| import OAI.Combinatorics.GraphThreshold.Resampling |
| import OAI.Combinatorics.GraphThreshold.FiniteProbability |
| import OAI.Combinatorics.GraphThreshold.BernoulliCoupling |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace LeanBlast.SecondKahnKalai |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| variable {X B : Type*} [Fintype X] [DecidableEq X] [Fintype B] |
|
|
| def resamplingExperimentLaw (ν : FiniteLaw B) (A : B → Finset X) (ρ : ℝ) |
| (hν : ν.IsPositive) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) : |
| FiniteLaw (Finset X × (B × B)) where |
| weight ω := bernoulliWeight ρ ω.1 * resamplingWeight ν A ρ ω.1 ω.2 |
| nonneg ω := mul_nonneg (bernoulliWeight_nonneg hρ.le hρ1 ω.1) |
| (resamplingWeight_nonneg ν A hρ.le ω.1 ω.2) |
| total := by |
| rw [Fintype.sum_prod_type] |
| simp only [← Finset.mul_sum, resamplingWeight_total ν A hν hρ, mul_one] |
| exact bernoulliWeight_sum ρ |
|
|
| theorem resamplingExperiment_expectation (ν : FiniteLaw B) (A : B → Finset X) |
| {ρ : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) |
| (F : Finset X → B × B → ℝ) : |
| (resamplingExperimentLaw ν A ρ hν hρ hρ1).expectation (fun ω => F ω.1 ω.2) = |
| bernoulliExpectation ρ (fun W => (resamplingLaw ν A ρ W hν hρ).expectation (F W)) := by |
| simp only [FiniteLaw.expectation, resamplingExperimentLaw, Fintype.sum_prod_type, |
| bernoulliExpectation, resamplingLaw_weight, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro W _ |
| apply Finset.sum_congr rfl |
| intro cb _ |
| ring_nf |
|
|
| /-- Mixture expansion before using the Bernoulli union identity. -/ |
| theorem bernoulliExpectation_resampling_source (ν : FiniteLaw B) (A : B → Finset X) |
| (ρ : ℝ) (F : Finset X → ℝ) : |
| bernoulliExpectation ρ (fun W => ν.expectation (fun c => F (resamplingY A W c))) = |
| ∑ c, ν.weight c * bernoulliExpectation ρ (fun W => F (W ∪ A c)) := by |
| simp only [bernoulliExpectation, FiniteLaw.expectation, resamplingY, Finset.mul_sum] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro c _ |
| apply Finset.sum_congr rfl |
| intro W _ |
| ring_nf |
|
|
| /-- Expand the proposed density into the finite mixture of conditional |
| Bernoulli densities. This is an algebraic identity, not a normalization premise. -/ |
| theorem bernoulliExpectation_resamplingZ (ν : FiniteLaw B) (A : B → Finset X) |
| (ρ : ℝ) (F : Finset X → ℝ) : |
| bernoulliExpectation ρ (fun Y => resamplingZ ν A ρ Y * F Y) = |
| ∑ c, ν.weight c * bernoulliExpectation ρ |
| (fun Y => if A c ⊆ Y then F Y / ρ ^ (A c).card else 0) := by |
| classical |
| simp only [bernoulliExpectation, resamplingZ, Finset.sum_mul, Finset.mul_sum] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro c _ |
| apply Finset.sum_congr rfl |
| intro Y _ |
| by_cases hY : A c ⊆ Y |
| · simp only [resamplingNumerator, ite_eq_left hY, div_eq_mul_inv, inv_pow] |
| ring_nf |
| · simp only [resamplingNumerator, ite_eq_right hY, zero_mul, mul_zero] |
|
|
| theorem resampling_source_density (ν : FiniteLaw B) (A : B → Finset X) |
| {ρ : ℝ} (hρ : ρ ≠ 0) (F : Finset X → ℝ) : |
| bernoulliExpectation ρ (fun W => ν.expectation (fun c => F (resamplingY A W c))) = |
| bernoulliExpectation ρ (fun Y => resamplingZ ν A ρ Y * F Y) := by |
