AutoSamplingTheory.TechnicalLemmas.InformationTheory.Renyi

- The real-valued Renyi/Hellinger-style density integrand `p^a q^(1-a)`.

noncomputable def renyiIntegrand (a p q : ℝ) : ℝ :=
  p ^ a * q ^ (1 - a)

/-- The `ℝ≥0∞` version used for lintegral contracts. -/

AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:27published source at 7bcd37294df1

- The `ℝ≥0∞` version used for lintegral contracts.

noncomputable def renyiIntegrandENNReal (a : ℝ) (p q : α → ℝ) : α → ℝ≥0∞ :=
  fun x => ENNReal.ofReal (renyiIntegrand a (p x) (q x))

/-- Nonnegative input densities give a nonnegative Renyi integrand. -/

AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:31published source at 7bcd37294df1

- Nonnegative input densities give a nonnegative Renyi integrand.

theorem renyiIntegrand_nonneg {a p q : ℝ}
    (hp : 0 ≤ p) (hq : 0 ≤ q) :
    0 ≤ renyiIntegrand a p q := by
  exact mul_nonneg (Real.rpow_nonneg hp a) (Real.rpow_nonneg hq (1 - a))

/-- Positive input densities give a positive Renyi integrand. -/

AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:35published source at 7bcd37294df1

- Positive input densities give a positive Renyi integrand.

theorem renyiIntegrand_pos {a p q : ℝ}
    (hp : 0 < p) (hq : 0 < q) :
    0 < renyiIntegrand a p q := by
  exact mul_pos (Real.rpow_pos_of_pos hp a) (Real.rpow_pos_of_pos hq (1 - a))

variable [MeasurableSpace α]

/-- A measurable pair of real densities gives a measurable Renyi integrand for
orders `a ∈ [0,1]`. -/

AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:41published source at 7bcd37294df1Open detailed card

- A measurable pair of real densities gives a measurable Renyi integrand for orders `a ∈ [0,1]`.

theorem measurable_renyiIntegrand {a : ℝ} {p q : α → ℝ}
    (ha0 : 0 ≤ a) (ha1 : a ≤ 1)
    (hp : Measurable p) (hq : Measurable q) :
    Measurable fun x => renyiIntegrand a (p x) (q x) := by
  have hpPow : Measurable fun x => p x ^ a :=
    (Real.continuous_rpow_const ha0).measurable.comp hp
  have hqPow : Measurable fun x => q x ^ (1 - a) :=
    (Real.continuous_rpow_const (sub_nonneg.mpr ha1)).measurable.comp hq
  change Measurable ((fun x => p x ^ a) * fun x => q x ^ (1 - a))
  exact hpPow.mul hqPow

/-- Measurability of the `ℝ≥0∞` Renyi integrand used in lintegrals. -/

AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:50published source at 7bcd37294df1

- Measurability of the `ℝ≥0∞` Renyi integrand used in lintegrals.

theorem measurable_renyiIntegrandENNReal {a : ℝ} {p q : α → ℝ}
    (ha0 : 0 ≤ a) (ha1 : a ≤ 1)
    (hp : Measurable p) (hq : Measurable q) :
    Measurable (renyiIntegrandENNReal a p q) := by
  change Measurable (fun x => ENNReal.ofReal (renyiIntegrand a (p x) (q x)))
  exact (measurable_renyiIntegrand (α := α) ha0 ha1 hp hq).ennreal_ofReal

/-- A finite envelope gives a finite Renyi lintegral. -/

AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:62published source at 7bcd37294df1Open detailed card

- A finite envelope gives a finite Renyi lintegral.

theorem lintegral_renyiIntegrandENNReal_ne_top_of_ae_le
    (μ : MeasureTheory.Measure α) (a : ℝ) (p q : α → ℝ) (g : α → ℝ≥0∞)
    (hle : ∀ᵐ x ∂μ, renyiIntegrandENNReal a p q x ≤ g x)
    (hgfin : ∫⁻ x, g x ∂μ ≠ ∞) :
    ∫⁻ x, renyiIntegrandENNReal a p q x ∂μ ≠ ∞ :=
  ne_top_of_le_ne_top hgfin (lintegral_mono_ae hle)

/-- Pointwise derivative rule for the Renyi density integrand.  Positivity,
domination, and differentiating under the integral are deliberately outside
this leaf. -/

AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:70published source at 7bcd37294df1Open detailed card

- Pointwise derivative rule for the Renyi density integrand. Positivity, domination, and differentiating under the integral are deliberately outside this leaf.

theorem hasDerivAt_renyiIntegrand {a : ℝ} {p q : ℝ → ℝ} {t pdot qdot : ℝ}
    (hp : HasDerivAt p pdot t) (hq : HasDerivAt q qdot t)
    (hp0 : p t ≠ 0) (hq0 : q t ≠ 0) :
    HasDerivAt (fun s => renyiIntegrand a (p s) (q s))
      ((pdot * a * (p t) ^ (a - 1)) * (q t) ^ (1 - a) +
        (p t) ^ a * (qdot * (1 - a) * (q t) ^ (-a))) t := by
  have hpPow :
      HasDerivAt (fun s => p s ^ a) (pdot * a * (p t) ^ (a - 1)) t :=
    hp.rpow_const (p := a) (Or.inl hp0)
  have hqPow :
      HasDerivAt (fun s => q s ^ (1 - a))
        (qdot * (1 - a) * (q t) ^ ((1 - a) - 1)) t :=
    hq.rpow_const (p := 1 - a) (Or.inl hq0)
  change HasDerivAt ((fun s => p s ^ a) * fun s => q s ^ (1 - a))
    ((pdot * a * (p t) ^ (a - 1)) * (q t) ^ (1 - a) +
      (p t) ^ a * (qdot * (1 - a) * (q t) ^ (-a))) t
  have hexponent : (1 - a) - 1 = -a := by ring
  rw [hexponent] at hqPow
  exact hpPow.mul hqPow

end Renyi
end InformationTheory
end TechnicalLemmas
end AutoSamplingTheory

AutoSamplingTheory/TechnicalLemmas/InformationTheory/Renyi.lean:80published source at 7bcd37294df1Open detailed card