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cdec422 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 | import OAI.LinearAlgebra.MatrixMultiplication.FieldGroups.PairWindows
import OAI.LinearAlgebra.MatrixMultiplication.JointExtraction.ConditionalCountEntropy
namespace OAI
/-! Group assignments, orbit counts and extraction capacities. -/
noncomputable section
namespace MatrixMultiplication.AllFieldGroupPairEntropy
open MatrixMultiplication.Foundation AllFieldParameters AllFieldHistory AllFieldActiveLaws
open AllFieldActiveCapacity AllFieldHistoryChildLaws AllFieldGroupOrbitData
open AllFieldGroupDegrees AllFieldGroupNativeLaws AllFieldGroupPairWindows
open JointCompatibilityScaling JointCompatibilityRateLimit JointPairClassWindows
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
theorem sum_conditionalCenter_entropy {A B : Type*} [Fintype A] [Fintype B]
[DecidableEq B] (p : FiniteLaw A) (f : A → B) :
(∑ b, (p.map f).mass b * finiteEntropy
(conditionalCenter f b p.mass ((p.map f).mass b))) =
finiteEntropy p.mass - finiteEntropy (p.map f).mass := by
have ht (b : B) :
(p.map f).mass b * finiteEntropy
(conditionalCenter f b p.mass ((p.map f).mass b)) =
(p.map f).mass b * finiteEntropy (p.conditional f b).mass := by
by_cases hz : (p.map f).mass b = 0
· simp only [hz, zero_mul]
· congr 1
congr 1
funext a
simp only [conditionalCenter, FiniteLaw.conditional, dite_eq_right hz]
split_ifs <;> simp
simp_rw [ht]
have he := p.entropy_eq_map_add_conditional f
linarith
variable {K tick : ℕ} {sigma : Placement}
theorem baseSize_eq_parentBase_mul_sideMass (allocation : Allocation)
(sigma : Placement) (side : Fin 3) (h : ActiveOrder K tick sigma) (k : Fin 17) :
(baseSize (base allocation sigma side) (h, k) : ℝ) =
(parentBase allocation h : ℝ) *
JointPopulationRates.sideMass (orderLaw h).mass (sigma side) k := by
simp only [baseSize, base, JointPopulationCompatibility.classCounts,
Nat.cast_sum, Nat.cast_ite, Nat.cast_zero, JointPopulationRates.sideMass,
Finset.mul_sum]
apply Finset.sum_congr rfl
intro u _
by_cases hu : JointPopulation.shapeSide (sigma side) u = k
· simp only [hu, ite_true]
exact jointCounts_cast allocation 1 h.val.val u
· simp only [hu, ite_false, mul_zero]
theorem classMass_eq_sideMass (allocation : Allocation) (sigma : Placement)
(side : Fin 3) (c : Class (K := K) (tick := tick) sigma) :
classMass allocation sigma side c =
JointPopulationRates.sideMass (orderLaw c.1).mass (sigma side) c.2 := by
rcases c with ⟨h, k⟩
unfold classMass
rw [baseSize_eq_parentBase_mul_sideMass]
exact mul_div_cancel_left₀ _ (by exact_mod_cast (parentBase_pos allocation h).ne')
theorem nativePair_weight_marginal {K : ℕ} (w : Work K) (side : Fin 3)
(hs : w.stage ≠ 2) :
((nativePairLaw w side).map (fun a => weightCode w a.1)).mass =
canonicalSideMass w side := by
cases w with
| stageA h =>
rw [canonicalSideMass_stageA]
exact congrArg FiniteLaw.mass
(AllFieldPairMarginals.stageAPairLaw_leftWeight_marginal _ h.property side)
| stageB h =>
rw [canonicalSideMass_stageB]
exact congrArg FiniteLaw.mass
(AllFieldPairMarginals.stageBPairLaw_leftWeight_marginal _ h.property side)
| stageC h => exact False.elim (hs rfl)
