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a2ee19a | 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 | import Mathlib
import OAI.Probability.Perceptron.Variational.BlockFocus
namespace OAI
noncomputable section
namespace SphericalPerceptronFreeEnergy
open MeasureTheory ProbabilityTheory Set Filter
open scoped Classical ENNReal NNReal BigOperators Topology
lemma indexedDepthLaw_block_mass (k : ℕ) (z : Fin k → ℝ)
(r : ℕ) (A : FiniteBlock (Fin (k+1)) r) :
(indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1)))) {R | finiteBlock r R = A} =
∫⁻ b, indexedBlockSample k A b ∂indexedCascadeBaseLaw k z := by
have hE : MeasurableSet {R : FiniteOverlap (Fin (k+1)) | finiteBlock r R = A} :=
measurableSet_eq_fun (finiteBlock_measurable r) measurable_const
rw [← lintegral_indicator_one hE]
simp only [Set.indicator_apply, Pi.one_apply, Set.mem_ofPred_eq]
rw [indexedDepthLaw_lintegral_prefix k z r (fun B => if B = A then 1 else 0)]
apply lintegral_congr
intro b
rw [indexedBlockSample_eq_measure,← lintegral_indicator_one ((Set.to_countable _).measurableSet)]
simp only [Set.indicator_apply, Pi.one_apply, Set.mem_ofPred_eq]
lemma indexedDepthLaw_focused_mass (k : ℕ) (z : Fin k → ℝ) (r d : ℕ)
(A : FiniteBlock (Fin (k+1)) r) (i : Fin r) :
(indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1))))
{R | finiteBlock r R = A ∧ d ≤ (R i r).val} =
∫⁻ b, indexedBlockFocused k d A i b ∂indexedCascadeBaseLaw k z := by
let E : Set (FiniteOverlap (Fin (k+1))) := {R | finiteBlock r R = A ∧ d ≤ (R i r).val}
have hE : MeasurableSet E := by
exact (measurableSet_eq_fun (finiteBlock_measurable r) measurable_const).inter
((Set.to_countable {l : Fin (k+1) | d ≤ l.val}).measurableSet.preimage (by fun_prop))
change (indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1)))) E = _
rw [← lintegral_indicator_one hE]
rw [indexedDepthLaw_lintegral k z (E.indicator 1) (measurable_const.indicator hE)]
apply lintegral_congr
intro b
let S : Set ((Fin r → IndexedLeaf k) × IndexedLeaf k) :=
{p | (fun a c => indexedCommonDepth k (p.1 a) (p.1 c)) = A ∧
d ≤ (indexedCommonDepth k (p.1 i) p.2).val}
have hh := (infiniteReplica_prefix_fresh_preserving (indexedLeafProbability k b) r).lintegral_comp
(f := S.indicator (fun _ => (1 : ℝ≥0∞)))
(measurable_const.indicator ((Set.to_countable S).measurableSet))
calc
_ = ∫⁻ xs, S.indicator 1 ((fun j : Fin r => xs j.val),xs r)
∂Measure.infinitePi (fun _ : ℕ => indexedLeafProbability k b) := rfl
_ = ∫⁻ p, S.indicator 1 p
∂(Measure.pi (fun _ : Fin r => indexedLeafProbability k b)).prod (indexedLeafProbability k b) := hh
_ = _ := by
rw [lintegral_indicator_one ((Set.to_countable S).measurableSet),indexedBlockFocused_eq_measure]
theorem indexedDepthLaw_focused_formula (k : ℕ) (z : Fin k → ℝ)
(hz : StrictMono z) (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1)
(r d : ℕ) (hd : d ≤ k) (A : FiniteBlock (Fin (k+1)) r) (i : Fin r) :
(indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1))))
{R | finiteBlock r R = A ∧ d ≤ (R i r).val} =
(indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1)))) {R | finiteBlock r R = A} *
ENNReal.ofReal (((∑ j, if d ≤ (A i j).val then (1:ℝ) else 0) -
(if h : d = 0 then 0 else z ⟨d-1,by omega⟩)) / r) := by
rw [indexedDepthLaw_focused_mass,indexedDepthLaw_block_mass,
indexedBlockFocused_integral k z hz hz0 hz1 d hd A i,Fintype.card_fin]
lemma indexedDepthLaw_one_block_ae (k : ℕ) (z : Fin k → ℝ) :
∀ᵐ R ∂(indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1)))),
finiteBlock 1 R = fun _ _ => Fin.last k := by
filter_upwards [indexedDepthLaw_geometry k z] with R hR
funext i j
have hi : i = 0 := Subsingleton.elim _ _
