openAI-math-dataset / lean /OAI /Probability /PosteriorReplicas /Information /RandomProjectionInformation.lean
| import OAI.Probability.PosteriorReplicas.Information.ParametricKernel | |
| namespace OAI | |
| open MeasureTheory ProbabilityTheory InformationTheory Set Filter | |
| open scoped ENNReal NNReal | |
| noncomputable section | |
| namespace PosteriorReplicas | |
| section | |
| variable {T Y Z : Type*} [MeasurableSpace T] [MeasurableSpace Y] [MeasurableSpace Z] | |
| [StandardBorelSpace Y] [StandardBorelSpace Z] [Nonempty Y] [Nonempty Z] | |
| {n : ℕ} | |
| omit [StandardBorelSpace Y] [Nonempty Y] [Nonempty Z] in | |
| lemma measurable_projected_totalCorrelation (ν : Measure (Fin n → Y)) [IsProbabilityMeasure ν] | |
| {g : T × Y → Z} (hg : Measurable g) : | |
| Measurable (fun b => totalCorrelation (ν.map (fun u i => g (b,u i)))) := by | |
| let h : T × (Fin n → Y) → (Fin n → Z) := fun z i => g (z.1,z.2 i) | |
| have hh : Measurable h := by fun_prop | |
| let κ := parametricMapKernel ν h | |
| have : IsMarkovKernel κ := parametricMapKernel_markov ν h hh | |
| have hc := measurable_kernel_klDiv κ (piKernel (coordKernel κ)) | |
| simp only [piKernel_apply, coordKernel_apply] at hc | |
| change Measurable (fun b => totalCorrelation (κ b)) at hc | |
| simpa only [κ, parametricMapKernel_apply ν hh, h] using hc | |
| end | |
| section | |
| variable {T W Y Z : Type*} [MeasurableSpace T] [MeasurableSpace W] | |
| [MeasurableSpace Y] [MeasurableSpace Z] | |
| [StandardBorelSpace W] [Nonempty W] [Fintype W] [MeasurableSingletonClass W] | |
| [StandardBorelSpace Y] [Nonempty Y] [StandardBorelSpace Z] [Nonempty Z] | |
| {n : ℕ} (β : Measure T) [IsProbabilityMeasure β] | |
| (μ : Measure W) [IsProbabilityMeasure μ] | |
| (κ : Kernel W (Fin n → Y)) [IsMarkovKernel κ] | |
| {g : T × Y → Z} (hg : Measurable g) | |
| include hg | |
| lemma random_replica_information_le_entropy : | |
| (∑ i, cmi (randomSideLaw β g (coordinateJoint μ κ i))) + | |
| (∫⁻ b, totalCorrelation ((μ ⊗ₘ κ).snd.map (fun u i => g (b,u i))) ∂β) ≤ | |
| totalCorrelation (μ ⊗ₘ κ).snd + ENNReal.ofReal (finiteEntropy μ) := by | |
| have hi (i : Fin n) := measurable_cmi_attachSide hg (coordinateJoint μ κ i) | |
| have ht := measurable_projected_totalCorrelation (μ ⊗ₘ κ).snd hg | |
| have h := lintegral_mono (μ := β) (fun b => | |
| replica_information_le_entropy μ κ (g := fun y => g (b,y)) (by fun_prop)) | |
| rw [lintegral_add_aux (Finset.measurable_sum _ (fun i _ => hi i)) ht, | |
| lintegral_finsetSum _ (fun i _ => hi i)] at h | |
| simp only [lintegral_const, measure_univ, mul_one] at h | |
| simpa only [cmi_randomSideLaw β hg] using h | |
| lemma random_replica_information_identical (q : Measure (Y × W)) [IsProbabilityMeasure q] | |
| (hq : ∀ i, coordinateJoint μ κ i = q) : | |
| (n : ℝ≥0∞) * cmi (randomSideLaw β g q) + | |
| (∫⁻ b, totalCorrelation ((μ ⊗ₘ κ).snd.map (fun u i => g (b,u i))) ∂β) ≤ | |
| totalCorrelation (μ ⊗ₘ κ).snd + ENNReal.ofReal (finiteEntropy μ) := by | |
| have h := random_replica_information_le_entropy β μ κ hg | |
| have hs : (∑ i, cmi (randomSideLaw β g (coordinateJoint μ κ i))) = | |
| (n : ℝ≥0∞) * cmi (randomSideLaw β g q) := by | |
| calc | |
| _ = ∑ _i : Fin n, cmi (randomSideLaw β g q) := by | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| cases hq i | |
| rfl | |
| _ = _ := by simp | |
| rw [hs] at h | |
| exact h | |
| end | |
| lemma conditionalInformation_eq_cmi {d ℓ N : ℕ} [NeZero N] | |
| (q : Measure (Data d ℓ × (Vec d × Fin N))) [IsFiniteMeasure q] : | |
| conditionalInformation q = cmi q := rfl | |
| lemma sideLaw_eq_randomSideLaw {d ℓ N : ℕ} (q : Measure (Vec d × Fin N)) : | |
| sideLaw ℓ q = randomSideLaw (gaussianRows d ℓ) (fun z => labels z.1 z.2) q := rfl | |
| lemma messageEntropy_eq_finiteEntropy {d N : ℕ} (q : Measure (Vec d × Fin N)) : | |
| messageEntropy q = finiteEntropy q.snd := rfl | |
| end PosteriorReplicas | |
| end | |
| end OAI | |