| import Mathlib |
| import OAI.Probability.GaussianInformation.Dyadic.ChannelProducts |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
| open MeasureTheory Metric Set |
| open scoped ENNReal NNReal |
| open MeasureTheory Set Filter |
| open scoped ENNReal Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
| open InnerProductSpace Matrix MeasureTheory Set |
| open scoped RealInnerProductSpace ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace RepeatedLocalization |
| lemma finite_entropy_le_subprob_reference {ι : Type*} [Fintype ι] |
| (p b : ι → ℝ) (hp : ∀ i,0 ≤ p i) (hpsum : ∑ i,p i = 1) |
| (hb : ∀ i,0 < b i) (hbsum : ∑ i,b i ≤ 1) : |
| -(∑ i,xlog (p i)) ≤ -∑ i,p i*Real.log (b i) := by |
| have hi (i : ι) : p i-b i ≤ p i*Real.log (p i/b i) := by |
| have h := log_ratio_lower (hp i) (hb i).le (by norm_num : (0:ℝ) < 1) |
| (fun hz => False.elim ((hb i).ne' hz)) |
| simpa [xlog_one] using h |
| have hs := Finset.sum_le_sum (s := Finset.univ) (fun i _ => hi i) |
| simp_rw [log_ratio_mul (hb _).ne'] at hs |
| simp only [Finset.sum_sub_distrib,hpsum] at hs |
| linarith |
|
|
| lemma weighted_log_max_ratio_le {p θ g : ℝ} (hp : 0 ≤ p) (hθ : 0 < θ) |
| (hg : 0 < g) (hg1 : g ≤ 1) : |
| p*Real.log (max 1 (θ/p)) ≤ -p*Real.log g+g*θ := by |
| by_cases hpz : p = 0 |
| · simp only [hpz,zero_mul,neg_zero,zero_add] |
| positivity |
| have hpp : 0 < p := lt_of_le_of_ne hp (Ne.symm hpz) |
| by_cases hθp : θ ≤ p |
| · rw [max_eq_left ((div_le_one hpp).mpr hθp),Real.log_one,mul_zero] |
| have hl := Real.log_nonpos hg.le hg1 |
| have hd := mul_nonpos_of_nonneg_of_nonpos hp hl |
| nlinarith [mul_pos hg hθ] |
| · rw [max_eq_right ((one_le_div hpp).mpr (le_of_not_ge hθp))] |
| have harg : 0 < θ*g/p := div_pos (mul_pos hθ hg) hpp |
| have h := Real.log_le_sub_one_of_pos harg |
| rw [Real.log_div (mul_pos hθ hg).ne' hpz,Real.log_mul hθ.ne' hg.ne'] at h |
| rw [Real.log_div hθ.ne' hpz] |
| have hh := mul_le_mul_of_nonneg_left h hp |
| have he : p*(θ*g/p) = θ*g := by field_simp |
| nlinarith |
|
|
| lemma finite_log_concentration_le {ι : Type*} [Fintype ι] |
| (p θ g : ι → ℝ) (hp : ∀ i,0 ≤ p i) (hθ : ∀ i,0 < θ i) |
| (hg : ∀ i,0 < g i) (hg1 : ∀ i,g i ≤ 1) |
| (hbound : ∑ i,θ i*g i ≤ 1) : |
| (∑ i,p i*Real.log (max 1 (θ i/p i))) ≤ -(∑ i,p i*Real.log (g i))+1 := by |
| have h := Finset.sum_le_sum (s := Finset.univ) (fun i _ => weighted_log_max_ratio_le (hp i) (hθ i) (hg i) (hg1 i)) |
| simp only [Finset.sum_add_distrib,neg_mul,Finset.sum_neg_distrib,mul_comm (g _) (θ _)] at h |
| linarith |
|
|
| noncomputable def depthGeometric (j : ℕ) : ℝ := (1/2:ℝ)^(j+1) |
| lemma depthGeometric_pos (j : ℕ) : 0 < depthGeometric j := by unfold depthGeometric; positivity |
