| import Mathlib.Probability.Distributions.Binomial |
| import OAI.Combinatorics.BalancedRyser.Candidates.CandidateMultiplicity |
|
|
| namespace OAI |
|
|
| namespace Balanced |
| open scoped _root_.Balanced |
|
|
| open Finset |
|
|
| section Counting |
|
|
| variable {I F : Type*} [Fintype I] [Fintype F] |
| variable [DecidableEq I] [DecidableEq F] |
|
|
| theorem card_exact_coordinate_set (target : I → F) (s : Finset I) : |
| (Finset.univ.filter (fun ω : I → F => |
| Finset.univ.filter (fun i => ω i = target i) = s)).card = |
| (Fintype.card F - 1) ^ (Fintype.card I - s.card) := by |
| classical |
| have hset : Finset.univ.filter (fun ω : I → F => |
| Finset.univ.filter (fun i => ω i = target i) = s) = |
| Fintype.piFinset (fun i => |
| if i ∈ s then {target i} else Finset.univ.erase (target i)) := by |
| ext ω |
| simp only [Finset.mem_filter, Finset.mem_univ, true_and, Fintype.mem_piFinset] |
| constructor |
| · intro h i |
| have hi : ω i = target i ↔ i ∈ s := by |
| simpa using (Finset.ext_iff.mp h i) |
| by_cases his : i ∈ s |
| · simp [his, hi.mpr his] |
| · have hne : ω i ≠ target i := fun he => his (hi.mp he) |
| simp [his, hne] |
| · intro h |
| ext i |
| specialize h i |
| by_cases his : i ∈ s <;> simp_all |
| rw [hset, Fintype.card_piFinset] |
| simp only [apply_ite Finset.card, Finset.card_singleton, |
| Finset.card_erase_of_mem (Finset.mem_univ _), Finset.card_univ] |
| rw [Finset.prod_ite] |
| simp [← Finset.compl_eq_univ_sdiff, Finset.filter_not, Finset.card_compl] |
|
|
| theorem card_exact_coordinate_matches (target : I → F) (m : ℕ) : |
| (Finset.univ.filter (fun ω : I → F => |
| (Finset.univ.filter (fun i => ω i = target i)).card = m)).card = |
| (Fintype.card I).choose m * |
| (Fintype.card F - 1) ^ (Fintype.card I - m) := by |
| classical |
| have hsplit := Finset.sum_card_fiberwise_eq_card_filter |
| (Finset.univ : Finset (I → F)) ((Finset.univ : Finset I).powersetCard m) |
| (fun ω => Finset.univ.filter (fun i => ω i = target i)) |
| have hmem : ∀ ω : I → F, |
| (Finset.univ.filter (fun i => ω i = target i)) ∈ |
| (Finset.univ : Finset I).powersetCard m ↔ |
| (Finset.univ.filter (fun i => ω i = target i)).card = m := by |
| intro ω |
| simp only [Finset.mem_powersetCard, Finset.subset_univ, true_and] |
| simp only [hmem] at hsplit |
| rw [← hsplit] |
| calc |
| _ = ∑ _s ∈ (Finset.univ : Finset I).powersetCard m, |
| (Fintype.card F - 1) ^ (Fintype.card I - m) := by |
| apply Finset.sum_congr rfl |
| intro s hs |
| rw [card_exact_coordinate_set, (Finset.mem_powersetCard.mp hs).2] |
| _ = _ := by simp |
|
|
| theorem uniform_exact_coordinate_matches [Nonempty F] (target : I → F) (m : ℕ) : |
| ((PMF.uniformOfFintype (I → F)).toOuterMeasure |
| {ω | (Finset.univ.filter (fun i => ω i = target i)).card = m}).toReal = |
| ((Fintype.card I).choose m : ℝ) * |
| ((Fintype.card F - 1 : ℕ) : ℝ) ^ (Fintype.card I - m) / |
| (Fintype.card F : ℝ) ^ Fintype.card I := by |
| classical |
| rw [PMF.toOuterMeasure_uniformOfFintype_apply] |
| simp only [ENNReal.toReal_div, ENNReal.toReal_pow, ENNReal.toReal_natCast, |
