LeanMachineLearning exposition

Bandits.ArrayModel.measurable_algFunction๐Ÿ”—

This page has the declaration's own card below, then its dependency graph, then a card for each dependency (type dependencies first, then the rest of the transitive closure). For a theorem, the graph and the dependency cards only follow its statement's dependencies (its proof is replaced by sorry, so what it proves doesn't depend on how); for everything else, both the type and the body/value are followed, since their content is part of what later declarations build on.

Minimal Lean file

measurable_algFunction๐Ÿ”—

LemmaBandits.ArrayModel.measurable_algFunction
Details

No docstring.

๐Ÿ”—theorem
Bandits.ArrayModel.measurable_algFunction.{u_1, u_2} {๐“ : Type u_1}
  {R : Type u_2} {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R}
  [Nonempty ๐“] [StandardBorelSpace ๐“] (alg : Learning.Algorithm ๐“ R)
  (n : โ„•) : Measurable (Function.uncurry (algFunction alg n))
Bandits.ArrayModel.measurable_algFunction.{u_1,
    u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“]
  (alg : Learning.Algorithm ๐“ R) (n : โ„•) :
  Measurable
    (Function.uncurry (algFunction alg n))

Code

lemma measurable_algFunction (alg : Algorithm ๐“ R) (n : โ„•) :
    Measurable (Function.uncurry (algFunction alg n))
Type uses (2)
Body uses (1)
Used by (4)

Actions: Source ยท Open Issue

Proof
(Kernel.exists_measurable_map_eq_unitInterval (alg.policy n)).choose_spec.1

Dependency graph

Type dependencies (2)

Algorithm๐Ÿ”—

StructureLearning.Algorithm
Details

A stochastic, sequential algorithm.

๐Ÿ”—structure
Learning.Algorithm.{u_4, u_5} (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] : Type (max u_4 u_5)
Learning.Algorithm.{u_4, u_5}
  (๐“ : Type u_4) (๐“จ : Type u_5)
  [MeasurableSpace ๐“]
  [MeasurableSpace ๐“จ] : Type (max u_4 u_5)

Code

structure Algorithm (๐“ ๐“จ : Type*) [MeasurableSpace ๐“] [MeasurableSpace ๐“จ] where
  /-- Policy or sampling rule: distribution of the next action. -/
  policy : (n : โ„•) โ†’ Kernel (Iic n โ†’ ๐“ ร— ๐“จ) ๐“
  /-- The policy is a Markov kernel. -/
  [h_policy : โˆ€ n, IsMarkovKernel (policy n)]
  /-- Distribution of the first action. -/
  p0 : Measure ๐“
  /-- The first action distribution is a probability measure. -/
  [hp0 : IsProbabilityMeasure p0]
Used by (216)

Actions: Source ยท Open Issue

algFunction๐Ÿ”—

DefinitionBandits.ArrayModel.algFunction
Details

The next action is the image of the history and a uniform random variable by this function.

๐Ÿ”—def
Bandits.ArrayModel.algFunction.{u_1, u_2} {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“} {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“] (alg : Learning.Algorithm ๐“ R) (n : โ„•) :
  (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) โ†’ โ†‘unitInterval โ†’ ๐“
Bandits.ArrayModel.algFunction.{u_1, u_2}
  {๐“ : Type u_1} {R : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {mR : MeasurableSpace R} [Nonempty ๐“]
  [StandardBorelSpace ๐“]
  (alg : Learning.Algorithm ๐“ R) (n : โ„•) :
  (โ†ฅ(Finset.Iic n) โ†’ ๐“ ร— R) โ†’
    โ†‘unitInterval โ†’ ๐“

Code

noncomputable
def algFunction (alg : Algorithm ๐“ R) (n : โ„•) :
    (Iic n โ†’ ๐“ ร— R) โ†’ I โ†’ ๐“ :=
  (Kernel.exists_measurable_map_eq_unitInterval (alg.policy n)).choose
Type uses (1)
Body uses (1)
Used by (17)

Actions: Source ยท Open Issue

All dependencies, transitively (1)

instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy๐Ÿ”—

InstanceLearning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy
Details

No docstring.

๐Ÿ”—theorem
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} (alg : Algorithm ๐“ ๐“จ) (n : โ„•) :
  ProbabilityTheory.IsMarkovKernel (Algorithm.policy alg n)
Learning.instIsMarkovKernelForallSubtypeNatMemFinsetIicProdPolicy.{u_1,
    u_2}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ}
  (alg : Algorithm ๐“ ๐“จ) (n : โ„•) :
  ProbabilityTheory.IsMarkovKernel
    (Algorithm.policy alg n)

Code

instance (alg : Algorithm ๐“ ๐“จ) (n : โ„•) : IsMarkovKernel (alg.policy n)
Type uses (1)
Used by (14)

Actions: Source ยท Open Issue

Proof
alg.h_policy n