theorem orderPair_weight_marginal (h : ActiveOrder K tick sigma) (side : Fin 3)
(hs : h.val.val.1.stage ≠ 2) :
((nativePairLaw h.val.val.1 (h.val.val.1.priority side)).map
(fun a => weightCode h.val.val.1 a.1)).mass =
JointPopulationRates.sideMass (orderLaw h).mass (sigma side) := by
rw [nativePair_weight_marginal _ _ hs]
have hm := placedLaw_priority_sideMass h.val.val side
rw [h.property] at hm
exact hm.symm
theorem q_eq_native_conditionalCenter (allocation : Allocation) (sigma : Placement)
(side : Fin 3) (c : Class (K := K) (tick := tick) sigma)
(hs : c.1.val.val.1.stage ≠ 2) :
q allocation sigma side c = conditionalCenter
(fun a => weightCode c.1.val.val.1 a.1) c.2
(nativePairLaw c.1.val.val.1 (c.1.val.val.1.priority side)).mass
(((nativePairLaw c.1.val.val.1 (c.1.val.val.1.priority side)).map
(fun a => weightCode c.1.val.val.1 a.1)).mass c.2) := by
rw [q_eq_conditionalCenter allocation sigma side c hs, classMass_eq_sideMass,
orderPair_weight_marginal _ _ hs]
have hp : parentPair allocation sigma side c.1 =
(nativePairLaw c.1.val.val.1 (c.1.val.val.1.priority side)).mass := by
funext a
rw [parentPair_eq_native, orderPairMixture_eq_native]
rw [hp]
theorem history_pair_entropy_of_sharing (allocation : Allocation) (sigma : Placement)
(side : Fin 3) (h : ActiveOrder K tick sigma) (hs : h.val.val.1.stage ≠ 2) :
(∑ k : Fin 17, (baseSize (base allocation sigma side) (h, k) : ℝ) *
finiteEntropy (q allocation sigma side (h, k))) =
(parentBase allocation h : ℝ) *
(finiteEntropy (nativePairLaw h.val.val.1 (h.val.val.1.priority side)).mass -
finiteEntropy (JointPopulationRates.sideMass (orderLaw h).mass (sigma side))) := by
have hq (k : Fin 17) := q_eq_native_conditionalCenter allocation sigma side (h, k) hs
simp_rw [baseSize_eq_parentBase_mul_sideMass, hq, mul_assoc]
rw [← Finset.mul_sum]
simpa only [orderPair_weight_marginal h side hs] using
congrArg (fun x : ℝ => (parentBase allocation h : ℝ) * x)
(sum_conditionalCenter_entropy
(nativePairLaw h.val.val.1 (h.val.val.1.priority side))
(fun a => weightCode h.val.val.1 a.1))
theorem history_pair_entropy_of_ordinary (allocation : Allocation) (sigma : Placement)
(side : Fin 3) (h : ActiveOrder K tick sigma) (hs : h.val.val.1.stage = 2) :
(∑ k : Fin 17, (baseSize (base allocation sigma side) (h, k) : ℝ) *
finiteEntropy (q allocation sigma side (h, k))) = 0 := by
have : Subsingleton (Statistic h.val) := by
change Subsingleton h.val.val.1.Statistic
cases he : h.val.val.1 with
| stageA a => simp [he, Work.stage] at hs
| stageB a => simp [he, Work.stage] at hs
| stageC a => dsimp [Work.Statistic]; infer_instance
apply Finset.sum_eq_zero
intro k _
by_cases hz : baseSize (base allocation sigma side) (h, k) = 0
· simp only [hz, Nat.cast_zero, zero_mul]
· have hq : q allocation sigma side (h, k) = fun _ => 1 := by
funext a
exact q_eq_one_of_subsingleton allocation sigma side (h, k)
(Nat.pos_of_ne_zero hz) a
simp only [hq, finiteEntropy, entropyTerm, Real.log_one, mul_zero,
Finset.sum_const_zero]
theorem pairReferenceEntropy_eq (allocation : Allocation) (sigma : Placement)
(side : Fin 3) :
pairReferenceEntropy (base (K := K) (tick := tick) allocation sigma side)