| lemma depthGeometric_le_one (j : ℕ) : depthGeometric j ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) |
| lemma log_depthGeometric (j : ℕ) : Real.log (depthGeometric j) = -((j:ℝ)+1)*Real.log 2 := by |
| simp only [depthGeometric,Real.log_pow,Real.log_div one_ne_zero (by norm_num : (2:ℝ) ≠ 0), |
| Real.log_one,zero_sub,Nat.cast_add,Nat.cast_one] |
| ring |
|
|
| lemma depthGeometric_sum_le_one (D : ℕ) : ∑ j ∈ Finset.range (D+1),depthGeometric j ≤ 1 := by |
| have hsum : ∀ d,∑ j ∈ Finset.range d,depthGeometric j = 1-(1/2:ℝ)^d := by |
| intro d |
| induction d with |
| | zero => simp |
| | succ d ih => rw [Finset.sum_range_succ,ih]; unfold depthGeometric; rw [pow_succ]; ring |
| rw [hsum] |
| have := pow_nonneg (show (0:ℝ) ≤ 1/2 by norm_num) (D+1) |
| linarith |
|
|
| lemma entropy_and_concentration_depth {ι : Type*} [Fintype ι] |
| (n : ℕ) (p : ι → ℝ) (j : ι → ℕ) (hp : ∀ i,0 ≤ p i) (hpsum : ∑ i,p i = 1) |
| (hbudget : ∑ i,(2:ℝ)^(-(n:ℝ)*(j i:ℝ)/2)*depthGeometric (j i) ≤ 1) : |
| -(∑ i,xlog (p i)) ≤ ((n:ℝ)/2+1)*Real.log 2*(∑ i,p i*(j i:ℝ))+Real.log 2 ∧ |
| (∑ i,p i*Real.log (max 1 ((2:ℝ)^(-(n:ℝ)*(j i:ℝ)/2)/p i))) ≤ |
| Real.log 2*(∑ i,p i*(j i:ℝ))+Real.log 2+1 := by |
| have hθ (i : ι) : 0 < (2:ℝ)^(-(n:ℝ)*(j i:ℝ)/2) := Real.rpow_pos_of_pos (by norm_num) _ |
| have hlog (i : ι) : Real.log ((2:ℝ)^(-(n:ℝ)*(j i:ℝ)/2)*depthGeometric (j i)) = |
| -((n:ℝ)/2+1)*Real.log 2*(j i:ℝ)-Real.log 2 := by |
| rw [Real.log_mul (hθ i).ne' (depthGeometric_pos _).ne',Real.log_rpow (by norm_num : (0:ℝ) < 2),log_depthGeometric] |
| ring |
| have he := finite_entropy_le_subprob_reference p (fun i => (2:ℝ)^(-(n:ℝ)*(j i:ℝ)/2)*depthGeometric (j i)) |
| hp hpsum (fun i => mul_pos (hθ i) (depthGeometric_pos _)) hbudget |
| have hb := finite_log_concentration_le p (fun i => (2:ℝ)^(-(n:ℝ)*(j i:ℝ)/2)) |
| (fun i => depthGeometric (j i)) hp hθ (fun i => depthGeometric_pos _) (fun i => depthGeometric_le_one _) hbudget |
| have halg (i : ι) : p i*(-((n:ℝ)/2+1)*Real.log 2*(j i:ℝ)-Real.log 2) = |
| -(((n:ℝ)/2+1)*Real.log 2)*(p i*(j i:ℝ))-Real.log 2*p i := by ring |
| have hbalg (i : ι) : p i*(-((j i:ℝ)+1)*Real.log 2) = |
| -(Real.log 2)*(p i*(j i:ℝ))-Real.log 2*p i := by ring |
| simp_rw [hlog,halg] at he |
| simp_rw [log_depthGeometric,hbalg] at hb |
| simp only [Finset.sum_sub_distrib,← Finset.mul_sum,hpsum,mul_one] at he hb |
| constructor <;> linarith |
|
|
| structure HeavyRefinement (n : ℕ) (μ : Measure (Coordinate n)) (Q : Cell n) where |
| Index : Type |
| [finiteIndex : Fintype Index] |
| channel : FiniteChannel (Coordinate n) Index |
| cell : Index → Cell n |
| B : Index → ℝ |
| B_one : ∀ i, 1 ≤ B i |
| select : Coordinate n → Index |
