (p : ((Sphere d × W) × Rows m d) × Rows r d) : |
| (Rows (m+r) d × W) × (Fin (m+r) → ℝ) := |
| ((Fin.append p.1.2 p.2,p.1.1.2), label (Fin.append p.1.2 p.2) (p.1.1.1 : Vector d)) |
| |
| lemma measurable_actualRowsSignal {d m r : ℕ} {W : Type*} [MeasurableSpace W] : |
| Measurable (actualRowsSignal : ((Sphere d×W)×Rows m d)×Rows r d → _) := |
| ((measurable_append (E:=Vector d) m r).comp (measurable_fst.snd.prodMk measurable_snd)).prodMk |
| measurable_fst.fst.fst |
| |
| lemma measurable_actualObservation {d m r : ℕ} {W : Type*} [MeasurableSpace W] : |
| Measurable (actualObservation : ((Sphere d×W)×Rows m d)×Rows r d → _) := by |
| have hG : Measurable (fun p : ((Sphere d×W)×Rows m d)×Rows r d => Fin.append p.1.2 p.2) := (measurable_append (E:=Vector d) m r).comp (measurable_fst.snd.prodMk measurable_snd) |
| have hY : Measurable (fun p : ((Sphere d×W)×Rows m d)×Rows r d => label (Fin.append p.1.2 p.2) (p.1.1.1:Vector d)) := (measurable_own_label (k:=m+r) (d:=d)).comp measurable_actualRowsSignal |
| exact (hG.prodMk measurable_fst.fst.snd).prodMk hY |
| |
| instance actualObservation_isProbabilityMeasure {d m r : ℕ} {W : Type*} [MeasurableSpace W] |
| (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] : |
| IsProbabilityMeasure ((actualExperiment (r:=r) P).map actualObservation) := |
| inferInstance |
| |
| lemma map_actualRowsSignal {d m r : ℕ} {W : Type*} [MeasurableSpace W] |
| (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)) : |
| (actualExperiment (r:=r) P).map actualRowsSignal = (gaussianRows (m+r) d).prod ν := by |
| let f : (Sphere d×W)×Rows m d → Sphere d×Rows m d := fun p => (p.1.1,p.2) |
| have hf : Measurable f := measurable_fst.fst.prodMk measurable_snd |
| have hBase : P.map f = ν.prod (gaussianRows m d) := by |
| rw [Measure.fst, Measure.map_map measurable_fst swapLast.measurable] at hInd |
| exact hInd |
| let T := Prod.map f (id : Rows r d → Rows r d) |
| have hT : Measurable T := hf.prodMap measurable_id |
| have hpre : (actualExperiment (r:=r) P).map T = |
| (ν.prod (gaussianRows m d)).prod (gaussianRows r d) := by |
| change (P.prod (gaussianRows r d)).map (Prod.map f id) = _ |
| rw [← Measure.map_prod_map _ _ hf measurable_id, Measure.map_id, hBase] |
| let U : (Sphere d×Rows m d)×Rows r d → (Rows m d×Rows r d)×Sphere d := |
| fun p => ((p.1.2,p.2),p.1.1) |
| have hU : Measurable U := (measurable_fst.snd.prodMk measurable_snd).prodMk measurable_fst.fst |
| have hmid : ((ν.prod (gaussianRows m d)).prod (gaussianRows r d)).map U = |
| ((gaussianRows m d).prod (gaussianRows r d)).prod ν := by |
| change Measure.map (Prod.swap ∘ MeasurableEquiv.prodAssoc) _ = _ |
| rw [← Measure.map_map measurable_swap MeasurableEquiv.prodAssoc.measurable, |
| Measure.prodAssoc_prod, Measure.prod_swap] |
| let V : (Rows m d×Rows r d)×Sphere d → Rows (m+r) d×Sphere d := |
