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Bandits.ArrayModel.instIsProbabilityMeasureProbSpaceArrayMeasureOfIsMarkovKernel🔗

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instIsProbabilityMeasureProbSpaceArrayMeasureOfIsMarkovKernel🔗

InstanceBandits.ArrayModel.instIsProbabilityMeasureProbSpaceArrayMeasureOfIsMarkovKernel
Details

No docstring.

🔗theorem
Bandits.ArrayModel.instIsProbabilityMeasureProbSpaceArrayMeasureOfIsMarkovKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν] :
  MeasureTheory.IsProbabilityMeasure (arrayMeasure ν)
Bandits.ArrayModel.instIsProbabilityMeasureProbSpaceArrayMeasureOfIsMarkovKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν] :
  MeasureTheory.IsProbabilityMeasure
    (arrayMeasure ν)

Code

instance (ν : Kernel 𝓐 R) [IsMarkovKernel ν] : IsProbabilityMeasure (arrayMeasure ν)
Type uses (3)
Body uses (2)
Used by (10)

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Proof
Measure.prod.instIsProbabilityMeasure _ _

Dependency graph

Type dependencies (3)

probSpace🔗

DefinitionBandits.ArrayModel.probSpace
Details

Probability space for the array model of stochastic bandits.

🔗def
Bandits.ArrayModel.probSpace.{u_1, u_2} (𝓐 : Type u_1) (R : Type u_2) :
  Type (max u_1 u_2)
Bandits.ArrayModel.probSpace.{u_1, u_2}
  (𝓐 : Type u_1) (R : Type u_2) :
  Type (max u_1 u_2)

Code

def probSpace : Type _ := (ℕ → I) × (ℕ → 𝓐 → R)
Used by (64)

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instMeasurableSpaceProbSpace🔗

InstanceBandits.ArrayModel.instMeasurableSpaceProbSpace
Details

No docstring.

🔗def
Bandits.ArrayModel.instMeasurableSpaceProbSpace.{u_3, u_4}
  {𝓐 : Type u_3} {R : Type u_4} [MeasurableSpace R] :
  MeasurableSpace (probSpace 𝓐 R)
Bandits.ArrayModel.instMeasurableSpaceProbSpace.{u_3,
    u_4}
  {𝓐 : Type u_3} {R : Type u_4}
  [MeasurableSpace R] :
  MeasurableSpace (probSpace 𝓐 R)

Code

instance {𝓐 R : Type*} [MeasurableSpace R] : MeasurableSpace (probSpace 𝓐 R)
Type uses (1)
Used by (41)

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Proof
inferInstanceAs (MeasurableSpace ((ℕ → I) × (ℕ → 𝓐 → R)))

arrayMeasure🔗

DefinitionBandits.ArrayModel.arrayMeasure
Details

Probability measure for the array model of stochastic bandits.

🔗def
Bandits.ArrayModel.arrayMeasure.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (probSpace 𝓐 R)
Bandits.ArrayModel.arrayMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (probSpace 𝓐 R)

Code

noncomputable
def arrayMeasure (ν : Kernel 𝓐 R) : Measure (probSpace 𝓐 R) :=
  (Measure.infinitePi fun _ ↦ volume).prod (streamMeasure ν)
Type uses (2)
Body uses (1)
Used by (29)

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All dependencies, transitively (2)

streamMeasure🔗

DefinitionBandits.streamMeasure
Details

Measure of an infinite stream of rewards from each action.

🔗def
Bandits.streamMeasure.{u_1, u_2} {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐} {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) : MeasureTheory.Measure (ℕ → 𝓐 → R)
Bandits.streamMeasure.{u_1, u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R) :
  MeasureTheory.Measure (ℕ → 𝓐 → R)

Code

noncomputable
def streamMeasure (ν : Kernel 𝓐 R) : Measure (ℕ → 𝓐 → R) :=
  Measure.infinitePi fun _ ↦ Measure.infinitePi ν
Used by (56)

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instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel🔗

InstanceBandits.instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel
Details

No docstring.

🔗theorem
Bandits.instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2} {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R} (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν] :
  MeasureTheory.IsProbabilityMeasure (streamMeasure ν)
Bandits.instIsProbabilityMeasureForallNatForallStreamMeasureOfIsMarkovKernel.{u_1,
    u_2}
  {𝓐 : Type u_1} {R : Type u_2}
  {m𝓐 : MeasurableSpace 𝓐}
  {mR : MeasurableSpace R}
  (ν : ProbabilityTheory.Kernel 𝓐 R)
  [ProbabilityTheory.IsMarkovKernel ν] :
  MeasureTheory.IsProbabilityMeasure
    (streamMeasure ν)

Code

instance (ν : Kernel 𝓐 R) [IsMarkovKernel ν] : IsProbabilityMeasure (streamMeasure ν)
Type uses (1)
Used by (22)

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Proof
by
  unfold streamMeasure
  infer_instance