| import OAI.Combinatorics.GraphThreshold.BernoulliUnion |
|
|
| namespace OAI |
|
|
| /-! |
| # Monotonicity of finite Bernoulli probabilities |
|
|
| For `p ≤ q`, adjoin an independent Bernoulli set to a Bernoulli-`p` set. |
| The resulting union has parameter `q` and contains the first set. This gives |
| expectation and event monotonicity, including both endpoints of `[0, 1]`. |
| -/ |
|
|
| noncomputable section |
| attribute [local instance] Classical.propDecidable |
|
|
| namespace LeanBlast.SecondKahnKalai |
|
|
| variable {X : Type*} [Fintype X] [DecidableEq X] |
|
|
| /-- A Bernoulli set with a larger parameter has larger expectation for every |
| function monotone under inclusion. The function need not be nonnegative. -/ |
| theorem bernoulliExpectation_mono_parameter {p q : ℝ} |
| (hp0 : 0 ≤ p) (hpq : p ≤ q) (hq1 : q ≤ 1) (F : Finset X → ℝ) |
| (hF : ∀ U V, U ⊆ V → F U ≤ F V) : |
| bernoulliExpectation p F ≤ bernoulliExpectation q F := by |
| have hp1 : p ≤ 1 := hpq.trans hq1 |
| by_cases hp : p = 1 |
| · have hq : q = 1 := by linarith |
| simp [hp, hq] |
| have hp_lt : p < 1 := by |
| rcases lt_or_eq_of_le hp1 with hlt | heq |
| · exact hlt |
| · exact (hp heq).elim |
| have hd : 0 < 1 - p := sub_pos.mpr hp_lt |
| let r := (q - p) / (1 - p) |
| have hr0 : 0 ≤ r := div_nonneg (sub_nonneg.mpr hpq) hd.le |
| have hr1 : r ≤ 1 := (div_le_one hd).2 (by linarith) |
| have hparam : p + r - p * r = q := by |
| dsimp [r] |
| field_simp [hd.ne'] |
| ring |
| calc |
| bernoulliExpectation p F = |
| bernoulliExpectation p (fun U => bernoulliExpectation r (fun _ => F U)) := by |
| simp |
| _ ≤ bernoulliExpectation p (fun U => bernoulliExpectation r (fun V => F (U ∪ V))) := |
| bernoulliExpectation_mono hp0 hp1 fun U => |
| bernoulliExpectation_mono hr0 hr1 fun V => hF U (U ∪ V) Finset.subset_union_left |
| _ = bernoulliExpectation q F := by |
| rw [bernoulliExpectation_union, hparam] |
|
|
| /-- Inclusion-monotone events become more likely as the Bernoulli parameter |
| increases. The proof couples actual independent finite Bernoulli sets. -/ |
| theorem bernoulliProbability_mono_parameter {p q : ℝ} |
| (hp0 : 0 ≤ p) (hpq : p ≤ q) (hq1 : q ≤ 1) (P : Finset X → Prop) |
| (hP : ∀ U V, U ⊆ V → P U → P V) : |
| bernoulliProbability p P ≤ bernoulliProbability q P := by |
| rw [bernoulliProbability_eq_expectation, bernoulliProbability_eq_expectation] |
| apply bernoulliExpectation_mono_parameter hp0 hpq hq1 |
| intro U V hUV |
| by_cases hu : P U |
| · simp [hu, hP U V hUV hu] |
| · by_cases hv : P V <;> simp [hu, hv] |
|
|
| end LeanBlast.SecondKahnKalai |
|
|
| end |
|
|
| end OAI |
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|
