`Learning.IsAlgEnvSeq.klDiv_map_history_compProd`🔗

Lemma

From the authors

Chain rule for histories of a single algorithm versus two stationary environments.

Types

𝓐 : Type u_2m𝓐 : MeasurableSpace 𝓐A measurable space is a space equipped with a σ-algebra.
𝓨 : Type u_3m𝓨 : MeasurableSpace 𝓨
Ω : Type u_4mΩ : MeasurableSpace Ω
Ω' : Type u_5mΩ' : MeasurableSpace Ω'

Given

P : MeasureTheory.Measure ΩA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure.MeasureTheory.IsProbabilityMeasure PA measure μ is called a probability measure if μ univ = 1.
P' : MeasureTheory.Measure Ω'MeasureTheory.IsProbabilityMeasure P'
A : ℕ → Ω → 𝓐
Y : ℕ → Ω → 𝓨
A' : ℕ → Ω' → 𝓐
Y' : ℕ → Ω' → 𝓨
O : ℕ → Ω → Unit
O' : ℕ → Ω' → Unit
alg : Algorithm Unit 𝓐 𝓨A stochastic, sequential algorithm.
κ : ProbabilityTheory.Kernel 𝓐 𝓨A kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β.ProbabilityTheory.IsMarkovKernel κA kernel is a Markov kernel if every measure in its image is a probability measure.
κ' : ProbabilityTheory.Kernel 𝓐 𝓨ProbabilityTheory.IsMarkovKernel κ'
M : ℕ

Assuming

h : IsAlgEnvSeq O A Y alg (stationaryEnv κ) PAn algorithm-environment sequence: a sequence of observations, actions and feedbacks generated by an algorithm interacting with an environment.
h' : IsAlgEnvSeq O' A' Y' alg (stationaryEnv κ') P'

Then

InformationTheory.klDiv (MeasureTheory.Measure.map (history O A Y M) P)
    (MeasureTheory.Measure.map (history O' A' Y' M) P') =
  ∑ t ∈ Finset.range M,
    InformationTheory.klDiv ((MeasureTheory.Measure.map (A t) P).compProd κ)
      ((MeasureTheory.Measure.map (A t) P).compProd κ')

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

MeasureTheory.IsProbabilityMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure `μ` is called a probability measure if `μ univ = 1`.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

A measure is defined to be an outer measure that is countably additive on
measurable sets, with the additional assumption that the outer measure is the canonical
extension of the restricted measure.

The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version
is written `μ.real s`.

Nat : Type

The natural numbers, starting at zero.

This type is special-cased by both the kernel and the compiler, and overridden with an efficient
implementation. Both use a fast arbitrary-precision arithmetic library (usually
[GMP](https://gmplib.org/)); at runtime, `Nat` values that are sufficiently small are unboxed.

Unit : Type

The canonical type with one element. This element is written `()`.

`Unit` has a number of uses:
* It can be used to model control flow that returns from a function call without providing other
  information.
* Monadic actions that return `Unit` have side effects without computing values.
* In polymorphic types, it can be used to indicate that no data is to be stored in a particular
  field.

Learning.Algorithm : (𝓞 : Type u_5) →
  (𝓐 : Type u_6) →
    (𝓨 : Type u_7) → [MeasurableSpace 𝓞] → [MeasurableSpace 𝓐] → [MeasurableSpace 𝓨] → Type (max (max u_5 u_6) u_7)

A stochastic, sequential algorithm.
At each round, it sees an observation in `𝓞`, then takes an action in `𝓐`, and finally receives
feedback in `𝓨`. The action is a random function of the past rounds and the current observation.

Go to its page

ProbabilityTheory.IsMarkovKernel : {α : Type u_1} →
  {β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → Prop

A kernel is a Markov kernel if every measure in its image is a probability measure.

ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)

A kernel from a measurable space `α` to another measurable space `β` is a measurable function
`κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by
`MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable
iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.