| Prod.map (fun p => Fin.append p.1 p.2) id |
| have hV : Measurable V := (measurable_append (E:=Vector d) m r).prodMap measurable_id |
| have hlast : (((gaussianRows m d).prod (gaussianRows r d)).prod ν).map V = |
| (gaussianRows (m+r) d).prod ν := by |
| rw [show V = Prod.map (fun p : Rows m d×Rows r d => Fin.append p.1 p.2) id from rfl, |
| ← Measure.map_prod_map _ _ (measurable_append (E:=Vector d) m r) measurable_id, |
| 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] |
| have hfun : actualRowsSignal = V ∘ U ∘ T := rfl |
| rw [hfun, ← Measure.map_map hV (hU.comp hT), ← Measure.map_map hU hT, |
| hpre, hmid, hlast] |
| |
| lemma actualObservation_ac {d m r q : ℕ} {W : Type*} [MeasurableSpace W] |
| [Countable W] [MeasurableSingletonClass W] |
| (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) (hH : 0 < H) (hν : ν ≤ ENNReal.ofReal H • uniformSphere d) : |
| (actualExperiment (r:=r) P).map actualObservation ≪ |
| ((gaussianRows (m+r) d).prod (Measure.count : Measure W)).prod |
| (volume : Measure (Fin (m+r) → ℝ)) := by |
| have hB := dominatedSphere_growth ν hH.le hν (show 1 ≤ d by have := hdim.2.2.2.1; omega) |
| (show m+r ≤ d-1 by have := hdim.2.2.2.1; omega) |
| have hD := dominatedSphere_growth ν hH.le hν (show 1 ≤ d by have := hdim.2.2.2.1; omega) le_rfl |
| have hac := ae_labelLaw_ac_of_growth hdim ν (H*(3*4^d)) (H*(3*4^d)) |
| (by positivity) (by positivity) hB hD |
| have hGY := map_fiber_ac_prod (gaussianRows (m+r) d) ν volume |
| (fun gs : Rows (m+r) d×Sphere d => label gs.1 (gs.2:Vector d)) measurable_own_label hac |
| let J := (actualExperiment (r:=r) P).map |
| (fun p => ((Fin.append p.1.2 p.2,label (Fin.append p.1.2 p.2) (p.1.1.1:Vector d)),p.1.1.2)) |
| have hJm : Measurable (fun p : ((Sphere d×W)×Rows m d)×Rows r d => |
| ((Fin.append p.1.2 p.2,label (Fin.append p.1.2 p.2) (p.1.1.1:Vector d)),p.1.1.2)) := |
| (measurable_actualRowsSignal.fst.prodMk |
| (measurable_own_label.comp measurable_actualRowsSignal)).prodMk measurable_fst.fst.snd |
| let : IsProbabilityMeasure (actualExperiment (r:=r) P) := inferInstanceAs (IsProbabilityMeasure (P.prod (gaussianRows r d))) |
| let : IsProbabilityMeasure J := inferInstance |
| have hJfst : J.fst = ((gaussianRows (m+r) d).prod ν).map |
| (fun gs : Rows (m+r) d×Sphere d => (gs.1,label gs.1 (gs.2:Vector d))) := by |
| rw [← map_actualRowsSignal P ν hInd] |
| rw [Measure.map_map (measurable_fst.prodMk measurable_own_label) measurable_actualRowsSignal] |
| change ((actualExperiment P).map _).fst = _ |
| rw [Measure.fst, Measure.map_map measurable_fst hJm] |
| rfl |
| have hjac : J.fst ≪ (gaussianRows (m+r) d).prod (volume : Measure (Fin (m+r) → ℝ)) := by |
| rw [hJfst]; exact hGY |
| have hh := adjoin_countable_ac J (gaussianRows (m+r) d) volume hjac |
| change (J.map swapLast) ≪ _ at hh |
| dsimp only [J] at hh |
| rw [Measure.map_map swapLast.measurable hJm] at hh |
