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e5a97c6 | 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 | import OAI.Combinatorics.MatchingEntropy.Polyhedron
namespace OAI
noncomputable section
open scoped BigOperators Topology
open Filter
universe uV uE uI uJ uA uB uC uW uF uD uι
namespace MatchingEntropyBounds
variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E]
def fiberMass {A : Type uA} {C : Type uC} [Fintype A] [DecidableEq C]
(p : A → ℝ) (f : A → C) (c : C) : ℝ := ∑ a, if f a = c then p a else 0
theorem fiberMass_nonneg {A : Type uA} {C : Type uC} [Fintype A] [DecidableEq C]
{p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (f : A → C) (c : C) :
0 ≤ fiberMass p f c := by
apply Finset.sum_nonneg
intro a _
split <;> simp_all
theorem le_fiberMass_self {A : Type uA} {C : Type uC} [Fintype A] [DecidableEq C]
{p : A → ℝ} (hp : ∀ a, 0 ≤ p a) (f : A → C) (a : A) :
p a ≤ fiberMass p f (f a) := by
have h := Finset.single_le_sum (s := Finset.univ) (a := a)
(f := fun b => if f b = f a then p b else 0)
(by intro b _; split <;> simp_all) (Finset.mem_univ a)
simpa only [fiberMass, ite_true] using h
def jointLaw {A : Type uA} {B : Type uB} {C : Type uC} [Fintype A] [Fintype B] [DecidableEq C]
(p : A → ℝ) (q : B → ℝ) (f : A → C) (g : B → C) (z : A × B) : ℝ :=
if f z.1 = g z.2 then p z.1*q z.2 / fiberMass p f (f z.1) else 0
theorem jointLaw_nonneg {A : Type uA} {B : Type uB} {C : Type uC} [Fintype A] [Fintype B] [DecidableEq C]
{p : A → ℝ} {q : B → ℝ} (hp : ∀ a, 0 ≤ p a) (hq : ∀ b, 0 ≤ q b)
(f : A → C) (g : B → C) (z : A × B) : 0 ≤ jointLaw p q f g z := by
unfold jointLaw
split
· exact div_nonneg (mul_nonneg (hp _) (hq _)) (fiberMass_nonneg hp f _)
· rfl
theorem jointLaw_first {A : Type uA} {B : Type uB} {C : Type uC} [Fintype A] [Fintype B] [DecidableEq C]
{p : A → ℝ} {q : B → ℝ} (hp : ∀ a, 0 ≤ p a)
(f : A → C) (g : B → C) (hcompat : ∀ c, fiberMass p f c = fiberMass q g c)
(a : A) : (∑ b, jointLaw p q f g (a,b)) = p a := by
have heq : ∀ b, jointLaw p q f g (a,b) =
(p a / fiberMass p f (f a)) * (if g b=f a then q b else 0) := by
intro b
by_cases h : f a=g b
· simp only [jointLaw, h, ite_true]
ring
· simp [jointLaw, h, Ne.symm h]
simp_rw [heq]
rw [← Finset.mul_sum]
change p a / fiberMass p f (f a) * fiberMass q g (f a) = p a
rw [← hcompat]
by_cases hz : fiberMass p f (f a) = 0
· have ha : p a = 0 := le_antisymm (by simpa [hz] using le_fiberMass_self hp f a) (hp a)
simp [ha]
· exact div_mul_cancel₀ _ hz
theorem jointLaw_second {A : Type uA} {B : Type uB} {C : Type uC} [Fintype A] [Fintype B] [DecidableEq C]
{p : A → ℝ} {q : B → ℝ} (hq : ∀ b, 0 ≤ q b)
(f : A → C) (g : B → C) (hcompat : ∀ c, fiberMass p f c = fiberMass q g c)
(b : B) : (∑ a, jointLaw p q f g (a,b)) = q b := by
have heq : ∀ a, jointLaw p q f g (a,b) =
(q b / fiberMass q g (g b)) * (if f a=g b then p a else 0) := by
intro a
by_cases h : f a=g b
· simp only [jointLaw, h, ite_true, ← hcompat]
ring
· simp [jointLaw, h]
simp_rw [heq]
rw [← Finset.mul_sum]
change q b / fiberMass q g (g b) * fiberMass p f (g b) = q b
rw [hcompat]
by_cases hz : fiberMass q g (g b) = 0
· have hb : q b = 0 := le_antisymm (by simpa [hz] using le_fiberMass_self hq g b) (hq b)
simp [hb]
· exact div_mul_cancel₀ _ hz
theorem jointLaw_mem_simplex {A : Type uA} {B : Type uB} {C : Type uC} [Fintype A] [Fintype B] [DecidableEq C]
{p : A → ℝ} {q : B → ℝ} (hp : p ∈ probabilitySimplex A) (hq : q ∈ probabilitySimplex B)
(f : A → C) (g : B → C) (hcompat : ∀ c, fiberMass p f c = fiberMass q g c) :
jointLaw p q f g ∈ probabilitySimplex (A × B) := by
refine ⟨jointLaw_nonneg hp.1 hq.1 f g, ?_⟩
rw [Fintype.sum_prod_type]
simp_rw [jointLaw_first hp.1 f g hcompat]
exact hp.2
