| import OAI.Probability.MatroidSecretary.Secretary.PrefixObservationModel |
| import OAI.Probability.MatroidSecretary.Secretary.SourceMask |
| import OAI.Probability.MatroidSecretary.Secretary.Prefix |
| import OAI.Probability.MatroidSecretary.Secretary.BinomialMixture |
| import Mathlib.MeasureTheory.Integral.Prod |
|
|
| namespace OAI |
|
|
| /-! |
| # The concrete source observation law after a uniform random prefix |
|
|
| The initially sampled source seed determines the branch and prefix length. |
| An independent uniform permutation supplies the actual observed prefix. |
| Averaging the length, rather than conditioning on the complete initial seed, |
| recovers the full joint law of the branch and sacrificed source mask. |
|
|
| This is the actual source-law input to conditional-table reconstruction in |
| `sections/secretary.tex`, equations `eq:secretary-prefix-mask` and its following |
| kernel argument (SHA256 |
| `3554fe0f7296782edf63cc7f1305c377e7a7b9d8f8948b9510ea4ad02a702d63`). |
| It does not assume a caller-given source transport law. |
| -/ |
|
|
| namespace MatroidProphet.Secretary |
|
|
| open MeasureTheory Finset |
|
|
| /-- A predrawn Bernoulli mask supplies only the length; the independent random |
| prefix has the original product-mask law after averaging that length. -/ |
| theorem predrawn_mask_prefix_integral {n : ℕ} (p : ℝ) |
| (f : Finset (Fin n) → ℝ) : |
| bitsExpectation (fun _ : Fin n => p) univ (fun B => |
| ∫ σ : ArrivalOrder n, |
| f (univ.filter fun e : Fin n => (σ.symm e).val < B.card) |
| ∂(PMF.uniformOfFintype (ArrivalOrder n)).toMeasure) = |
| bitsExpectation (fun _ : Fin n => p) univ f := by |
| have hB (B : Finset (Fin n)) : |
| (∫ σ : ArrivalOrder n, |
| f (univ.filter fun e : Fin n => (σ.symm e).val < B.card) |
| ∂(PMF.uniformOfFintype (ArrivalOrder n)).toMeasure) = |
| (∑ S ∈ (univ : Finset (Fin n)).powersetCard B.card, f S) / |
| Nat.choose n B.card := |
| integral_uniform_prefix (by simpa using Finset.card_le_univ B) f |
| simp_rw [hB] |
| simpa only [Finset.card_univ, Fintype.card_fin, div_eq_mul_inv, mul_comm] using |
| SecretaryBinomialMixture.predrawn_card_uniform_subset p (univ : Finset (Fin n)) f |
|
|
| /-- Exact test-function identity for the manuscript's concrete full source seed. |
| The branch, prefix length and every unused source coin are sampled before the |
| independent uniform permutation. -/ |
| theorem integral_source_random_prefix {n : ℕ} (M : Matroid (Fin n)) |
| (hE : M.E = Set.univ) (f : Bool → Finset (Fin n) → ℝ) : |
| (∫ r, ∫ σ : ArrivalOrder n, |
| f (mainBranch r) |
| (univ.filter fun e : Fin n => |
| (σ.symm e).val < ((completeHiddenRule M hE).mask r).card) |
| ∂(PMF.uniformOfFintype (ArrivalOrder n)).toMeasure ∂sourceSeedLaw n) = |
| ∫ r, f (mainBranch r) ((completeHiddenRule M hE).mask r) ∂sourceSeedLaw n := by |
| rw [integral_source_branch_mask M hE (fun j B => |
| ∫ σ : ArrivalOrder n, |
| f j (univ.filter fun e : Fin n => (σ.symm e).val < B.card) |
| ∂(PMF.uniformOfFintype (ArrivalOrder n)).toMeasure)] |
| rw [integral_source_branch_mask M hE f] |
| simp_rw [predrawn_mask_prefix_integral] |
| |
| /-- The full branch/prefix pushforward equals the branch/sacrificed-mask law of |
| our concrete source algorithm. This is the exact input used by the finite |
| conditional-table reconstruction, including empty and full prefixes. -/ |
| theorem source_random_prefix_joint_law {n : ℕ} (M : Matroid (Fin n)) |
| (hE : M.E = Set.univ) : |
| ((sourceSeedLaw n).prod |
| (PMF.uniformOfFintype (ArrivalOrder n)).toMeasure).map |
| (fun z : Seed (mainSeedBits n) × ArrivalOrder n => |
| (mainBranch z.1, orderPrefix z.2 ((completeHiddenRule M hE).mask z.1).card)) = |
| (sourceSeedLaw n).map |
| (fun r => (mainBranch r, (completeHiddenRule M hE).mask r)) := by |
| classical |
| let index : Seed (mainSeedBits n) × ArrivalOrder n → Bool × Finset (Fin n) := |
| fun z => (mainBranch z.1, orderPrefix z.2 ((completeHiddenRule M hE).mask z.1).card) |
| let obs : Seed (mainSeedBits n) → Bool × Finset (Fin n) := |
| fun r => (mainBranch r, (completeHiddenRule M hE).mask r) |
| let μ := (sourceSeedLaw n).prod (PMF.uniformOfFintype (ArrivalOrder n)).toMeasure |
| have htest (f : Bool × Finset (Fin n) → ℝ) : |
| (∫ z, f (index z) ∂μ) = ∫ r, f (obs r) ∂sourceSeedLaw n := by |
| rw [integral_prod _ Integrable.of_finite] |
| simpa only [index, obs, orderPrefix] using |
| integral_source_random_prefix M hE (fun j P => f (j, P)) |
| change μ.map index = (sourceSeedLaw n).map obs |
| apply Measure.ext_of_singleton |
| intro b |
| have hreal : (μ.map index).real {b} = ((sourceSeedLaw n).map obs).real {b} := by |
| rw [← integral_indicator_one (measurableSet_singleton b), |
| ← integral_indicator_one (measurableSet_singleton b)] |
| rw [integral_map_of_stronglyMeasurable (measurable_of_countable index) |
| (measurable_of_countable _).stronglyMeasurable, |
| integral_map_of_stronglyMeasurable (measurable_of_countable obs) |
| (measurable_of_countable _).stronglyMeasurable] |
| exact htest _ |
| calc |
| μ.map index {b} = ENNReal.ofReal ((μ.map index).real {b}) := |
| (ofReal_measureReal).symm |
| _ = ENNReal.ofReal (((sourceSeedLaw n).map obs).real {b}) := |
| congrArg ENNReal.ofReal hreal |
| _ = (sourceSeedLaw n).map obs {b} := ofReal_measureReal |
| |
| end MatroidProphet.Secretary |
| |
| end OAI |
| |