| exact hh |
| |
| |
| def selectedCube {k d : ℕ} {W : Type*} (ρ : W → Measure (Sphere d)) (δ : ℝ) |
| (z : (Rows k d×W)×(Fin k → ℝ)) : ℝ≥0∞ := cubeAverage (ρ z.1.2) z.1.1 z.2 δ |
| |
| def selectedExact {k d : ℕ} {W : Type*} (ρ : W → Measure (Sphere d)) |
| (z : (Rows k d×W)×(Fin k → ℝ)) : ℝ≥0∞ := exactDensity (ρ z.1.2) z.1.1 z.2 |
| |
| lemma measurable_countable_selection {A B T W : Type*} [MeasurableSpace A] |
| [MeasurableSpace B] [MeasurableSpace T] [MeasurableSpace W] |
| [Countable W] [MeasurableSingletonClass W] (f : W → A×B → T) |
| (hf : ∀ w, Measurable (f w)) : |
| Measurable (fun z : (A×W)×B => f z.1.2 (z.1.1,z.2)) := by |
| have hh : Measurable (fun z : (A×B)×W => f z.2 z.1) := |
| measurable_from_prod_countable_left hf |
| exact hh.comp ((measurable_fst.fst.prodMk measurable_snd).prodMk measurable_fst.snd) |
| |
| lemma measurable_selectedCube {k d : ℕ} {W : Type*} [MeasurableSpace W] |
| [Countable W] [MeasurableSingletonClass W] (ρ : W → Measure (Sphere d)) |
| [∀ w, SFinite (ρ w)] (δ : ℝ) : Measurable (selectedCube (k:=k) ρ δ) := |
| measurable_countable_selection (fun w gy => cubeAverage (ρ w) gy.1 gy.2 δ) |
| (fun w => measurable_cubeAverage (ρ w) δ) |
| |
| lemma measurable_selectedExact {k d : ℕ} {W : Type*} [MeasurableSpace W] |
| [Countable W] [MeasurableSingletonClass W] (ρ : W → Measure (Sphere d)) |
| [∀ w, SFinite (ρ w)] : Measurable (selectedExact (k:=k) ρ) := |
| measurable_countable_selection (fun w gy => exactDensity (ρ w) gy.1 gy.2) |
| (fun w => measurable_exactDensity (ρ w)) |
| |
| lemma exactDensity_ne_top {k d : ℕ} (ρ : Measure (Sphere d)) (G : Rows k d) (y : Fin k → ℝ) : |
| exactDensity ρ G y ≠ ∞ := by |
| unfold exactDensity |
| dsimp only |
| split_ifs <;> simp |
| |
| lemma actual_cube_limit {d m r q : ℕ} {W : Type*} [MeasurableSpace W] |
| [Countable W] [MeasurableSingletonClass W] |
| (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) (hH : 0 < H) (hν : ν ≤ ENNReal.ofReal H • uniformSphere d) |
| (ρ : W → Measure (Sphere d)) [∀ w, IsFiniteMeasure (ρ w)] : |
| ∀ᵐ p ∂actualExperiment (r:=r) P, |
| Tendsto (fun j : ℕ => selectedCube ρ (1/(j:ℝ)) (actualObservation p)) atTop |
| (𝓝 (selectedExact ρ (actualObservation p))) := by |
| have hh := countable_null_transfer ((actualExperiment (r:=r) P).map actualObservation) |
| (gaussianRows (m+r) d) (volume : Measure (Fin (m+r) → ℝ)) |
| (actualObservation_ac hdim P ν hInd H hH hν) |
| (fun g w y => Tendsto (fun j : ℕ => cubeAverage (ρ w) g y (1/(j:ℝ))) atTop |
| (𝓝 (exactDensity (ρ w) g y))) |
| (fun w => measurableSet_tendsto_fun (fun j : ℕ => measurable_cubeAverage (ρ w) (1/(j:ℝ))) |
| (measurable_exactDensity (ρ w))) |
| (fun w g => volume_ae_cube_limit (ρ w) g) |
| exact ae_of_ae_map measurable_actualObservation.aemeasurable hh |
| |
| lemma actual_conditional_label_ac {d m r q : ℕ} {W : Type*} [MeasurableSpace W] |