| select_fiber : ∀ i, MeasurableSet {z | select z = i} |
| chosen : ∀ i z, 0 < channel.prob i z → select z = i |
| cell_subset : ∀ i, cellSet (cell i) ⊆ cellSet Q |
| select_mem : ∀ z ∈ cellSet Q, z ∈ cellSet (cell (select z)) |
| level_ge : ∀ i,Q.1 ≤ (cell i).1 |
| support : ∀ i,0 < scoreMass μ (channel.prob i) → |
| scorePosterior μ (channel.prob i) (cellSet (cell i)) = 1 |
| spread : ∀ i,0 < scoreMass μ (channel.prob i) → |
| DyadicSpread (scorePosterior μ (channel.prob i)) (cell i) (B i) |
| entropy_bound : finiteInformation μ channel.prob ≤ |
| ((n:ℝ)/2+1)*Real.log 2*(∑ i,scoreMass μ (channel.prob i)*((cell i).1-Q.1:ℕ))+Real.log 2 |
| concentration_bound : (∑ i,scoreMass μ (channel.prob i)*Real.log (B i)) ≤ |
| Real.log 2*(∑ i,scoreMass μ (channel.prob i)*((cell i).1-Q.1:ℕ))+Real.log 2+1 |
|
|
| attribute [instance] HeavyRefinement.finiteIndex |
|
|
| lemma exists_heavy_refinement {n : ℕ} (hn : 0 < n) |
| (μ : Measure (Coordinate n)) [IsProbabilityMeasure μ] (Q : Cell n) |
| (hQ : μ (cellSet Q) = 1) {C : ℝ} (hC : 0 ≤ C) |
| (hdom : ∀ R : Cell n,μ.real (cellSet R) ≤ C*(1/2:ℝ)^(n*R.1)) : |
| Nonempty (HeavyRefinement n μ Q) := by |
| classical |
| obtain ⟨D,hD⟩ := exists_heavy_cutoff hn μ hC hdom Q |
| let s := (finite_range_deepHeavyCell μ Q D).toFinset |
| let κ := {R : Cell n // R ∈ s} |
| let f : Coordinate n → κ := fun z => ⟨deepHeavyCell μ Q D z,by |
| simpa only [s,Set.Finite.mem_toFinset] using (Set.mem_range_self z (f := deepHeavyCell μ Q D))⟩ |
| have hf : Measurable f := (measurable_deepHeavyCell μ Q D).subtype_mk |
| let q := FiniteChannel.deterministic f hf |
| let j : κ → ℕ := fun i => i.val.1-Q.1 |
| let P : κ → ℝ := fun i => scoreMass μ (q.prob i) |
| let B : κ → ℝ := fun i => max 1 (heavyThreshold n (j i)/P i) |
| have hQa : ∀ᵐ z ∂μ,z ∈ cellSet Q := (mem_ae_iff_prob_eq_one (measurableSet_cellSet Q)).mpr hQ |
| have hQr : μ.real (cellSet Q) = 1 := by simp [measureReal_def,hQ] |
| have hprop (R : Cell n) (hR : R ∈ s) := |
| heavy_selection_range_properties hn μ Q hQr D (by simpa only [s,Set.Finite.mem_toFinset] using hR) |
| have hbudget : ∑ i : κ,heavyThreshold n (j i)*depthGeometric (j i) ≤ 1 := by |
| change (∑ i : {R : Cell n // R ∈ s}, heavyThreshold n (i.val.1-Q.1)*depthGeometric (i.val.1-Q.1)) ≤ 1 |
| rw [Finset.sum_coe_sort s (fun R : Cell n => heavyThreshold n (R.1-Q.1)*depthGeometric (R.1-Q.1))] |
| exact heavy_cell_weight_budget hn μ Q D s (fun R hR => (hprop R hR).1) |
| (fun R hR => (hprop R hR).2.1) (fun R hR => (hprop R hR).2.2) |
| depthGeometric (fun j => (depthGeometric_pos j).le) (depthGeometric_sum_le_one D) |
| have hest := entropy_and_concentration_depth n P j |
| (fun i => scoreMass_nonneg μ _ (q.nonneg i)) (q.mass_total μ) hbudget |
