Learning.Algorithm.p0
From the authors
Distribution of the first action given the first observation: the policy at time 0 applied to
the empty history.
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๐ : Type u_1m๐ : MeasurableSpace ๐A measurable space is a space equipped with a ฯ-algebra. -
๐ : Type u_2m๐ : MeasurableSpace ๐ -
๐จ : Type u_3m๐จ : MeasurableSpace ๐จ
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alg : Algorithm ๐ ๐ ๐จA stochastic, sequential algorithm.
ProbabilityTheory.Kernel ๐ ๐ฮฑ to another measurable space ฮฒ is a measurable function ฮบ : ฮฑ โ Measure ฮฒ.(alg.policy 0).sectR defaultMeasurableSpace : Type u_6 โ Type u_6A measurable space is a space equipped with a ฯ-algebra.
Learning.Algorithm : (๐ : Type u_5) โ
(๐ : Type u_6) โ
(๐จ : Type u_7) โ [MeasurableSpace ๐] โ [MeasurableSpace ๐] โ [MeasurableSpace ๐จ] โ Type (max (max u_5 u_6) u_7)A stochastic, sequential algorithm. At each round, it sees an observation in `๐`, then takes an action in `๐`, and finally receives feedback in `๐จ`. The action is a random function of the past rounds and the current observation.Go to its page
ProbabilityTheory.Kernel : (ฮฑ : Type u_1) โ (ฮฒ : Type u_2) โ [MeasurableSpace ฮฑ] โ [MeasurableSpace ฮฒ] โ Type (max u_1 u_2)A kernel from a measurable space `ฮฑ` to another measurable space `ฮฒ` is a measurable function `ฮบ : ฮฑ โ Measure ฮฒ`. The measurable space structure on `MeasureTheory.Measure ฮฒ` is given by `MeasureTheory.Measure.instMeasurableSpace`. A map `ฮบ : ฮฑ โ MeasureTheory.Measure ฮฒ` is measurable iff `โ s : Set ฮฒ, MeasurableSet s โ Measurable (fun a โฆ ฮบ a s)`.
ProbabilityTheory.Kernel.sectR : {ฮฑ : Type u_1} โ
{ฮฒ : Type u_2} โ
{mฮฑ : MeasurableSpace ฮฑ} โ
{mฮฒ : MeasurableSpace ฮฒ} โ
{ฮณ : Type u_4} โ
{mฮณ : MeasurableSpace ฮณ} โ ProbabilityTheory.Kernel (ฮฑ ร ฮฒ) ฮณ โ ฮฑ โ ProbabilityTheory.Kernel ฮฒ ฮณDefine a `Kernel ฮฒ ฮณ` from a `Kernel (ฮฑ ร ฮฒ) ฮณ` by taking the comap of `fun b โฆ (a, b)` for a given `a : ฮฑ`.
Inhabited.default : {ฮฑ : Sort u} โ [self : Inhabited ฮฑ] โ ฮฑ`default` is a function that produces a "default" element of any `Inhabited` type. This element does not have any particular specified properties, but it is often an all-zeroes value.
Code
noncomputable def Algorithm.p0 (alg : Algorithm ๐ ๐ ๐จ) : Kernel ๐ ๐ := (alg.policy 0).sectR default deriving IsMarkovKernel
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 ยท open it in the Lean web editor.
Dependency graph
Audit surface: 3 project declarations, 14 external constants
โ 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.