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f0f05de | 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 | import OAI.Combinatorics.BalancedRyser.Local.AlignedFreeCoordinates
import OAI.Combinatorics.BalancedRyser.Local.AlignmentNoBias
import OAI.Combinatorics.BalancedRyser.Probability.FiniteConditionalRatios
namespace OAI
namespace Balanced.AlignedListProbability
open Local PartialAlignment
noncomputable section
attribute [local instance] Classical.propDecidable
variable {F : Type*} [Field F] [Fintype F] [DecidableEq F]
theorem conditioned_alignment_suffix_observed {ι : Type*} [DecidableEq ι]
(T : ι → FourTuple F) (L : ι → Line F) (post pre : List ι)
(hT : ∀ i ∈ post ++ pre, Function.Injective (T i).generator)
(hg : ∀ i ∈ post ++ pre, ∀ j, (T i).generator j ≠ (L i).1)
(horder : SmallReuseOrder (fun i => generatorSet (T i)) (post ++ pre))
(H E : MergedPairs F → Prop)
(hH : ∀ M N, (∀ d ∈ pre.toFinset.biUnion (fun i => generatorSet (T i)),
M d = N d) → (H M ↔ H N))
(hE : ∀ M N, (∀ d ∈ pre.toFinset.biUnion (fun i => generatorSet (T i)),
M d = N d) → (E M ↔ E N)) :
conditionedOffsetProbability (fun M => ListAligned T L (post ++ pre) M ∧ H M) E =
conditionedOffsetProbability (fun M => ListAligned T L pre M ∧ H M) E := by
have hnum := list_alignment_append_weighted T L post pre hT hg horder
(fun M => H M ∧ E M) (fun M N h => and_congr (hH M N h) (hE M N h))
have hden := list_alignment_append_weighted T L post pre hT hg horder H hH
have hn :
offsetProbability (fun M => (ListAligned T L (post ++ pre) M ∧ H M) ∧ E M) =
((Fintype.card F : ℝ)⁻¹ ^ 4) ^ post.length *
offsetProbability (fun M => (ListAligned T L pre M ∧ H M) ∧ E M) := by
simpa only [and_assoc, and_left_comm, and_comm] using hnum
have hd : offsetProbability (fun M => ListAligned T L (post ++ pre) M ∧ H M) =
((Fintype.card F : ℝ)⁻¹ ^ 4) ^ post.length *
offsetProbability (fun M => ListAligned T L pre M ∧ H M) := by
simpa only [and_comm] using hden
unfold conditionedOffsetProbability
rw [hn, hd]
have hc : ((Fintype.card F : ℝ)⁻¹ ^ 4) ^ post.length ≠ 0 := by positivity
exact mul_div_mul_left _ _ hc
theorem conditioned_alignment_step {ι : Type*} [DecidableEq ι]
(T : ι → FourTuple F) (L : ι → Line F) (post pre : List ι) (i : ι)
(hT : ∀ k ∈ post ++ i :: pre, Function.Injective (T k).generator)
(hg : ∀ k ∈ post ++ i :: pre, ∀ j, (T k).generator j ≠ (L k).1)
(horder : SmallReuseOrder (fun k => generatorSet (T k)) (post ++ i :: pre))
(a : MergedPairs F) (ha : ListAligned T L pre a)
(E : MergedPairs F → Prop)
(hE : ∀ M N, (∀ d ∈ (i :: pre).toFinset.biUnion (fun k => generatorSet (T k)),
M d = N d) → (E M ↔ E N)) :
conditionedOffsetProbability
(fun M => ListAligned T L (post ++ i :: pre) M ∧
∀ d ∈ pre.toFinset.biUnion (fun k => generatorSet (T k)), M d = a d) E =
conditionedOffsetProbability
(fun M => (∀ j, OnLine (tuplePoint M (T i) j) (L i)) ∧
∀ d ∈ pre.toFinset.biUnion (fun k => generatorSet (T k)), M d = a d) E := by
let H : MergedPairs F → Prop :=
fun M => ∀ d ∈ pre.toFinset.biUnion (fun k => generatorSet (T k)), M d = a d
have hH (M N : MergedPairs F)
(h : ∀ d ∈ (i :: pre).toFinset.biUnion (fun k => generatorSet (T k)),
M d = N d) : H M ↔ H N := by
have heq (d) (hd : d ∈ pre.toFinset.biUnion (fun k => generatorSet (T k))) :