| [Countable W] [MeasurableSingletonClass W] |
| (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) (hH : 0 < H) (hν : ν ≤ ENNReal.ofReal H • uniformSphere d) : |
| ∀ᵐ gw ∂((actualExperiment (r:=r) P).map actualObservation).fst, |
| ((actualExperiment (r:=r) P).map actualObservation).condKernel gw ≪ volume := by |
| let : IsProbabilityMeasure (actualExperiment (r:=r) P) := |
| inferInstanceAs (IsProbabilityMeasure (P.prod (gaussianRows r d))) |
| let : IsProbabilityMeasure ((actualExperiment (r:=r) P).map actualObservation) := |
| inferInstance |
| exact ae_condKernel_ac_of_joint_ac _ _ _ (actualObservation_ac hdim P ν hInd H hH hν) |
| |
| |
| def momentBound (C : ℝ) (d : ℕ) (B D : ℝ) : ℝ := |
| Real.exp (C*d)*B*(1+max (Real.log (D/B)) 0) |
| |
| lemma momentBound_nonneg (C : ℝ) (d : ℕ) {B D : ℝ} (hB : 0 ≤ B) : |
| 0 ≤ momentBound C d B D := by |
| unfold momentBound |
| positivity |
| |
| lemma cubeAverage_inner_finite {d m r : ℕ} (ρ : Measure (Sphere d)) [IsFiniteMeasure ρ] |
| (s : Sphere d) (X : Rows m d) (δ : ℝ) : |
| ∫⁻ Z, cubeAverage ρ (Fin.append X Z) (label (Fin.append X Z) (s:Vector d)) δ |
| ∂gaussianRows r d ≠ ∞ := by |
| apply ne_top_of_le_ne_top (b:=ENNReal.ofReal ((2*δ)⁻¹^(m+r))*ρ univ) |
| · finiteness |
| · calc |
| _ ≤ ∫⁻ _Z : Rows r d, ENNReal.ofReal ((2*δ)⁻¹^(m+r))*ρ univ ∂gaussianRows r d := by |
| apply lintegral_mono; intro Z |
| exact mul_le_mul' le_rfl (measure_mono (Set.subset_univ _)) |
| _ = _ := by simp |
| |
| lemma actual_smoothed_log_bound {d m r q : ℕ} {W : Type*} [MeasurableSpace W] |
| [Countable W] [MeasurableSingletonClass W] |
| (hdim : AdmissibleDimensions d m r q) |
| (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] |
| (hKL : rowInformation P (gaussianRows m d) ≠ ∞) |
| (ρ : W → Measure (Sphere d)) [∀ w, IsFiniteMeasure (ρ w)] |
| (B D : W → ℝ) (hB : ∀ w, 0 < B w) (hD : ∀ w, 0 < D w) |
| (hgB : ∀ w, LocalGrowth (ρ w) (m+r) (B w)) |
| (hgD : ∀ w, LocalGrowth (ρ w) (d-1) (D w)) |
| (hCI : Integrable (fun sw : Sphere d×W => Real.log (1+momentBound 9 d (B sw.2) (D sw.2))) P.fst) |
| {δ : ℝ} (hδ : 0 < δ) : |
| ∫⁻ p, ENNReal.ofReal (Real.log (1+(selectedCube ρ δ (actualObservation p)).toReal)) |
| ∂actualExperiment (r:=r) P ≤ |
| ENNReal.ofReal ((rowInformation P (gaussianRows m d)).toReal/q + |
| ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst) := by |
| apply mixed_entropy_log_bound P (gaussianRows m d) (gaussianRows r d) hKL |
| (fun p => selectedCube ρ δ (actualObservation p)) |
| ((measurable_selectedCube ρ δ).comp measurable_actualObservation) |
| (fun p => (cubeAverage_lt_top _ _ _ _).ne) |
| (fun p => cubeAverage_inner_finite (ρ p.1.2) p.1.1 p.2 δ) |
| (fun sw : Sphere d×W => momentBound 9 d (B sw.2) (D sw.2)) |
| ((measurable_of_countable (f:=fun w => momentBound 9 d (B w) (D w))).comp measurable_snd) |
