| import OAI.Combinatorics.BalancedRyser.Local.InjectionAuxiliaryExposure |
|
|
| namespace OAI |
|
|
| namespace Balanced.InjectionExposure |
| noncomputable section |
| open scoped Classical BigOperators |
| variable {I J Z : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] |
|
|
| theorem mem_completions_symm (K : Finset I) (e f : I ↪ J) : |
| e ∈ completions K f ↔ f ∈ completions K e := by |
| simp only [mem_completions] |
| exact ⟨fun h j hj => (h j hj).symm, fun h j hj => (h j hj).symm⟩ |
|
|
| theorem completions_eq_of_mem (K : Finset I) (e f : I ↪ J) |
| (h : e ∈ completions K f) : completions K e = completions K f := by |
| ext a |
| simp only [mem_completions] at h ⊢ |
| exact ⟨fun ha j hj => (ha j hj).trans (h j hj), |
| fun ha j hj => (ha j hj).trans (h j hj).symm⟩ |
|
|
| variable [Nonempty (I ↪ J)] |
|
|
| theorem uniform_bind_completions (K : Finset I) : |
| (PMF.uniformOfFintype (I ↪ J)).bind |
| (fun e => PMF.uniformOfFinset (completions K e) (completions_nonempty K e)) = |
| PMF.uniformOfFintype (I ↪ J) := by |
| ext f |
| rw [PMF.bind_apply, tsum_fintype] |
| simp only [PMF.uniformOfFintype_apply] |
| rw [← Finset.mul_sum] |
| suffices hs : (∑ e : I ↪ J, |
| PMF.uniformOfFinset (completions K e) (completions_nonempty K e) f) = 1 by |
| rw [hs, mul_one] |
| have heq : (∑ e : I ↪ J, |
| PMF.uniformOfFinset (completions K e) (completions_nonempty K e) f) = |
| ∑ e : I ↪ J, if e ∈ completions K f then |
| ((completions K f).card : ENNReal)⁻¹ else 0 := by |
| apply Finset.sum_congr rfl |
| intro e _ |
| rw [PMF.uniformOfFinset_apply] |
| by_cases h : e ∈ completions K f |
| · have hf : f ∈ completions K e := (mem_completions_symm K e f).mp h |
| rw [ite_eq_left hf, ite_eq_left h, completions_eq_of_mem K e f h] |
| · have hf : f ∉ completions K e := fun hf => h ((mem_completions_symm K f e).mp hf) |
| rw [ite_eq_right hf, ite_eq_right h] |
| rw [heq] |
| have hfilter : Finset.univ.filter (fun e => e ∈ completions K f) = completions K f := by |
| ext e |
| simp |
| rw [← Finset.sum_filter, hfilter, Finset.sum_const, nsmul_eq_mul] |
| exact ENNReal.mul_inv_cancel |
| (by exact_mod_cast (completions_nonempty K f).card_pos.ne') (by simp) |
| |
| theorem uniform_bind_auxiliaryLaw (K : Finset I) |
| (noise : (I ↪ J) → PMF Z) : |
| (PMF.uniformOfFintype (I ↪ J)).bind (fun e => auxiliaryLaw K e noise) = |
| (PMF.uniformOfFintype (I ↪ J)).bind |
| (fun e => (noise e).map (fun z => (e,z))) := by |
| unfold auxiliaryLaw |
| rw [← PMF.bind_bind, uniform_bind_completions] |
| |
| theorem original_probability_eq_auxiliary_average (K : Finset I) |
| (noise : (I ↪ J) → PMF Z) (E : Set ((I ↪ J) × Z)) : |
| (((PMF.uniformOfFintype (I ↪ J)).bind |
| (fun e => (noise e).map (fun z => (e,z)))).toOuterMeasure E).toReal = |
| (∑ e : I ↪ J, ((auxiliaryLaw K e noise).toOuterMeasure E).toReal) / |
| Fintype.card (I ↪ J) := by |
| rw [← uniform_bind_auxiliaryLaw K noise, PMF.toOuterMeasure_bind_apply, |
| tsum_fintype, ENNReal.toReal_sum] |
| · simp_rw [ENNReal.toReal_mul, PMF.uniformOfFintype_apply, |
| ENNReal.toReal_inv, ENNReal.toReal_natCast] |
| rw [← Finset.mul_sum] |
| ring |
| · intro e _ |
| apply ENNReal.mul_ne_top ((PMF.uniformOfFintype (I ↪ J)).apply_ne_top e) |
| rw [PMF.toOuterMeasure_apply] |
| exact PMF.tsum_coe_indicator_ne_top _ _ |
| |
| theorem original_probability_le_of_auxiliary (K : Finset I) |
| (noise : (I ↪ J) → PMF Z) (E : Set ((I ↪ J) × Z)) (B : ℝ) |
| (h : ∀ e, ((auxiliaryLaw K e noise).toOuterMeasure E).toReal ≤ B) : |
| (((PMF.uniformOfFintype (I ↪ J)).bind |
| (fun e => (noise e).map (fun z => (e,z)))).toOuterMeasure E).toReal ≤ B := by |
| rw [original_probability_eq_auxiliary_average K noise E] |
| have hc : (0 : ℝ) < Fintype.card (I ↪ J) := by exact_mod_cast Fintype.card_pos |
| apply (div_le_iff₀ hc).mpr |
| calc |
| _ ≤ ∑ _e : I ↪ J, B := Finset.sum_le_sum (fun e _ => h e) |
| _ = B * Fintype.card (I ↪ J) := by simp [mul_comm] |
| |
| end |
| end Balanced.InjectionExposure |
| |
| end OAI |
| |