M d = N d := by
apply h
rcases Finset.mem_biUnion.mp hd with ⟨k, hk, hdk⟩
exact Finset.mem_biUnion.mpr ⟨k, by simpa using Or.inr hk, hdk⟩
constructor <;> intro hm d hd
· exact (heq d hd).symm.trans (hm d hd)
· exact (heq d hd).trans (hm d hd)
have hs := conditioned_alignment_suffix_observed T L post (i :: pre)
hT hg horder H E hH hE
have hcollapse : (fun M => ListAligned T L (i :: pre) M ∧ H M) =
(fun M => (∀ j, OnLine (tuplePoint M (T i) j) (L i)) ∧ H M) := by
funext M
apply propext
constructor
· rintro ⟨hal, hfix⟩
exact ⟨hal i (by simp), hfix⟩
· rintro ⟨hcur, hfix⟩
refine ⟨?_, hfix⟩
have hp : ListAligned T L pre M :=
(listAligned_agree T L pre M a hfix).mpr ha
intro k hk
rcases List.mem_cons.mp hk with rfl | hk
· exact hcur
· exact hp k hk
rw [hcollapse] at hs
exact hs
theorem current_tuple_conditional_ratio
(T : FourTuple F) (hT : Function.Injective T.generator)
(g : Direction F) (c : F) (D : Finset (Direction F)) (a : MergedPairs F)
(E : (Fin 4 → Bool → F) → Prop) :
conditionedOffsetProbability
(fun M => (∀ d ∈ D, M d = a d) ∧ ∀ j, OnLine (tuplePoint M T j) (g,c))
(fun M => E (fun j => M (T.generator j))) =
((PMF.uniformOfFintype (PartialOffsets (exposedTupleIndices T D)
(fun j => a (T.generator j)))).toOuterMeasure
{o | (∀ j, OnLine (TupleAlignment.tuplePositionEquiv T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side o.1 j) (g,c)) ∧ E o.1}).toReal /
((PMF.uniformOfFintype (PartialOffsets (exposedTupleIndices T D)
(fun j => a (T.generator j)))).toOuterMeasure
{o | ∀ j, OnLine (TupleAlignment.tuplePositionEquiv T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side o.1 j) (g,c)}).toReal := by
let P := PartialOffsets (exposedTupleIndices T D) (fun j => a (T.generator j))
let B : Set (Fin 4 → Bool → F) :=
{o | ∀ j, OnLine (TupleAlignment.tuplePositionEquiv T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side o j) (g,c)}
have hm := global_exposed_tuple_law T hT D a
rw [partialOffsets_conditional_law] at hm
have hr := marginal_conditional_ratio (PMF.uniformOfFintype (MergedPairs F))
((PMF.uniformOfFintype P).map Subtype.val)
{M | ∀ d ∈ D, M d = a d} (ConditionalRestriction.fixed_supported (fun d => d ∈ D) a)
(fun M j => M (T.generator j)) hm B {o | E o}
rw [PMF.toOuterMeasure_map_apply, PMF.toOuterMeasure_map_apply] at hr
have hp (K : MergedPairs F → Prop) : offsetProbability K =
((PMF.uniformOfFintype (MergedPairs F)).toOuterMeasure {M | K M}).toReal :=
(FiniteUniformFibers.probability_eq_sum K).symm
unfold conditionedOffsetProbability
rw [hp, hp]
simpa only [P, B, Set.preimage_ofPred_eq, ← Set.ofPred_and, and_assoc,
tuplePoint, constituent, TupleAlignment.tuplePositionEquiv_apply] using hr
abbrev CurrentFreeCoordinates (T : FourTuple F) (D : Finset (Direction F)) :=
{j : Fin 4 // j ∈ freePositions (exposedTupleIndices T D)} → F
def generateCurrentTupleOffsets
(T : FourTuple F) (hT : Function.Injective T.generator)
(g : Direction F) (hg : ∀ j, T.generator j ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) (x : CurrentFreeCoordinates T D) : Fin 4 → Bool → F :=
((alignedFreeCoordinatesEquiv T.generator (TupleAlignment.adjacent_ne_of_injective T hT)
T.side g hg c (exposedTupleIndices T D) hD (fun j => a (T.generator j))).symm x).1.1
