| import OAI.Probability.MatroidProphet.Maximum.Probability |
| import OAI.Probability.MatroidProphet.SeedLaw |
| import OAI.Probability.MatroidProphet.Simulation |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet.Maximum |
|
|
| open MeasureTheory Finset Set |
|
|
| variable {n : ℕ} {K : Type*} [Countable K] [MeasurableSpace K] |
| [MeasurableSingletonClass K] [LinearOrder K] |
|
|
| lemma le_hiddenWorstReward {bits : ℕ} (A : HiddenRule n bits) (w : Weights n) |
| (r : Seed bits) (c : ℝ) (h : ∀ π, c ≤ hiddenReward A r w π) : |
| c ≤ hiddenWorstReward A w r := by |
| classical |
| apply Finset.le_inf' |
| intro π _ |
| exact h π |
|
|
| theorem maximum_integral_top (M : Matroid (Fin n)) |
| (key : ℝ → K) (hkey : Measurable key) (active : K → Prop) |
| (w : Weights n) (hw : ∀ e, 0 ≤ w e) (g : Fin n) |
| (hg : M.Indep ({g} : Set (Fin n))) (hag : active (key (w g))) |
| (htop : ∀ f, M.Indep ({f} : Set (Fin n)) → active (key (w f)) → f ≠ g → |
| higher (g, key (w g)) (f, key (w f))) : |
| w g / 4 ≤ ∫ r, hiddenWorstReward |
| (maximumHidden M key hkey active (seedSet (bits := n))) w r |
| ∂bernoulliSeedLaw (fun _ => (1 / 2 : ℝ)) |
| (fun _ => by norm_num) (fun _ => by norm_num) := by |
| classical |
| let A := maximumHidden M key hkey active (seedSet (bits := n)) |
| let F : Finset (Fin n) → ℝ := fun H => hiddenWorstReward A w (setSeed H) |
| rw [integral_bernoulliSeedLaw] |
| change w g / 4 ≤ bitsExpectation (fun _ => (1 / 2 : ℝ)) Finset.univ F |
| have hF : ∀ H ⊆ Finset.univ, 0 ≤ F H := |
| fun H _ => hiddenWorstReward_nonneg A w hw (setSeed H) |
| let P : Finset (Fin n) := Finset.univ.filter fun f => |
| M.Indep ({f} : Set (Fin n)) ∧ active (key (w f)) ∧ f ≠ g |
| by_cases hP : P.Nonempty |
| · obtain ⟨h, hhP, hsecond⟩ := exists_top (fun f => key (w f)) P hP |
| have hh := (Finset.mem_filter.mp hhP).2 |
| apply maximum_expectation_two Finset.univ g h (Finset.mem_univ _) (Finset.mem_univ _) |
| (Ne.symm hh.2.2) F (w g) hF |
| intro H _ hgH hhH |
| apply le_hiddenWorstReward |
| intro π |
| have hqual := threshold_top_two M active (seedSet (bits := n)) (setSeed H) |
| (fun f => key (w f)) g h hg hag hh.1 hh.2.1 htop |
| (fun f hf haf hfg hfh => hsecond f (Finset.mem_filter.mpr ⟨Finset.mem_univ _, hf, haf, hfg⟩) hfh) |
| (by simpa using hgH) (by simpa using hhH) |
| exact (maximumHidden_single_qualifier M key hkey active seedSet w (setSeed H) g |
| (by simpa using hgH) hqual π).symm.le |
| · apply maximum_expectation_one Finset.univ g (Finset.mem_univ _) F (w g) (hw g) hF |
| intro H _ hgH |
| apply le_hiddenWorstReward |
| intro π |
| have honly : ∀ f, M.Indep ({f} : Set (Fin n)) → active (key (w f)) → f = g := by |
| intro f hf haf |
| by_contra hfg |
| exact hP ⟨f, Finset.mem_filter.mpr ⟨Finset.mem_univ _, hf, haf, hfg⟩⟩ |
| have hqual := threshold_only_positive M active (seedSet (bits := n)) (setSeed H) |
| (fun f => key (w f)) g hg hag honly (by simpa using hgH) |
| exact (maximumHidden_single_qualifier M key hkey active seedSet w (setSeed H) g |
| (by simpa using hgH) hqual π).symm.le |
|
|
| end MatroidProphet.Maximum |
|
|
| end OAI |
|
|
