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e320e82 | 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 | import OAI.Combinatorics.ExpectationThreshold.SelectorDefinitions
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Basic.Real.Basic
namespace OAI
/-!
# Likelihood of an actual move of colors
The full guarded profile records every changed coordinate exactly once.
Grouping the coordinate likelihood factors by that profile gives the precise
ratio used by the selector estimate.
-/
noncomputable section
open scoped BigOperators
namespace TalagrandThreshold
open Selector
variable {α : Type*} [Fintype α] [DecidableEq α] {s : ℕ}
omit [DecidableEq α] in
/-- Group a product over changed vertices by the actual full profile. -/
theorem prod_profile_eq_prod_changes (a z : Coloring α s)
(F : Color s → Color s → ℝ) :
(∏ i : Fin s, ∏ h : Color s, F i.castSucc h ^ profile a z i h) =
∏ x ∈ changes a z, F (z x) (a x) := by
classical
let n (c h : Color s) : ℕ :=
(Finset.univ.filter
(fun x : α => z x = c ∧ a x = h ∧ c.val < h.val)).card
have hfiber (c h : Color s) :
(changes a z).filter (fun x => (z x, a x) = (c, h)) =
Finset.univ.filter
(fun x : α => z x = c ∧ a x = h ∧ c.val < h.val) := by
ext x
simp only [changes, Finset.mem_filter, Finset.mem_univ, true_and, Prod.mk.injEq,
Fin.lt_def]
constructor
· rintro ⟨hx, hz, ha⟩
exact ⟨hz, ha, by simpa only [hz, ha] using hx⟩
· rintro ⟨hz, ha, hch⟩
exact ⟨by simpa only [hz, ha] using hch, hz, ha⟩
have hgroup :
(∏ c : Color s, ∏ h : Color s, F c h ^ n c h) =
∏ x ∈ changes a z, F (z x) (a x) := by
calc
_ = ∏ ch : Color s × Color s,
F ch.1 ch.2 ^
((changes a z).filter (fun x => (z x, a x) = ch)).card := by
rw [Fintype.prod_prod_type]
apply Finset.prod_congr rfl
intro c _
apply Finset.prod_congr rfl
intro h _
rw [hfiber c h]
_ = ∏ x ∈ changes a z, F (z x) (a x) := by
simpa only [Finset.prod_const] using
(Finset.prod_fiberwise' (changes a z)
(fun x => (z x, a x))
(fun ch : Color s × Color s => F ch.1 ch.2))
have hlast : (∏ h : Color s, F (Fin.last s) h ^ n (Fin.last s) h) = 1 := by
apply Finset.prod_eq_one
intro h _
have hnot : ¬ (Fin.last s).val < h.val :=
Nat.not_lt_of_ge (Nat.le_of_lt_succ h.isLt)
simp only [n, hnot, and_false, Finset.filter_false, Finset.card_empty, pow_zero]
have hcount (i : Fin s) (h : Color s) : n i.castSucc h = profile a z i h := by
simp only [n, profile, Fin.ext_iff, Fin.val_castSucc]
rw [Fin.prod_univ_castSucc, hlast, mul_one] at hgroup
simpa only [hcount] using hgroup
omit [DecidableEq α] in
/-- The profile exponent is the product of one p-factor per actual changed vertex. -/
theorem pow_profileSize_eq_prod_changes (p : ℝ) (a z : Coloring α s) :
p ^ profileSize (profile a z) = ∏ _x ∈ changes a z, p := by
calc
_ = ∏ i : Fin s, ∏ h : Color s, p ^ profile a z i h := by
simp only [Finset.prod_pow_eq_pow_sum, profileSize, rowCount]
_ = ∏ _x ∈ changes a z, p :=
prod_profile_eq_prod_changes a z (fun _ _ => p)
omit [DecidableEq α] in
/-- Exact likelihood regrouping for the original independent color law and an
actual move to earlier colors. Each source row retains all original-color counts. -/
theorem colorMass_profile_likelihood (p : ℝ) (s : ℕ) (hp : 0 < p)
(hs : p * scaleSum s ≤ 1 / 2) (a z : Coloring α s) (haz : Selector.Earlier a z) :
p ^ Selector.profileSize (Selector.profile a z) * colorMass p s a =
colorMass p s z * ∏ i : Fin s,
((p / colorProb p s i.castSucc) ^ Selector.rowCount (Selector.profile a z) i *
∏ h : Color s, (colorProb p s h) ^ Selector.profile a z i h) := by
classical
have hpowers : p ^ profileSize (profile a z) =
∏ x : α, if z x < a x then p else 1 := by
calc
_ = ∏ _x ∈ changes a z, p := pow_profileSize_eq_prod_changes p a z
_ = ∏ x : α, if z x < a x then p else 1 :=
Finset.prod_filter (fun x => z x < a x) (fun _ => p)
have hratios :
(∏ x ∈ changes a z, (p / colorProb p s (z x)) * colorProb p s (a x)) =
∏ x : α, if z x < a x then
(p / colorProb p s (z x)) * colorProb p s (a x) else 1 :=
Finset.prod_filter (fun x => z x < a x)
(fun x => (p / colorProb p s (z x)) * colorProb p s (a x))
have hpoint (x : α) :
(if z x < a x then p else 1) * colorProb p s (a x) =
colorProb p s (z x) * (if z x < a x then
(p / colorProb p s (z x)) * colorProb p s (a x) else 1) := by
by_cases h : z x < a x
· rw [ite_eq_left h, ite_eq_left h, ← mul_assoc,
mul_div_cancel₀ p (ne_of_gt (colorProb_pos p s hp hs (z x)))]
· have heq : z x = a x := le_antisymm (haz x) (le_of_not_gt h)
simp only [heq, lt_self_iff_false, ite_false, one_mul, mul_one]
calc
_ = ∏ x : α, (if z x < a x then p else 1) * colorProb p s (a x) := by
rw [hpowers, colorMass, Finset.prod_mul_distrib]
_ = ∏ x : α, colorProb p s (z x) * (if z x < a x then
(p / colorProb p s (z x)) * colorProb p s (a x) else 1) := by
apply Finset.prod_congr rfl
intro x _
exact hpoint x
_ = colorMass p s z *
∏ x ∈ changes a z, (p / colorProb p s (z x)) * colorProb p s (a x) := by
rw [Finset.prod_mul_distrib, ← hratios, colorMass]
_ = colorMass p s z *
∏ i : Fin s, ∏ h : Color s,
((p / colorProb p s i.castSucc) * colorProb p s h) ^ profile a z i h := by
rw [prod_profile_eq_prod_changes a z
(fun c h => (p / colorProb p s c) * colorProb p s h)]
_ = colorMass p s z * ∏ i : Fin s,
((p / colorProb p s i.castSucc) ^ rowCount (profile a z) i *
∏ h : Color s, (colorProb p s h) ^ profile a z i h) := by
congr 1
apply Finset.prod_congr rfl
intro i _
simp only [mul_pow, Finset.prod_mul_distrib, Finset.prod_pow_eq_pow_sum, rowCount]
end TalagrandThreshold
end
end OAI
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