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b317718 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 | 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
end OAI
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