File size: 3,721 Bytes
a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | 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
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