| import Mathlib |
| import OAI.Probability.GaussianInformation.Dyadic.StateInformation |
|
|
| 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 |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal |
|
|
| structure ExtensionWitness {n N : ℕ} (s : AnalysisState n) |
| (q : s.Index → FiniteChannel (Coordinate n) (Fin N)) |
| (s' : AnalysisState n) (f : s'.Index → s.Index × Fin N) where |
| choose : (s.Index × Fin N) → Coordinate n → s'.Index |
| choose_fiber : ∀ i j, MeasurableSet {z | choose i z = j} |
| erase_choose : ∀ i z, f (choose i z) = i |
| chosen : ∀ j z, 0 < s'.channel.prob j z → choose (f j) z = j |
| positive_parent : ∀ j z, 0 < s'.channel.prob j z → 0 < s.channel.prob (f j).1 z |
| contains : ∀ i z, z ∈ cellSet (s.cell i.1) → z ∈ cellSet (s'.cell (choose i z)) |
| subset : ∀ j, cellSet (s'.cell j) ⊆ cellSet (s.cell (f j).1) |
|
|
| lemma analysis_extension_realized {n N : ℕ} (hn : 0 < n) (s : AnalysisState n) |
| (q : s.Index → FiniteChannel (Coordinate n) (Fin N)) (L : ℝ) |
| (hcost : ∀ i,0 < scoreMass (cubePrior n) (s.channel.prob i) → |
| finiteInformation (scorePosterior (cubePrior n) (s.channel.prob i)) (q i).prob ≤ L+Real.log (s.B i)) : |
| ∃ s' : AnalysisState n, ∃ f : s'.Index → (s.Index × Fin N), |
| (s'.channel.coarsen f).prob = (s.channel.bind q).prob ∧ |
| s'.potential ≤ s.potential+L+2*Real.log 2+1 ∧ Nonempty (ExtensionWitness s q s' f) := by |
| classical |
| let μ := cubePrior n |
| have := cubePrior_isProbabilityMeasure hn |
| let r := s.channel.bind q |
| have hparentsupport (i : s.Index × Fin N) (hi : 0 < scoreMass μ (r.prob i)) : |
| scorePosterior μ (r.prob i) (cellSet (s.cell i.1)) = 1 := by |
| have hP := s.channel.bind_positive_parent q μ i hi |
| have := scorePosterior_probability μ _ (r.measurable i) (r.nonneg i) (r.le_one i) hi |
| have := scorePosterior_probability μ _ (s.channel.measurable i.1) (s.channel.nonneg i.1) (s.channel.le_one i.1) hP |
| have hz := (scorePosterior_null_iff μ _ (s.channel.measurable i.1) (s.channel.nonneg i.1) hP |
| _ (measurableSet_cellSet (s.cell i.1)).compl).mp |
| ((prob_compl_eq_zero_iff (measurableSet_cellSet (s.cell i.1))).mpr (s.support i.1 hP)) |
| apply (prob_compl_eq_zero_iff (measurableSet_cellSet (s.cell i.1))).mp |
| apply (scorePosterior_null_iff μ _ (r.measurable i) (r.nonneg i) hi _ |
| (measurableSet_cellSet (s.cell i.1)).compl).mpr |
| filter_upwards [hz] with x hx |
| intro hnot |
| change s.channel.prob i.1 x*(q i.1).prob i.2 x = 0 |
| rw [hx hnot,zero_mul] |
| have hex (i : s.Index × Fin N) : Nonempty (HeavyRefinement n (scorePosterior μ (r.prob i)) (s.cell i.1)) := by |
| by_cases h : 0 < scoreMass μ (r.prob i) |
| · have := scorePosterior_probability μ _ (r.measurable i) (r.nonneg i) (r.le_one i) h |
| exact exists_heavy_refinement hn (scorePosterior μ (r.prob i)) (s.cell i.1) (hparentsupport i h) |
| (div_nonneg zero_le_one h.le) (posterior_cube_domination hn r i h) |