| (fun sw => momentBound_nonneg 9 d (hB sw.2).le) hCI hdim.2.2.1 |
| intro sw |
| exact mixed_cube_moment hdim (ρ sw.2) (B sw.2) (D sw.2) |
| (hB sw.2) (hD sw.2) (hgB sw.2) (hgD sw.2) sw.1 δ hδ |
| |
| lemma log_lintegral_bound_of_ae_tendsto {A : Type*} [MeasurableSpace A] |
| (μ : Measure A) (f : ℕ → A → ℝ≥0∞) (F : A → ℝ≥0∞) |
| (hf : ∀ j, Measurable (f j)) (hFt : ∀ a, F a ≠ ∞) |
| (ht : ∀ᵐ a ∂μ, Tendsto (fun j => f j a) atTop (𝓝 (F a))) |
| (R : ℝ≥0∞) (hR : ∀ j ≥ 1, ∫⁻ a, ENNReal.ofReal (Real.log (1+(f j a).toReal)) ∂μ ≤ R) : |
| ∫⁻ a, ENNReal.ofReal (Real.log (1+(F a).toReal)) ∂μ ≤ R := by |
| let g : ℕ → A → ℝ≥0∞ := fun j a => ENNReal.ofReal (Real.log (1+(f j a).toReal)) |
| have hlim : ∀ᵐ a ∂μ, liminf (fun j => g j a) atTop = ENNReal.ofReal (Real.log (1+(F a).toReal)) := by |
| filter_upwards [ht] with a ha |
| have htr := (ENNReal.continuousAt_toReal (hFt a)).tendsto.comp ha |
| have hlog := (tendsto_const_nhds.add htr).log (show 1+(F a).toReal ≠ 0 by positivity) |
| exact (ENNReal.continuous_ofReal.tendsto _ |>.comp hlog).liminf_eq |
| calc |
| _ = ∫⁻ a, liminf (fun j => g j a) atTop ∂μ := lintegral_congr_ae (hlim.mono fun a ha => ha.symm) |
| _ ≤ liminf (fun j => ∫⁻ a, g j a ∂μ) atTop := |
| lintegral_liminf_le (fun j => (measurable_log_one_add (hf j)).ennreal_ofReal) |
| _ ≤ R := liminf_le_of_frequently_le (hu_le := by isBoundedDefault) |
| ((eventually_ge_atTop 1).mono fun j hj => hR j hj).frequently |
| |
| def logPositive (x : ℝ≥0∞) : ℝ := max (Real.log x.toReal) 0 |
| |
| def logNegative (x : ℝ≥0∞) : ℝ≥0∞ := if x = 0 then ∞ else ENNReal.ofReal (-Real.log x.toReal) |
| |
| |
| def extendedLogExpectation {A : Type*} [MeasurableSpace A] (μ : Measure A) |
| (f : A → ℝ≥0∞) : EReal := |
| ((∫⁻ a, ENNReal.ofReal (logPositive (f a)) ∂μ) : EReal) - |
| ((∫⁻ a, logNegative (f a) ∂μ) : EReal) |
| |
| lemma logPositive_le_log_one_add (x : ℝ≥0∞) : logPositive x ≤ Real.log (1+x.toReal) := by |
| apply max_le _ (log_one_add_nonneg x) |
| by_cases h : x.toReal = 0 |
| · simp only [h, Real.log_zero, add_zero, Real.log_one, le_refl] |
| · exact Real.log_le_log (lt_of_le_of_ne ENNReal.toReal_nonneg (Ne.symm h)) (le_add_of_nonneg_left zero_le_one) |
| |
| lemma log_bound_consequences {A : Type*} [MeasurableSpace A] (μ : Measure A) |
| (f : A → ℝ≥0∞) (hf : Measurable f) (R : ℝ) (hR0 : 0 ≤ R) |
| (hR : ∫⁻ a, ENNReal.ofReal (Real.log (1+(f a).toReal)) ∂μ ≤ ENNReal.ofReal R) : |
| (Integrable (fun a => Real.log (1+(f a).toReal)) μ ∧ |
| ∫ a, Real.log (1+(f a).toReal) ∂μ ≤ R) ∧ |
| Integrable (fun a => logPositive (f a)) μ ∧ extendedLogExpectation μ f ≤ (R:EReal) := by |
| have hLi : Integrable (fun a => Real.log (1+(f a).toReal)) μ := |
| (lintegral_ofReal_ne_top_iff_integrable (measurable_log_one_add hf).aestronglyMeasurable |
| (Filter.Eventually.of_forall fun _ => log_one_add_nonneg _)).mp |
| (ne_top_of_le_ne_top ENNReal.ofReal_ne_top hR) |
