| import Mathlib.Probability.Kernel.Composition.MeasureCompProd |
| import Mathlib.MeasureTheory.Measure.WithDensity |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
|
|
| namespace Invariant233.FiniteKernelMixture |
|
|
| variable {I X Y : Type*} [Fintype I] [MeasurableSpace X] [MeasurableSpace Y] |
|
|
| noncomputable def weightedKernel (κ : I → Kernel X Y) (w : I → X → ℝ≥0∞) |
| (hw : ∀ i, Measurable (w i)) : Kernel X Y where |
| toFun x := ∑ i, w i x • κ i x |
| measurable' := by |
| apply Measure.measurable_of_measurable_coe |
| intro s hs |
| simp only [Measure.finsetSum_apply, Measure.smul_apply, smul_eq_mul] |
| exact Finset.measurable_sum _ (fun i _ => (hw i).mul (Kernel.measurable_coe (κ i) hs)) |
|
|
| noncomputable def normalizedKernel (κ : I → Kernel X Y) (w : I → X → ℝ≥0∞) |
| (hw : ∀ i, Measurable (w i)) : Kernel X Y := |
| weightedKernel κ (fun i x => w i x / ∑ j, w j x) |
| (fun i => (hw i).div (Finset.measurable_sum _ (fun j _ => hw j))) |
|
|
| noncomputable def diagonalReplacementKernel (q : ℝ≥0∞) (p : I → ℝ≥0∞) |
| (u : I → X → ℝ≥0∞) (hu : ∀ i, Measurable (u i)) |
| (ν : I → Measure X) : Kernel X X := |
| normalizedKernel |
| (fun o : Option I => match o with |
| | none => Kernel.id |
| | some i => Kernel.const X (ν i)) |
| (fun o x => match o with |
| | none => q |
| | some i => p i * u i x) |
| (fun o => by cases o with |
| | none => exact measurable_const |
| | some i => exact measurable_const.mul (hu i)) |
|
|
| end Invariant233.FiniteKernelMixture |
|
|
| end OAI |
|
|