| · have hz : scoreMass μ (r.prob i) = 0 := le_antisymm (le_of_not_gt h) (scoreMass_nonneg μ _ (r.nonneg i)) |
| rw [scorePosterior_of_mass_zero μ _ hz] |
| exact zero_heavy_refinement n (s.cell i.1) |
| let R := fun i => Classical.choice (hex i) |
| let a := r.dbind (fun i => (R i).channel) |
| have hpost (i : Sigma (fun i => (R i).Index)) (hi : 0 < scoreMass μ (a.prob i)) : |
| scorePosterior μ (a.prob i) = scorePosterior (scorePosterior μ (r.prob i.1)) ((R i.1).channel.prob i.2) := |
| (scorePosterior_assoc μ _ _ (r.measurable i.1) ((R i.1).channel.measurable i.2) |
| (r.nonneg i.1) ((R i.1).channel.nonneg i.2) |
| (r.dbind_positive_parent _ μ i hi) (r.dbind_positive_conditional _ μ i hi)).symm |
| let s' : AnalysisState n := { |
| Index := Sigma (fun i => (R i).Index) |
| channel := a |
| cell := fun i => (R i.1).cell i.2 |
| B := fun i => (R i.1).B i.2 |
| B_one := fun i => (R i.1).B_one i.2 |
| support := fun i hi => by |
| rw [hpost i hi] |
| exact (R i.1).support i.2 (r.dbind_positive_conditional _ μ i hi) |
| spread := fun i hi => by |
| rw [hpost i hi] |
| exact (R i.1).spread i.2 (r.dbind_positive_conditional _ μ i hi) } |
| refine ⟨s',Sigma.fst,?_,?_,?_⟩ |
| · exact r.coarsen_dbind_fst (fun i => (R i).channel) |
| · |
| let δ := fun i j => (((R i).cell j).1-(s.cell i.1).1:ℕ) |
| let D := r.expectation μ (fun i => (R i).channel.expectation (scorePosterior μ (r.prob i)) (fun j => δ i j)) |
| have hdepth : s'.depth = s.depth+D := by |
| have he : (fun i : Sigma (fun i => (R i).Index) => (((R i.1).cell i.2).1:ℝ)) = |
| fun i => ((s.cell i.1.1).1:ℝ)+(δ i.1 i.2:ℝ) := by |
| funext i |
| dsimp only [δ] |
| rw [Nat.cast_sub ((R i.1).level_ge i.2)] |
| ring |
| change a.expectation μ _ = s.depth+D |
| rw [he,FiniteChannel.expectation_add] |
| change (r.dbind (fun i => (R i).channel)).expectation μ _ + |
| (r.dbind (fun i => (R i).channel)).expectation μ _ = _ |
| rw [r.expectation_dbind_fst (fun i => (R i).channel) μ (fun i => ((s.cell i.1).1:ℝ)), |
| r.expectation_dbind (fun i => (R i).channel) μ] |
| change r.expectation μ (fun i => ((s.cell i.1).1:ℝ))+D = s.depth+D |
| change (s.channel.bind q).expectation μ (fun i => ((s.cell i.1).1:ℝ))+D = s.depth+D |
| rw [s.channel.expectation_bind_fst q μ (fun i => ((s.cell i).1:ℝ))] |
| rfl |
| have href (i : s.Index × Fin N) : |
| finiteInformation (scorePosterior μ (r.prob i)) (R i).channel.prob+ |
| (R i).channel.expectation (scorePosterior μ (r.prob i)) (fun j => Real.log ((R i).B j)) ≤ |
| ((n:ℝ)/2+2)*Real.log 2*((R i).channel.expectation (scorePosterior μ (r.prob i)) (fun j => δ i j))+2*Real.log 2+1 := by |
| have h1 := (R i).entropy_bound |
| have h2 := (R i).concentration_bound |
| change _ ≤ _ at h1 |
| change _ ≤ _ at h2 |
| dsimp only [FiniteChannel.expectation,δ] |
| linarith |
| have hrefsum := r.expectation_mono μ href |
| rw [r.expectation_add μ] at hrefsum |
| have hright : r.expectation μ (fun i => |
| ((n:ℝ)/2+2)*Real.log 2*((R i).channel.expectation (scorePosterior μ (r.prob i)) (fun j => δ i j))+2*Real.log 2+1) = |
