`Learning.IsBayesAlgEnvSeq.gap_le_of_mem_Icc`🔗

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Minimal Lean file

`gap_le_of_mem_Icc`🔗

LemmaLearning.IsBayesAlgEnvSeq.gap_le_of_mem_Icc

Details

No docstring.

theorem

Learning.IsBayesAlgEnvSeq.gap_le_of_mem_Icc.{u_1, u_2, u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2} {Ω : Type u_4} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [Nonempty 𝓐]
  {κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ} {E : Ω → 𝓔} {A : ℕ → Ω → 𝓐}
  {n : ℕ} {ω : Ω} {l u : ℝ}
  (h : ∀ (e : 𝓔) (a : 𝓐), ∫ (x : ℝ), id x ∂κ (e, a) ∈ Set.Icc l u) :
  gap κ E A n ω ≤ u - l
Learning.IsBayesAlgEnvSeq.gap_le_of_mem_Icc.{u_1,
    u_2, u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {Ω : Type u_4} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐] [Nonempty 𝓐]
  {κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ}
  {E : Ω → 𝓔} {A : ℕ → Ω → 𝓐} {n : ℕ}
  {ω : Ω} {l u : ℝ}
  (h :
    ∀ (e : 𝓔) (a : 𝓐),
      ∫ (x : ℝ), id x ∂κ (e, a) ∈
        Set.Icc l u) :
  gap κ E A n ω ≤ u - l

Code

lemma gap_le_of_mem_Icc [Nonempty 𝓐] {κ : Kernel (𝓔 × 𝓐) ℝ} {E : Ω → 𝓔} {A : ℕ → Ω → 𝓐} {n : ℕ}
    {ω : Ω} {l u : ℝ} (h : ∀ e a, (κ (e, a))[id] ∈ Set.Icc l u) : gap κ E A n ω ≤ u - l

Type uses (1)

gap

Body uses (1)

gap_le_of_mem_Icc

Used by (1)

integrable_gap

Actions: Source · Open Issue

Proof

Bandits.gap_le_of_mem_Icc (h (E ω))

Dependency graph

Type dependencies (1)

`gap`🔗

DefinitionLearning.IsBayesAlgEnvSeq.gap

Details

A random variable that gives the gap at time n.

def

Learning.IsBayesAlgEnvSeq.gap.{u_1, u_2, u_4} {𝓔 : Type u_1}
  {𝓐 : Type u_2} {Ω : Type u_4} [MeasurableSpace 𝓔] [MeasurableSpace 𝓐]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ) (E : Ω → 𝓔) (A : ℕ → Ω → 𝓐)
  (n : ℕ) (ω : Ω) : ℝ
Learning.IsBayesAlgEnvSeq.gap.{u_1, u_2,
    u_4}
  {𝓔 : Type u_1} {𝓐 : Type u_2}
  {Ω : Type u_4} [MeasurableSpace 𝓔]
  [MeasurableSpace 𝓐]
  (κ : ProbabilityTheory.Kernel (𝓔 × 𝓐) ℝ)
  (E : Ω → 𝓔) (A : ℕ → Ω → 𝓐) (n : ℕ)
  (ω : Ω) : ℝ

Code

noncomputable
def gap (κ : Kernel (𝓔 × 𝓐) ℝ) (E : Ω → 𝓔) (A : ℕ → Ω → 𝓐) (n : ℕ) (ω : Ω) : ℝ :=
  Bandits.gap (κ.sectR (E ω)) (A n ω)

Body uses (1)

gap

Used by (10)

Actions: Source · Open Issue

All dependencies, transitively (1)

`gap`🔗

DefinitionBandits.gap

Details

Gap of an action a: difference between the highest mean of the actions and the mean of a.

def

Bandits.gap.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ) (a : 𝓐) : ℝ
Bandits.gap.{u_1} {𝓐 : Type u_1}
  {m𝓐 : MeasurableSpace 𝓐}
  (ν : ProbabilityTheory.Kernel 𝓐 ℝ)
  (a : 𝓐) : ℝ

Code

noncomputable
def gap (ν : Kernel 𝓐 ℝ) (a : 𝓐) : ℝ := (⨆ i, (ν i)[id]) - (ν a)[id]

Used by (27)

Actions: Source · Open Issue

Learning.IsBayesAlgEnvSeq.gap_le_of_mem_Icc🔗

gap_le_of_mem_Icc🔗

gap🔗

gap🔗

`Learning.IsBayesAlgEnvSeq.gap_le_of_mem_Icc`🔗

`gap_le_of_mem_Icc`🔗

`gap`🔗

`gap`🔗