theorem sum_subtype_supported {A : Type uA} [Fintype A] (P : A → Prop) [DecidablePred P]
(w : A → ℝ) (hsupp : ∀ a, ¬P a → w a = 0) :
(∑ a : {a // P a}, w a.val) = ∑ a, w a := by
have hh : (∑ a : {a // ¬P a}, w a.val) = 0 := by
apply Finset.sum_eq_zero
intro a _
exact hsupp a.val a.property
simpa only [hh, add_zero] using (Fintype.sum_subtype_add_sum_subtype P w)
theorem compatible_jointLaw_mem_simplex {A : Type uA} {B : Type uB} {C : Type uC} [Fintype A] [Fintype B] [DecidableEq C]
{p : A → ℝ} {q : B → ℝ} (hp : p ∈ probabilitySimplex A) (hq : q ∈ probabilitySimplex B)
(f : A → C) (g : B → C) (hcompat : ∀ c, fiberMass p f c = fiberMass q g c) :
(fun z : {z : A × B // f z.1=g z.2} => jointLaw p q f g z.val) ∈
probabilitySimplex {z : A × B // f z.1=g z.2} := by
refine ⟨fun z => jointLaw_nonneg hp.1 hq.1 f g z.val, ?_⟩
rw [sum_subtype_supported (fun z : A × B => f z.1=g z.2) (jointLaw p q f g)
(by intro z hz; simp [jointLaw, hz])]
exact (jointLaw_mem_simplex hp hq f g hcompat).2
theorem fiberMass_sum {A : Type uA} {C : Type uC} [Fintype A] [Fintype C] [DecidableEq C]
(p : A → ℝ) (f : A → C) : ∑ c, fiberMass p f c = ∑ a, p a := by
unfold fiberMass
rw [Finset.sum_comm]
simp
theorem fiberMass_expect {A : Type uA} {C : Type uC} [Fintype A] [Fintype C] [DecidableEq C]
(p : A → ℝ) (f : A → C) (h : C → ℝ) :
∑ c, fiberMass p f c * h c = ∑ a, p a*h (f a) := by
unfold fiberMass
simp_rw [Finset.sum_mul]
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro a _
simp only [ite_mul, zero_mul]
simp
theorem fiberMass_mem_simplex {A : Type uA} {C : Type uC} [Fintype A] [Fintype C] [DecidableEq C]
{p : A → ℝ} (hp : p ∈ probabilitySimplex A) (f : A → C) :
fiberMass p f ∈ probabilitySimplex C :=
⟨fiberMass_nonneg hp.1 f, (fiberMass_sum p f).trans hp.2⟩
theorem jointLaw_expect_first {A : Type uA} {B : Type uB} {C : Type uC} [Fintype A] [Fintype B] [DecidableEq C]
{p : A → ℝ} {q : B → ℝ} (hp : ∀ a, 0 ≤ p a)
(f : A → C) (g : B → C) (hcompat : ∀ c, fiberMass p f c = fiberMass q g c)
(h : A → ℝ) : ∑ z : A × B, jointLaw p q f g z * h z.1 = ∑ a, p a*h a := by
rw [Fintype.sum_prod_type]
simp_rw [← Finset.sum_mul, jointLaw_first hp f g hcompat]
theorem jointLaw_expect_second {A : Type uA} {B : Type uB} {C : Type uC} [Fintype A] [Fintype B] [DecidableEq C]
{p : A → ℝ} {q : B → ℝ} (hq : ∀ b, 0 ≤ q b)
(f : A → C) (g : B → C) (hcompat : ∀ c, fiberMass p f c = fiberMass q g c)
(h : B → ℝ) : ∑ z : A × B, jointLaw p q f g z * h z.2 = ∑ b, q b*h b := by
rw [Fintype.sum_prod_type, Finset.sum_comm]
simp_rw [← Finset.sum_mul, jointLaw_second hq f g hcompat]
theorem compatible_jointLaw_expect_first {A : Type uA} {B : Type uB} {C : Type uC} [Fintype A] [Fintype B] [DecidableEq C]
{p : A → ℝ} {q : B → ℝ} (hp : ∀ a, 0 ≤ p a)
(f : A → C) (g : B → C) (hcompat : ∀ c, fiberMass p f c = fiberMass q g c)
(h : A → ℝ) :
∑ z : {z : A × B // f z.1=g z.2}, jointLaw p q f g z.val * h z.val.1 = ∑ a, p a*h a := by
rw [sum_subtype_supported (fun z : A × B => f z.1=g z.2)
(fun z => jointLaw p q f g z * h z.1)
(by intro z hz; simp [jointLaw, hz])]
exact jointLaw_expect_first hp f g hcompat h
theorem compatible_jointLaw_expect_second {A : Type uA} {B : Type uB} {C : Type uC} [Fintype A] [Fintype B] [DecidableEq C]
{p : A → ℝ} {q : B → ℝ} (hq : ∀ b, 0 ≤ q b)
(f : A → C) (g : B → C) (hcompat : ∀ c, fiberMass p f c = fiberMass q g c)
(h : B → ℝ) :
∑ z : {z : A × B // f z.1=g z.2}, jointLaw p q f g z.val * h z.val.2 = ∑ b, q b*h b := by
rw [sum_subtype_supported (fun z : A × B => f z.1=g z.2)
(fun z => jointLaw p q f g z * h z.2)
(by intro z hz; simp [jointLaw, hz])]
exact jointLaw_expect_second hq f g hcompat h
end MatchingEntropyBounds
end
end OAI
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