| have hev (i : κ) : q.prob i =ᵐ[μ] |
| {z | z ∈ cellSet Q ∧ deepHeavyCell μ Q D z = i.val}.indicator (fun _ => (1:ℝ)) := by |
| filter_upwards [hQa] with z hz |
| have he : f z = i ↔ deepHeavyCell μ Q D z = i.val := Subtype.ext_iff |
| simp [q,FiniteChannel.deterministic,he,hz,Set.indicator] |
| have hmass (i : κ) : P i = μ.real {z | z ∈ cellSet Q ∧ deepHeavyCell μ Q D z = i.val} := by |
| change (∫ z,q.prob i z ∂μ) = _ |
| rw [integral_congr_ae (hev i)] |
| exact scoreMass_indicator_one μ _ (measurable_heavySelectionEvent μ Q D i.val) |
| refine ⟨{ Index := κ |
| channel := q |
| cell := Subtype.val |
| B := B |
| B_one := fun i => le_max_left _ _ |
| select := f |
| select_fiber := fun i => hf (measurableSet_singleton i) |
| chosen := fun i z hz => by |
| by_contra h |
| simp [q,FiniteChannel.deterministic,h] at hz |
| cell_subset := fun i => by |
| have hmem : i.val ∈ Set.range (deepHeavyCell μ Q D) := by |
| simpa only [s,Set.Finite.mem_toFinset] using i.property |
| obtain ⟨z,hz⟩ := hmem |
| rw [← hz] |
| exact deepHeavyCell_subset hn μ Q D z |
| select_mem := fun z hz => deepHeavyCell_mem hn μ Q D hz |
| level_ge := fun i => (hprop i.val i.property).1 |
| support := ?_ |
| spread := ?_ |
| entropy_bound := ?_ |
| concentration_bound := ?_ }⟩ |
| · intro i hi |
| have := scorePosterior_probability μ _ (q.measurable i) (q.nonneg i) (q.le_one i) hi |
| apply (prob_compl_eq_zero_iff (measurableSet_cellSet i.val)).mp |
| apply (scorePosterior_null_iff μ _ (q.measurable i) (q.nonneg i) hi _ (measurableSet_cellSet i.val).compl).mpr |
| filter_upwards [hQa] with z hz |
| intro hnot |
| have he : f z ≠ i := by |
| intro he |
| have hm := deepHeavyCell_mem hn μ Q D hz |
| have hei : deepHeavyCell μ Q D z = i.val := congrArg Subtype.val he |
| rw [hei] at hm |
| exact hnot hm |
| simp [q,FiniteChannel.deterministic,he] |
| · intro i hi |
| have hip : 0 < μ.real {z | z ∈ cellSet Q ∧ deepHeavyCell μ Q D z = i.val} := by |
| rw [← hmass]; exact hi |
| have hposterior : scorePosterior μ (q.prob i) = |
| cond μ {z | z ∈ cellSet Q ∧ deepHeavyCell μ Q D z = i.val} := by |
| rw [scorePosterior_congr_ae μ (hev i)] |
| exact scorePosterior_indicator_one μ _ (measurable_heavySelectionEvent μ Q D i.val) hip |
| rw [hposterior] |
| have hlev : i.val.1 = Q.1+j i := by have := (hprop i.val i.property).1; dsimp [j]; omega |
| dsimp [B] |
| rw [hmass] |
| exact selected_posterior_spread hn μ Q D hD hlev hip |
| · rw [FiniteChannel.deterministic_information] |
| exact hest.1 |
| · exact hest.2 |
|
|
| end RepeatedLocalization |
|
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| end OAI |
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