LeanMachineLearning exposition

Learning.Algorithm.prodLeft_p0๐Ÿ”—

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

prodLeft_p0๐Ÿ”—

LemmaLearning.Algorithm.prodLeft_p0
Details

No docstring.

๐Ÿ”—theorem
Learning.Algorithm.prodLeft_p0.{u_1, u_2, u_4} {๐“ : Type u_1}
  {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (๐“ง : Type u_4) [MeasurableSpace ๐“ง] (alg : Algorithm ๐“ ๐“จ) :
  p0 (prodLeft ๐“ง alg) = p0 alg
Learning.Algorithm.prodLeft_p0.{u_1, u_2,
    u_4}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} (๐“ง : Type u_4)
  [MeasurableSpace ๐“ง]
  (alg : Algorithm ๐“ ๐“จ) :
  p0 (prodLeft ๐“ง alg) = p0 alg

Code

theorem prodLeft_p0 : โˆ€ {๐“ : Type u_1} {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ} (๐“ง : Type u_4)
  [inst : MeasurableSpace ๐“ง] (alg : Learning.Algorithm ๐“ ๐“จ), (Learning.Algorithm.prodLeft ๐“ง alg).p0 = alg.p0
Type uses (2)

Actions: Source ยท Open Issue

Proof
@[simps]

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

prodLeft๐Ÿ”—

DefinitionLearning.Algorithm.prodLeft
Details

An algorithm with observations in ๐“ง ร— ๐“จ obtained from an algorithm with observations in ๐“จ by ignoring the ๐“ง component of each observation.

๐Ÿ”—def
Learning.Algorithm.prodLeft.{u_1, u_2, u_4} {๐“ : Type u_1}
  {๐“จ : Type u_2} {m๐“ : MeasurableSpace ๐“} {m๐“จ : MeasurableSpace ๐“จ}
  (๐“ง : Type u_4) [MeasurableSpace ๐“ง] (alg : Algorithm ๐“ ๐“จ) :
  Algorithm ๐“ (๐“ง ร— ๐“จ)
Learning.Algorithm.prodLeft.{u_1, u_2,
    u_4}
  {๐“ : Type u_1} {๐“จ : Type u_2}
  {m๐“ : MeasurableSpace ๐“}
  {m๐“จ : MeasurableSpace ๐“จ} (๐“ง : Type u_4)
  [MeasurableSpace ๐“ง]
  (alg : Algorithm ๐“ ๐“จ) :
  Algorithm ๐“ (๐“ง ร— ๐“จ)

Code

def Algorithm.prodLeft (๐“ง : Type*) [MeasurableSpace ๐“ง] (alg : Algorithm ๐“ ๐“จ) :
    Algorithm ๐“ (๐“ง ร— ๐“จ) where
  policy n := (alg.policy n).comap (fun h i โ†ฆ ((h i).1, (h i).2.2)) (by fun_prop)
  p0 := alg.p0
Type uses (1)
Body uses (2)
Used by (6)

Actions: Source ยท Open Issue

All dependencies, transitively (2)

instIsProbabilityMeasureP0๐Ÿ”—

InstanceLearning.instIsProbabilityMeasureP0
Details

No docstring.

๐Ÿ”—theorem
Learning.instIsProbabilityMeasureP0.{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) :
  MeasureTheory.IsProbabilityMeasure (Algorithm.p0 alg)
Learning.instIsProbabilityMeasureP0.{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) :
  MeasureTheory.IsProbabilityMeasure
    (Algorithm.p0 alg)

Code

instance (alg : Algorithm ๐“ ๐“จ) : IsProbabilityMeasure alg.p0
Type uses (1)
Used by (13)

Actions: Source ยท Open Issue

Proof
alg.hp0

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