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Bandits.gap_le_of_mem_Icc🔗

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Minimal Lean file

gap_le_of_mem_Icc🔗

LemmaBandits.gap_le_of_mem_Icc
Details

No docstring.

🔗theorem
Bandits.gap_le_of_mem_Icc.{u_1} {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ} {a : 𝓐} [Nonempty 𝓐] {l u : ℝ}
  (h : ∀ (a : 𝓐), ∫ (x : ℝ), id x ∂ν a ∈ Set.Icc l u) : gap ν a ≤ u - l
Bandits.gap_le_of_mem_Icc.{u_1}
  {𝓐 : Type u_1} {m𝓐 : MeasurableSpace 𝓐}
  {ν : ProbabilityTheory.Kernel 𝓐 ℝ}
  {a : 𝓐} [Nonempty 𝓐] {l u : ℝ}
  (h :
    ∀ (a : 𝓐),
      ∫ (x : ℝ), id x ∂ν a ∈
        Set.Icc l u) :
  gap ν a ≤ u - l

Code

lemma gap_le_of_mem_Icc [Nonempty 𝓐] {l u : ℝ} (h : ∀ a, (ν a)[id] ∈ Set.Icc l u) :
    gap ν a ≤ u - l
Type uses (1)
Used by (1)

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Proof
by
  grind [gap, ciSup_le (fun i ↦ (h i).2)]

Dependency graph

Type dependencies (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)

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