have hj : j = 0 := Subsingleton.elim _ _
simpa only [hi,hj,finiteBlock,Fin.val_zero] using hR.2.1 0
lemma indexedDepthLaw_pair_tail (k : ℕ) (z : Fin k → ℝ)
(hz : StrictMono z) (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1)
(d : ℕ) (hd : d ≤ k) :
(indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1)))) {R | d ≤ (R 0 1).val} =
ENNReal.ofReal (1 - (if h : d = 0 then 0 else z ⟨d-1,by omega⟩)) := by
let μ := (indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1))))
let A : FiniteBlock (Fin (k+1)) 1 := fun _ _ => Fin.last k
have hB : μ {R | finiteBlock 1 R = A} = 1 := by
rw [← measure_univ (μ := μ)]
apply measure_congr
filter_upwards [indexedDepthLaw_one_block_ae k z] with R hR
change (finiteBlock 1 R = A) = True
exact propext ⟨fun _ => trivial, fun _ => hR⟩
have hC : μ {R | finiteBlock 1 R = A ∧ d ≤ (R (0:Fin 1) 1).val} =
μ {R | d ≤ (R 0 1).val} := by
apply measure_congr
filter_upwards [indexedDepthLaw_one_block_ae k z] with R hR
change (finiteBlock 1 R = A ∧ d ≤ (R 0 1).val) = (d ≤ (R 0 1).val)
exact propext ⟨fun h => h.2, fun h => ⟨hR,h⟩⟩
have h := indexedDepthLaw_focused_formula k z hz hz0 hz1 1 d hd A 0
change μ _ = μ _ * _ at h
rw [hB,hC] at h
simpa only [A,Fintype.sum_unique,Fin.val_last,ite_eq_left hd,Nat.cast_one,div_one,one_mul] using h
lemma measureReal_fin_atom_from_tails {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsFiniteMeasure μ] (k : ℕ)
(f : Ω → Fin (k+1)) (hf : Measurable f) (E : Set Ω) (hE : MeasurableSet E)
(c : Fin (k+1)) :
μ.real {x | x ∈ E ∧ f x = c} =
μ.real {x | x ∈ E ∧ c.val ≤ (f x).val} -
μ.real {x | x ∈ E ∧ c.val+1 ≤ (f x).val} := by
have hs : MeasurableSet {x | x ∈ E ∧ c.val+1 ≤ (f x).val} :=
hE.inter ((Set.to_countable {l : Fin (k+1) | c.val+1 ≤ l.val}).measurableSet.preimage hf)
have he : {x | x ∈ E ∧ f x = c} =
{x | x ∈ E ∧ c.val ≤ (f x).val} \ {x | x ∈ E ∧ c.val+1 ≤ (f x).val} := by
apply Set.ext
intro x
change (x ∈ E ∧ f x = c) ↔
(x ∈ E ∧ c.val ≤ (f x).val) ∧ ¬ (x ∈ E ∧ c.val+1 ≤ (f x).val)
constructor
· rintro ⟨hx,hxc⟩
simp only [hxc,le_refl,and_true,Nat.not_succ_le_self,and_false,not_false_eq_true]
exact hx
· intro hx
have hn : ¬ c.val+1 ≤ (f x).val := fun h => hx.2 ⟨hx.1.1,h⟩
have hl := hx.1.2
exact ⟨hx.1.1,Fin.ext (by omega)⟩
rw [he,measureReal_sdiff (by intro x hx; exact ⟨hx.1, Nat.le_trans (Nat.le_succ _) hx.2⟩) hs]
theorem finiteGG_of_depth_thresholds (k : ℕ)
(μ : Measure (FiniteOverlap (Fin (k+1)))) [IsProbabilityMeasure μ]
(hh : ∀ r, 1 ≤ r → ∀ (i : Fin r) (A : FiniteBlock (Fin (k+1)) r)
(d : ℕ), d ≤ k+1 →
μ.real {R | finiteBlock r R = A ∧ d ≤ (R i r).val} =
μ.real {R | finiteBlock r R = A} * μ.real {R | d ≤ (R 0 1).val} / r +
(∑ j ∈ Finset.univ.erase i,
μ.real {R | finiteBlock r R = A ∧ d ≤ (R i j).val}) / r) : FiniteGG μ := by
classical
intro r hr i A c
have hE : MeasurableSet {R : FiniteOverlap (Fin (k+1)) | finiteBlock r R = A} :=
measurableSet_eq_fun (finiteBlock_measurable r) measurable_const
have hm (i j : ℕ) : Measurable (fun R : FiniteOverlap (Fin (k+1)) => R i j) := by fun_prop
have hAtom (a b : ℕ) : μ.real {R | finiteBlock r R = A ∧ R a b = c} =
μ.real {R | finiteBlock r R = A ∧ c.val ≤ (R a b).val} -
μ.real {R | finiteBlock r R = A ∧ c.val+1 ≤ (R a b).val} :=
measureReal_fin_atom_from_tails μ k (fun R => R a b)
(hm a b) {R | finiteBlock r R = A} hE c
have hPair := measureReal_fin_atom_from_tails μ k (fun R => R 0 1)
(hm 0 1) univ MeasurableSet.univ c
simp only [Set.mem_univ,true_and] at hPair
simp_rw [hAtom]
rw [hPair,hh r hr i A c.val (by omega),hh r hr i A (c.val+1) (by omega),
Finset.sum_sub_distrib]
ring
end SphericalPerceptronFreeEnergy
end
end OAI
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