| have hLf : ∫ a, Real.log (1+(f a).toReal) ∂μ ≤ R := by |
| rw [← ofReal_integral_eq_lintegral_ofReal hLi |
| (Filter.Eventually.of_forall fun _ => log_one_add_nonneg _)] at hR |
| exact (ENNReal.ofReal_le_ofReal_iff hR0).mp hR |
| have hPi : Integrable (fun a => logPositive (f a)) μ := by |
| apply hLi.mono' ((hf.ennreal_toReal.log.max measurable_const).aestronglyMeasurable) |
| apply Filter.Eventually.of_forall; intro a |
| rw [Real.norm_eq_abs, abs_of_nonneg (le_max_right _ _)] |
| exact logPositive_le_log_one_add _ |
| refine ⟨⟨hLi,hLf⟩,hPi,?_⟩ |
| have hPL : ∫⁻ a, ENNReal.ofReal (logPositive (f a)) ∂μ ≤ ENNReal.ofReal R := |
| (lintegral_mono fun a => ENNReal.ofReal_le_ofReal (logPositive_le_log_one_add _)).trans hR |
| apply (EReal.sub_le_sub (EReal.coe_ennreal_le_coe_ennreal_iff.mpr hPL) |
| (EReal.coe_ennreal_nonneg (∫⁻ a, logNegative (f a) ∂μ))).trans |
| have he : (ENNReal.ofReal R : EReal) = (R:EReal) := |
| by simpa using (EReal.coe_toENNReal (x := (R:EReal)) (by exact_mod_cast hR0)) |
| simp only [sub_zero, he, le_refl] |
| |
| |
| |
| def PositiveMessage {W : Type*} [MeasurableSpace W] (π : Measure W) := {w : W // 0 < π {w}} |
| |
| def completeFamily {W : Type*} [MeasurableSpace W] {T : Type*} (π : Measure W) |
| (f : PositiveMessage π → T) (t₀ : T) (w : W) : T := |
| if h : 0 < π {w} then f ⟨w,h⟩ else t₀ |
| |
| instance completeFamily_finite {W : Type*} [MeasurableSpace W] {d : ℕ} |
| (π : Measure W) (ρ : PositiveMessage π → Measure (Sphere d)) |
| [hρ : ∀ w, IsFiniteMeasure (ρ w)] (w : W) : IsFiniteMeasure (completeFamily π ρ 0 w) := by |
| unfold completeFamily |
| split_ifs with h |
| · exact hρ ⟨w,h⟩ |
| · infer_instance |
| |
| lemma completeFamily_pos {W : Type*} [MeasurableSpace W] (π : Measure W) |
| (f : PositiveMessage π → ℝ) (hf : ∀ w, 0 < f w) (w : W) : |
| 0 < completeFamily π f 1 w := by |
| unfold completeFamily |
| split_ifs with h |
| · exact hf ⟨w,h⟩ |
| · norm_num |
| |
| lemma completeFamily_growth {W : Type*} [MeasurableSpace W] {d a : ℕ} |
| (π : Measure W) (ρ : PositiveMessage π → Measure (Sphere d)) |
| (B : PositiveMessage π → ℝ) (hg : ∀ w, LocalGrowth (ρ w) a (B w)) (w : W) : |
| LocalGrowth (completeFamily π ρ 0 w) a (completeFamily π B 1 w) := by |
| unfold completeFamily |
| split_ifs with h |
| · exact hg ⟨w,h⟩ |
| · intro z t ht |
| simp |
| |
| |
| structure ActualRowConclusion {d m r q : ℕ} {W : Type*} [MeasurableSpace W] |
| (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] |
| (ρ : W → Measure (Sphere d)) (B D : W → ℝ) : Prop where |
| row_finite : rowInformation P (gaussianRows m d) ≠ ∞ |
| row_le_entropy : (rowInformation P (gaussianRows m d)).toReal ≤ |
| (shannonEntropy (P.map swapLast).snd).toReal |
| conditional_density : ∀ᵐ gw ∂((actualExperiment (r:=r) P).map actualObservation).fst, |