| ((n:ℝ)/2+2)*Real.log 2*D+2*Real.log 2+1 := by |
| rw [r.expectation_add μ,r.expectation_add μ,r.expectation_const_mul μ,r.expectation_const μ,r.expectation_const μ] |
| rw [hright] at hrefsum |
| have hinfo : s'.information = finiteInformation μ r.prob+ |
| r.expectation μ (fun i => finiteInformation (scorePosterior μ (r.prob i)) (R i).channel.prob) := |
| r.information_dbind (fun i => (R i).channel) μ |
| have hconc : s'.concentration = r.expectation μ (fun i => |
| (R i).channel.expectation (scorePosterior μ (r.prob i)) (fun j => Real.log ((R i).B j))) := r.expectation_dbind (fun i => (R i).channel) μ (fun i => Real.log ((R i.1).B i.2)) |
| have hblock : finiteInformation μ r.prob ≤ s.information+s.concentration+L := by |
| rw [s.channel.information_bind q μ] |
| have hs : (∑ i,scoreMass μ (s.channel.prob i)*finiteInformation (scorePosterior μ (s.channel.prob i)) (q i).prob) ≤ |
| s.channel.expectation μ (fun i => L+Real.log (s.B i)) := by |
| apply Finset.sum_le_sum |
| intro i _ |
| by_cases h : scoreMass μ (s.channel.prob i) = 0 |
| · simp [h] |
| · exact mul_le_mul_of_nonneg_left (hcost i |
| (lt_of_le_of_ne (scoreMass_nonneg μ _ (s.channel.nonneg i)) (Ne.symm h))) |
| (scoreMass_nonneg μ _ (s.channel.nonneg i)) |
| rw [s.channel.expectation_add μ,s.channel.expectation_const μ] at hs |
| change _ ≤ L+s.concentration at hs |
| change s.information+_ ≤ _ |
| linarith |
| unfold AnalysisState.potential |
| rw [hinfo,hconc,hdepth] |
| linarith |
| · refine ⟨{ |
| choose := fun i z => ⟨i,(R i).select z⟩ |
| choose_fiber := ?_ |
| erase_choose := fun _ _ => rfl |
| chosen := ?_ |
| positive_parent := ?_ |
| contains := fun i z hz => (R i).select_mem z hz |
| subset := fun j => (R j.1).cell_subset j.2 }⟩ |
| · intro i j |
| obtain ⟨k,j⟩ := j |
| by_cases h : i = k |
| · subst i |
| change MeasurableSet {z | (⟨k,(R k).select z⟩ : Sigma (fun i => (R i).Index)) = ⟨k,j⟩} |
| simpa only [Sigma.mk.inj_iff, heq_eq_eq, true_and] using (R k).select_fiber j |
| · have he : {z | (⟨i,(R i).select z⟩ : Sigma (fun i => (R i).Index)) = ⟨k,j⟩} = ∅ := by |
| ext z |
| simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false] |
| intro h' |
| exact h (congrArg Sigma.fst h') |
| rw [he] |
| exact MeasurableSet.empty |
| · intro j z hz |
| change 0 < r.prob j.1 z*(R j.1).channel.prob j.2 z at hz |
| have hpos : 0 < (R j.1).channel.prob j.2 z := |
| pos_of_mul_pos_right hz (r.nonneg j.1 z) |
| exact congrArg (Sigma.mk j.1) ((R j.1).chosen j.2 z hpos) |
| · intro j z hz |
| change 0 < (s.channel.prob j.1.1 z*(q j.1.1).prob j.1.2 z)*(R j.1).channel.prob j.2 z at hz |
| have hr : 0 < s.channel.prob j.1.1 z*(q j.1.1).prob j.1.2 z := |
| pos_of_mul_pos_left hz ((R j.1).channel.nonneg j.2 z) |
| exact pos_of_mul_pos_left hr ((q j.1.1).nonneg j.1.2 z) |
| end RepeatedLocalization |
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| end OAI |
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