theorem current_tuple_generated_probability
(T : FourTuple F) (hT : Function.Injective T.generator)
(g : Direction F) (hg : ∀ j, T.generator j ≠ g) (c : F)
(D : Finset (Direction F)) (hD : (exposedTupleIndices T D).card ≤ 1)
(a : MergedPairs F) (E : (Fin 4 → Bool → F) → Prop) :
conditionedOffsetProbability
(fun M => (∀ d ∈ D, M d = a d) ∧ ∀ j, OnLine (tuplePoint M T j) (g,c))
(fun M => E (fun j => M (T.generator j))) =
((PMF.uniformOfFintype (CurrentFreeCoordinates T D)).toOuterMeasure
{x | E (generateCurrentTupleOffsets T hT g hg c D hD a x)}).toReal := by
rw [current_tuple_conditional_ratio T hT g c D a E]
have hm := alignedPartial_conditional_law T.generator
(TupleAlignment.adjacent_ne_of_injective T hT) T.side g hg c
(exposedTupleIndices T D) hD (fun j => a (T.generator j))
have h := congrArg (fun law => (law.toOuterMeasure {o | E o.1}).toReal) hm
rw [filter_outer_ratio, ENNReal.toReal_div, PMF.toOuterMeasure_map_apply] at h
exact h
omit [Field F] [Fintype F] in
theorem step_exposed_count {ι : Type*} [DecidableEq ι]
(T : ι → FourTuple F) (post pre : List ι) (i : ι)
(hi : Function.Injective (T i).generator)
(horder : SmallReuseOrder (fun k => generatorSet (T k)) (post ++ i :: pre)) :
(exposedTupleIndices (T i) (pre.toFinset.biUnion (fun k => generatorSet (T k)))).card ≤ 1 := by
rw [card_exposedTupleIndices (T i) hi]
exact (smallReuseOrder_suffix (fun k => generatorSet (T k)) post (i :: pre) horder).2
theorem full_aligned_step_generated_probability {ι : Type*} [DecidableEq ι]
(T : ι → FourTuple F) (L : ι → Line F) (post pre : List ι) (i : ι)
(hT : ∀ k ∈ post ++ i :: pre, Function.Injective (T k).generator)
(hg : ∀ k ∈ post ++ i :: pre, ∀ j, (T k).generator j ≠ (L k).1)
(horder : SmallReuseOrder (fun k => generatorSet (T k)) (post ++ i :: pre))
(a : MergedPairs F) (ha : ListAligned T L pre a)
(E : (Fin 4 → Bool → F) → Prop) :
let D := pre.toFinset.biUnion (fun k => generatorSet (T k))
conditionedOffsetProbability
(fun M => ListAligned T L (post ++ i :: pre) M ∧ ∀ d ∈ D, M d = a d)
(fun M => E (fun j => M ((T i).generator j))) =
((PMF.uniformOfFintype (CurrentFreeCoordinates (T i) D)).toOuterMeasure
{x | E (generateCurrentTupleOffsets (T i) (hT i (by simp)) (L i).1
(hg i (by simp)) (L i).2 D (step_exposed_count T post pre i (hT i (by simp)) horder)
a x)}).toReal := by
let D := pre.toFinset.biUnion (fun k => generatorSet (T k))
have hE (M N : MergedPairs F)
(h : ∀ d ∈ (i :: pre).toFinset.biUnion (fun k => generatorSet (T k)), M d = N d) :
E (fun j => M ((T i).generator j)) ↔ E (fun j => N ((T i).generator j)) := by
have heq : (fun j => M ((T i).generator j)) = (fun j => N ((T i).generator j)) := by
funext j
apply h
exact Finset.mem_biUnion.mpr ⟨i, by simp, by simp [generatorSet]⟩
rw [heq]
have hs := conditioned_alignment_step T L post pre i hT hg horder a ha
(fun M => E (fun j => M ((T i).generator j))) hE
have hc := current_tuple_generated_probability (T i) (hT i (by simp)) (L i).1
(hg i (by simp)) (L i).2 D (step_exposed_count T post pre i (hT i (by simp)) horder) a E
exact hs.trans (by simpa only [and_comm, Prod.eta] using hc)
end
end Balanced.AlignedListProbability
end OAI
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