Learning.IsAlgEnvSeq : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} →
      {Ω : Type u_4} →
        {m𝓞 : MeasurableSpace 𝓞} →
          {m𝓐 : MeasurableSpace 𝓐} →
            {m𝓨 : MeasurableSpace 𝓨} →
              {mΩ : MeasurableSpace Ω} →
                (ℕ → Ω → 𝓞) →
                  (ℕ → Ω → 𝓐) →
                    (ℕ → Ω → 𝓨) →
                      Learning.Algorithm 𝓞 𝓐 𝓨 →…

An algorithm-environment sequence: a sequence of observations, actions and feedbacks generated
by an algorithm interacting with an environment.

Go to its page

Learning.stationaryEnv : {𝓐 : Type u_2} →
  {𝓨 : Type u_3} →
    {m𝓐 : MeasurableSpace 𝓐} →
      {m𝓨 : MeasurableSpace 𝓨} →
        (ν : ProbabilityTheory.Kernel 𝓐 𝓨) → [ProbabilityTheory.IsMarkovKernel ν] → Learning.Environment Unit 𝓐 𝓨

A stationary environment without observations, in which the distribution of the next feedback
depends only on the last action.

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Eq : {α : Sort u_1} → α → α → Prop

The equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`.

InformationTheory.klDiv : {α : Type u_2} → {mα : MeasurableSpace α} → MeasureTheory.Measure α → MeasureTheory.Measure α → ENNReal

Kullback-Leibler divergence between two measures.

MeasureTheory.Measure.map : {α : Type u_4} →
  {β : Type u_5} →
    [inst : MeasurableSpace α] →
      [inst_1 : MeasurableSpace β] → (α → β) → MeasureTheory.Measure α → MeasureTheory.Measure β

The pushforward of a measure. If `f` is not an almost everywhere measurable function,
we define it to be `0` if `μ = 0`, and to be an arbitrary Dirac mass otherwise. That way
we always have `map f 0 = 0`, and the push-forward of a probability measure is always a
probability measure.

Learning.history : {𝓞 : Type u_1} →
  {𝓐 : Type u_2} →
    {𝓨 : Type u_3} → {Ω : Type u_4} → (ℕ → Ω → 𝓞) → (ℕ → Ω → 𝓐) → (ℕ → Ω → 𝓨) → (n : ℕ) → Ω → Learning.Hist 𝓞 𝓐 𝓨 n

History of the algorithm-environment sequence before time `n`: the rounds at
times `0, ..., n - 1`.

Go to its page

Finset.range : ℕ → Finset ℕ

`range n` is the set of natural numbers less than `n`.

MeasureTheory.Measure.compProd : {α : Type u_1} →
  {β : Type u_2} →
    {mα : MeasurableSpace α} →
      {mβ : MeasurableSpace β} → MeasureTheory.Measure α → ProbabilityTheory.Kernel α β → MeasureTheory.Measure (α × β)

The composition-product of a measure and a kernel.

Code

lemma IsAlgEnvSeq.klDiv_map_history_compProd (h : IsAlgEnvSeq O A Y alg (stationaryEnv κ) P)
    (h' : IsAlgEnvSeq O' A' Y' alg (stationaryEnv κ') P') (M : ℕ) :
    klDiv (P.map (history O A Y M)) (P'.map (history O' A' Y' M)) =
      ∑ t ∈ range M, klDiv (P.map (A t) ⊗ₘ κ) (P.map (A t) ⊗ₘ κ')

Proof

by
  rw [h.klDiv_map_history_stepKernel h']
  refine sum_congr rfl fun t _ ↦ ?_
  have h_obs := (h.hasCondDistrib_obs t).map_eq
  rw [obs_stationaryEnv] at h_obs
  rw [stepKernel_stationaryEnv, stepKernel_stationaryEnv,
    klDiv_compProd_compProd_compProd_prodMkLeft_eq_klDiv_comp_compProd, ← h_obs,
    ← (h.hasCondDistrib_action t).hasLaw_comp.map_eq]

Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10), the 2th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 8 project declarations, 40 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.

Learning.IsAlgEnvSeq.klDiv_map_history_compProd🔗

`Learning.IsAlgEnvSeq.klDiv_map_history_compProd`🔗