| Set.mem_ofPred_eq, Fintype.card_subtype, Fintype.card_fun, Nat.cast_pow] |
| rw [card_exact_coordinate_matches] |
| simp only [Nat.cast_mul, Nat.cast_pow] |
|
|
| theorem uniform_exact_coordinate_matches_binomial [Nonempty F] |
| (target : I → F) (m : ℕ) : |
| ((PMF.uniformOfFintype (I → F)).toOuterMeasure |
| {ω | (Finset.univ.filter (fun i => ω i = target i)).card = m}).toReal = |
| ((Fintype.card I).choose m : ℝ) * (1 / (Fintype.card F : ℝ)) ^ m * |
| (1 - 1 / (Fintype.card F : ℝ)) ^ (Fintype.card I - m) := by |
| rw [uniform_exact_coordinate_matches] |
| have hq : (Fintype.card F : ℝ) ≠ 0 := by |
| exact_mod_cast Fintype.card_ne_zero (α := F) |
| by_cases hm : m ≤ Fintype.card I |
| · have hbase : 1 - 1 / (Fintype.card F : ℝ) = |
| ((Fintype.card F : ℝ) - 1) / Fintype.card F := by |
| field_simp |
| rw [Nat.cast_sub Fintype.card_pos, Nat.cast_one, hbase, div_pow, div_pow, one_pow] |
| have hpow : (Fintype.card F : ℝ) ^ Fintype.card I = |
| (Fintype.card F : ℝ) ^ m * (Fintype.card F : ℝ) ^ (Fintype.card I - m) := by |
| rw [← pow_add, Nat.add_sub_of_le hm] |
| rw [hpow] |
| field_simp |
| · rw [Nat.choose_eq_zero_of_lt (Nat.lt_of_not_ge hm)] |
| simp |
|
|
| end Counting |
|
|
| section Affine |
|
|
| variable {F : Type*} [Field F] [Fintype F] [DecidableEq F] |
|
|
| noncomputable def multiplicityParameter (F : Type*) [Fintype F] [Nonempty F] : |
| unitInterval := |
| ⟨1 / (Fintype.card F : ℝ), by |
| constructor |
| · positivity |
| · have hq : (1 : ℝ) ≤ Fintype.card F := by |
| exact_mod_cast Fintype.card_pos (α := F) |
| simpa using one_div_le_one_div_of_le (by norm_num : (0 : ℝ) < 1) hq⟩ |
|
|
| theorem sampledMultiplicity_atom (p : Point F) (m : ℕ) : |
| offsetEventProbability {M : MergedPairs F | sampledMultiplicity M p = m} = |
| ((2 * (Fintype.card F + 1)).choose m : ℝ) * |
| (1 / (Fintype.card F : ℝ)) ^ m * |
| (1 - 1 / (Fintype.card F : ℝ)) ^ (2 * (Fintype.card F + 1) - m) := by |
| classical |
| let e := Equiv.curry (Direction F) Bool F |
| have he := uniform_toOuterMeasure_equiv e |
| {M : MergedPairs F | sampledMultiplicity M p = m} |
| have h := uniform_exact_coordinate_matches_binomial |
| (fun ds : Direction F × Bool => lineEval ds.1 p) m |
| unfold offsetEventProbability |
| rw [offsetLaw_eq_uniform, ← he] |
| simpa [e, Equiv.curry, Function.curry, sampledMultiplicity, OnLine, constituent, |
| eq_comm, Fintype.card_prod, Fintype.card_option, Nat.mul_comm] using h |
|
|
| theorem sampledMultiplicity_law (p : Point F) : |
| ((offsetLaw F).map (fun M => sampledMultiplicity M p)).toMeasure = |
| ProbabilityTheory.binomial (2 * (Fintype.card F + 1)) (multiplicityParameter F) := by |
| apply MeasureTheory.Measure.ext_of_measureReal_singleton |
| intro m |
| rw [ProbabilityTheory.binomial_real_singleton] |
| change (((offsetLaw F).map (fun M => sampledMultiplicity M p)).toMeasure {m}).toReal = _ |
| rw [PMF.toMeasure_apply_eq_toOuterMeasure_apply _ (measurableSet_singleton _), |
| PMF.toOuterMeasure_map_apply] |
| exact sampledMultiplicity_atom p m |
|
|
| end Affine |
|
|
| end Balanced |
|
|
| end OAI |
|
|