(q allocation sigma side) =
∑ h : ActiveOrder K tick sigma, (parentBase allocation h : ℝ) *
(if h.val.val.1.stage ≠ 2 then
finiteEntropy (nativePairLaw h.val.val.1 (h.val.val.1.priority side)).mass -
finiteEntropy (JointPopulationRates.sideMass (orderLaw h).mass (sigma side))
else 0) := by
unfold pairReferenceEntropy
rw [Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro h _
by_cases hs : h.val.val.1.stage ≠ 2
· rw [ite_eq_left hs]
exact history_pair_entropy_of_sharing allocation sigma side h hs
· rw [ite_eq_right hs, mul_zero]
exact history_pair_entropy_of_ordinary allocation sigma side h (not_not.mp hs)
theorem baseEntropy_eq (allocation : Allocation) (sigma : Placement) (side : Fin 3) :
baseEntropy (base (K := K) (tick := tick) allocation sigma side) =
∑ h : ActiveOrder K tick sigma, (parentBase allocation h : ℝ) *
(finiteEntropy (orderLaw h).mass -
finiteEntropy (JointPopulationRates.sideMass (orderLaw h).mass (sigma side))) := by
unfold baseEntropy
rw [Fintype.sum_prod_type]
apply Finset.sum_congr rfl
intro h _
have hb (k : Fin 17) : base allocation sigma side (h, k) =
JointConditionalCountEntropy.classCounts (Counts allocation 1 sigma h)
(JointPopulation.shapeSide (sigma side)) k := by
funext u
by_cases hu : JointPopulation.shapeSide (sigma side) u = k
· simp only [base, JointPopulationCompatibility.classCounts,
JointConditionalCountEntropy.classCounts, hu, ite_true]
· simp only [base, JointPopulationCompatibility.classCounts,
JointConditionalCountEntropy.classCounts, hu, ite_false]
have hs (k : Fin 17) : baseSize (base allocation sigma side) (h, k) =
JointConditionalCountEntropy.classSize (Counts allocation 1 sigma h)
(JointPopulation.shapeSide (sigma side)) k := by
simp only [baseSize, JointConditionalCountEntropy.classSize, hb k]
have hm : ((orderLaw h).map (JointPopulation.shapeSide (sigma side))).mass =
JointPopulationRates.sideMass (orderLaw h).mass (sigma side) := by
funext k
rw [FiniteLaw.map_mass]
rfl
have he := JointConditionalCountEntropy.sum_classSize_mul_entropy_of_law
(Counts allocation 1 sigma h) (JointPopulation.shapeSide (sigma side))
(orderLaw h) (parentBase allocation h : ℝ)
(fun u => jointCounts_cast allocation 1 h.val.val u)
rw [hm] at he
simpa only [hs, hb] using he
def observedEntropy (h : ActiveOrder K tick sigma) (side : Fin 3) : ℝ :=
if h.val.val.1.stage ≠ 2 then
finiteEntropy (nativePairLaw h.val.val.1 (h.val.val.1.priority side)).mass
else finiteEntropy (JointPopulationRates.sideMass (orderLaw h).mass (sigma side))
theorem baseEntropy_sub_pairReferenceEntropy (allocation : Allocation)
(sigma : Placement) (side : Fin 3) :
baseEntropy (base (K := K) (tick := tick) allocation sigma side) -
pairReferenceEntropy (base (K := K) (tick := tick) allocation sigma side)
(q allocation sigma side) =
∑ h : ActiveOrder K tick sigma, (parentBase allocation h : ℝ) *
(finiteEntropy (orderLaw h).mass - observedEntropy h side) := by
rw [baseEntropy_eq, pairReferenceEntropy_eq, ← Finset.sum_sub_distrib]
apply Finset.sum_congr rfl
intro h _
unfold observedEntropy
split_ifs <;> ring
end MatrixMultiplication.AllFieldGroupPairEntropy
end
end OAI
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