Error map eq hypothesis Error
Cslib.MachineLearning.PACLearning.error_map_eq_hypothesisError
Plain-language statement
Under a realizable distribution P.map (x ↦ (x, c(x))), the general 0-1 error coincides with the binary hypothesisError P h c, where h and c are viewed as subsets of α via the characteristic function decide (· ∈ ·).
Exact Lean statement
theorem error_map_eq_hypothesisError (P : Measure α) (h c : Set α)
(hh : MeasurableSet h) (hc : MeasurableSet c) :
error (P.map (fun x => (x, decide (x ∈ c)))) (fun x => decide (x ∈ h)) =
hypothesisError P h cFormal artifact
Lean source
theorem error_map_eq_hypothesisError (P : Measure α) (h c : Set α) (hh : MeasurableSet h) (hc : MeasurableSet c) : error (P.map (fun x => (x, decide (x ∈ c)))) (fun x => decide (x ∈ h)) = hypothesisError P h c := by simp only [error, hypothesisError] have hf : Measurable (fun x => (x, decide (x ∈ c))) := Measurable.prodMk measurable_id (measurable_to_bool (by convert hc using 1; ext x; simp [decide_eq_true_eq])) rw [Measure.map_apply_of_aemeasurable hf.aemeasurable] · congr 1; ext x by_cases hx : x ∈ h <;> simp_all [symmDiff_def] · convert (hh.prod (measurableSet_singleton false)).union (hh.compl.prod (measurableSet_singleton true)) using 1 ext ⟨x, b⟩; cases b <;> simp- Project
- Lean Computer Science Library
- License
- Apache-2.0
- Commit
- f36649cff2c9
- Source
- Cslib/MachineLearning/PACLearning/Defs.lean:509-522
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