| ((actualExperiment (r:=r) P).map actualObservation).condKernel gw ≪ volume |
| actual_limit : ∀ᵐ p ∂actualExperiment (r:=r) P, |
| Tendsto (fun j : ℕ => selectedCube ρ (1/(j:ℝ)) (actualObservation p)) atTop |
| (𝓝 (selectedExact ρ (actualObservation p))) |
| smoothed_bound : ∀ δ : ℝ, 0 < δ → |
| Integrable (fun p => Real.log (1+(selectedCube ρ δ (actualObservation p)).toReal)) |
| (actualExperiment (r:=r) P) ∧ |
| (∫ p, Real.log (1+(selectedCube ρ δ (actualObservation p)).toReal) ∂actualExperiment (r:=r) P) ≤ |
| (rowInformation P (gaussianRows m d)).toReal/q + |
| ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst |
| exact_integrable : Integrable (fun p => Real.log (1+(selectedExact ρ (actualObservation p)).toReal)) |
| (actualExperiment (r:=r) P) |
| exact_bound : (∫ p, Real.log (1+(selectedExact ρ (actualObservation p)).toReal) ∂actualExperiment (r:=r) P) ≤ |
| (rowInformation P (gaussianRows m d)).toReal/q + |
| ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst |
| entropy_bound : (rowInformation P (gaussianRows m d)).toReal/q + |
| ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst ≤ |
| (shannonEntropy (P.map swapLast).snd).toReal/q + |
| ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst |
| positive_log_integrable : Integrable (fun p => logPositive (selectedExact ρ (actualObservation p))) |
| (actualExperiment (r:=r) P) |
| extended_log_bound : extendedLogExpectation (actualExperiment (r:=r) P) |
| (fun p => selectedExact ρ (actualObservation p)) ≤ |
| (((rowInformation P (gaussianRows m d)).toReal/q + |
| ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B sw.2) (D sw.2)) ∂P.fst) : EReal) |
|
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|
|
| theorem actualRowMain {d m r q : ℕ} {W : Type*} [MeasurableSpace W] |
| [Countable W] [MeasurableSingletonClass W] |
| (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) |
| (hEntropy : shannonEntropy (P.map swapLast).snd ≠ ∞) |
| (ρ : PositiveMessage (P.map swapLast).snd → Measure (Sphere d)) |
| [∀ w, IsFiniteMeasure (ρ w)] |
| (_hρac : ∀ w, ρ w ≪ uniformSphere d) |
| (B D : PositiveMessage (P.map swapLast).snd → ℝ) |
| (hB : ∀ w, 0 < B w) (hD : ∀ w, 0 < D w) |
| (hgB : ∀ w, LocalGrowth (ρ w) (m+r) (B w)) |
| (hgD : ∀ w, LocalGrowth (ρ w) (d-1) (D w)) |
| (hCI : Integrable (fun sw : Sphere d×W => Real.log (1+momentBound 9 d |
| (completeFamily (P.map swapLast).snd B 1 sw.2) (completeFamily (P.map swapLast).snd D 1 sw.2))) P.fst) : |
| ActualRowConclusion (r:=r) (q:=q) P (completeFamily (P.map swapLast).snd ρ 0) |
| (completeFamily (P.map swapLast).snd B 1) (completeFamily (P.map swapLast).snd D 1) := by |
| let ρ' := completeFamily (P.map swapLast).snd ρ 0 |
| let B' := completeFamily (P.map swapLast).snd B 1 |
| let D' := completeFamily (P.map swapLast).snd D 1 |
| have hB' := completeFamily_pos (P.map swapLast).snd B hB |