| rw [bernoulliExpectation_resampling_source, bernoulliExpectation_resamplingZ] |
| apply Finset.sum_congr rfl |
| intro c _ |
| rw [bernoulliExpectation_union_fixed ρ hρ] |
|
|
| theorem resampling_density (ν : FiniteLaw B) (A : B → Finset X) |
| {ρ : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (F : Finset X → ℝ) : |
| bernoulliExpectation ρ (fun W => (resamplingLaw ν A ρ W hν hρ).expectation |
| (fun cb => F (resamplingY A W cb.1))) = |
| bernoulliExpectation ρ (fun Y => resamplingZ ν A ρ Y * F Y) := by |
| calc |
| _ = bernoulliExpectation ρ (fun W => ν.expectation |
| (fun c => F (resamplingY A W c))) := by |
| unfold bernoulliExpectation |
| apply Finset.sum_congr rfl |
| intro W _ |
| exact congrArg (fun x => bernoulliWeight ρ W * x) |
| (resampling_expectation_source ν A hν hρ W (fun c => F (resamplingY A W c))) |
| _ = _ := resampling_source_density ν A hρ.ne' F |
| |
| /-- Event form of the observed-set density formula, used for small-`Z` |
| failure events in the actual resampling procedure. -/ |
| theorem resampling_density_prob (ν : FiniteLaw B) (A : B → Finset X) |
| {ρ : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (P : Finset X → Prop) : |
| bernoulliExpectation ρ (fun W => (resamplingLaw ν A ρ W hν hρ).prob |
| (fun cb => P (resamplingY A W cb.1))) = |
| bernoulliExpectation ρ (fun Y => if P Y then resamplingZ ν A ρ Y else 0) := by |
| classical |
| simpa only [FiniteLaw.prob_eq_expectation, mul_ite, mul_one, mul_zero] using |
| resampling_density ν A hν hρ (fun Y => if P Y then 1 else 0) |
| |
| theorem resamplingExperiment_density (ν : FiniteLaw B) (A : B → Finset X) |
| {ρ : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) |
| (F : Finset X → ℝ) : |
| (resamplingExperimentLaw ν A ρ hν hρ hρ1).expectation |
| (fun ω => F (resamplingY A ω.1 ω.2.1)) = |
| bernoulliExpectation ρ (fun Y => resamplingZ ν A ρ Y * F Y) := by |
| exact (resamplingExperiment_expectation ν A hν hρ hρ1 |
| (fun W cb => F (resamplingY A W cb.1))).trans (resampling_density ν A hν hρ F) |
| |
| theorem resamplingExperiment_density_prob (ν : FiniteLaw B) (A : B → Finset X) |
| {ρ : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) |
| (P : Finset X → Prop) : |
| (resamplingExperimentLaw ν A ρ hν hρ hρ1).prob |
| (fun ω => P (resamplingY A ω.1 ω.2.1)) = |
| bernoulliExpectation ρ (fun Y => if P Y then resamplingZ ν A ρ Y else 0) := by |
| classical |
| simpa only [FiniteLaw.prob_eq_expectation, mul_ite, mul_one, mul_zero] using |
| resamplingExperiment_density ν A hν hρ hρ1 (fun Y => if P Y then 1 else 0) |
| |
| /-- Every atom of the observed-set law is exactly the Bernoulli atom times |
| the explicit finite density `Z`. -/ |
| theorem resamplingExperiment_observed_mass (ν : FiniteLaw B) (A : B → Finset X) |
| {ρ : ℝ} (hν : ν.IsPositive) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (Y : Finset X) : |
| (resamplingExperimentLaw ν A ρ hν hρ hρ1).prob |
| (fun ω => resamplingY A ω.1 ω.2.1 = Y) = |
| bernoulliWeight ρ Y * resamplingZ ν A ρ Y := by |
| classical |
| simpa [bernoulliExpectation, mul_ite] using |
| resamplingExperiment_density_prob ν A hν hρ hρ1 (fun Y' => Y' = Y) |
| |
| end LeanBlast.SecondKahnKalai |
| |
| end |
| |
| end OAI |
| |
