Theorems · Theorem · measure theory

MeasureTheory.llr_tilted_left

∀ {α : Type u_1} {mα : MeasurableSpace α} {μ ν : MeasureTheory.Measure α} {f : α → ℝ} [MeasureTheory.SigmaFinite μ]
  [MeasureTheory.SigmaFinite ν],
  μ.AbsolutelyContinuous ν →
    MeasureTheory.Integrable (fun x => Real.exp (f x)) μ →
      AEMeasurable f ν →
        MeasureTheory.llr (μ.tilted f) ν =ᵐ[μ] fun x =>
          f x - Real.log (∫ (z : α), Real.exp (f z) ∂μ) + MeasureTheory.llr μ ν x

Defined in

Mathlib.MeasureTheory.Measure.LogLikelihoodRatio

Cited by

2 results in Mathlib

Foundations

Depth 258 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

MeasureTheory.SigmaFiniteMeasureTheory.SigmaFinite