| have hD' := completeFamily_pos (P.map swapLast).snd D hD |
| have hgB' := completeFamily_growth (P.map swapLast).snd ρ B hgB |
| have hgD' := completeFamily_growth (P.map swapLast).snd ρ D hgD |
| have hH : 0 < H := by |
| by_contra hn |
| have hh := hν univ |
| simp [ENNReal.ofReal_of_nonpos (le_of_not_gt hn)] at hh |
| obtain ⟨hKL,hKE⟩ := rowInformation_le_entropy P ν (gaussianRows m d) hInd hEntropy |
| have hlim := actual_cube_limit hdim P ν hInd H hH hν ρ' |
| let R := (rowInformation P (gaussianRows m d)).toReal/q + |
| ∫ sw : Sphere d×W, Real.log (1+momentBound 9 d (B' sw.2) (D' sw.2)) ∂P.fst |
| have hR0 : 0 ≤ R := add_nonneg (div_nonneg ENNReal.toReal_nonneg (Nat.cast_nonneg _)) |
| (integral_nonneg fun sw => Real.log_nonneg (le_add_of_nonneg_right (momentBound_nonneg 9 d (hB' sw.2).le))) |
| have hsmooth (δ : ℝ) (hδ : 0 < δ) := |
| actual_smoothed_log_bound hdim P hKL ρ' B' D' hB' hD' hgB' hgD' hCI hδ |
| have hExact : ∫⁻ p, ENNReal.ofReal (Real.log (1+(selectedExact ρ' (actualObservation p)).toReal)) |
| ∂actualExperiment (r:=r) P ≤ ENNReal.ofReal R := by |
| apply log_lintegral_bound_of_ae_tendsto (actualExperiment (r:=r) P) |
| (fun j p => selectedCube ρ' (1/(j:ℝ)) (actualObservation p)) |
| (fun p => selectedExact ρ' (actualObservation p)) |
| (fun j => (measurable_selectedCube ρ' (1/(j:ℝ))).comp measurable_actualObservation) |
| (fun p => exactDensity_ne_top _ _ _) hlim (ENNReal.ofReal R) |
| intro j hj |
| exact hsmooth _ (by positivity) |
| have hc := log_bound_consequences (actualExperiment (r:=r) P) |
| (fun p => selectedExact ρ' (actualObservation p)) |
| ((measurable_selectedExact ρ').comp measurable_actualObservation) R hR0 hExact |
| refine ⟨hKL,hKE,actual_conditional_label_ac hdim P ν hInd H hH hν,hlim,?_,hc.1.1,hc.1.2,?_,hc.2.1,hc.2.2⟩ |
| · intro δ hδ |
| exact (log_bound_consequences (actualExperiment (r:=r) P) |
| (fun p => selectedCube ρ' δ (actualObservation p)) |
| ((measurable_selectedCube ρ' δ).comp measurable_actualObservation) R hR0 (hsmooth δ hδ)).1 |
| · exact add_le_add (div_le_div_of_nonneg_right hKE (Nat.cast_nonneg q)) le_rfl |
|
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|
|
| theorem actualRowRefinement {d m : ℕ} {W E : Type*} [MeasurableSpace W] |
| [MeasurableSpace E] [Countable W] [MeasurableSingletonClass W] |
| [Countable E] [MeasurableSingletonClass E] |
| (P : Measure ((Sphere d×W)×Rows m d)) [IsProbabilityMeasure P] |
| (e : Sphere d×W → E) (he : Measurable e) |
| (_hEntropy : shannonEntropy |
| ((P.map (fun p => ((p.1.1,(p.1.2,e p.1)),p.2))).map swapLast).snd ≠ ∞) : |
| rowInformation (P.map (fun p => ((p.1.1,(p.1.2,e p.1)),p.2))) (gaussianRows m d) = |
| rowInformation P (gaussianRows m d) := |
| rowInformation_refinement P (gaussianRows m d) e he |
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|
| end CurrentProjection |
|
|
| end |
|
|
| end OAI |
|
|
