Variance Addition Under Gaussian Smoothing

Proved

FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.smoothedLaw_variance_readable

by Lazyfill · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

functional-inequalitieskls-conjecturekls-modularkls-readablelog-concave

This is the variance-addition identity for Gaussian smoothing.

Let $E$ be a complete finite-dimensional real inner-product space with Borel measurable structure. Let $\mu$ be a probability measure on $E$ with finite second moment, let $s>0$ , and let $v\in E$ . For smoothedLaw μ s, obtained by convolving $\mu$ with gaussianLaw s,

\operatorname{Var}_{\mu*\gamma_s}\langle v,\cdot\rangle =\operatorname{Var}_{\mu}\langle v,\cdot\rangle +\frac{\|v\|^2}{2s}.

The result records the exact contribution of the independent Gaussian noise. The finite-second-moment assumption is expressed as MemLp id 2 μ.

Original Lean declaration header

Within namespace FunctionalInequalities.KLS.LogConcave.GaussianSmoothing. Shown in its original module context. Proof omitted.

theorem smoothedLaw_variance (μ : Measure E) [IsProbabilityMeasure μ]
    (hμ2 : MemLp id 2 μ) {s : ℝ} (hs : 0 < s) (v : E) :
    variance (fun y : E => ⟪v, y⟫) (smoothedLaw μ s) =
      variance (fun y : E => ⟪v, y⟫) μ + ‖v‖ ^ 2 / (2 * s)

Module navigation

Module: KLS/LogConcave/Density/GaussianSmoothing/Moments.

Definition modules: KLS/LogConcave/Density/GaussianSmoothing/Basic.

Supporting theorems: gaussianLaw_exp_norm_integrable, gaussianLaw_variance.

Previous: gaussianLaw_variance · Next: smoothedLaw_covarianceBilin.

Source excerpt, lines 256–259 · Frozen source 408915f.

Preamble

import Definitions.Def_KLSModular_FunctionalInequalities_KLS_LogConcave_Density_GaussianSmoothing_Moments

open MeasureTheory ProbabilityTheory Filter Topology
open scoped RealInnerProductSpace ContDiff
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [CompleteSpace E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

Formal statement


theorem FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.smoothedLaw_variance_readable (μ : Measure E) [IsProbabilityMeasure μ]
    (hμ2 : MemLp id 2 μ) {s : ℝ} (hs : 0 < s) (v : E) :
    variance (fun y : E => ⟪v, y⟫) (smoothedLaw μ s) =
      variance (fun y : E => ⟪v, y⟫) μ + ‖v‖ ^ 2 / (2 * s) := by sorry

Source

Lean source: https://github.com/human-ai-mathematics/kls-lean/blob/408915f/formalization/FunctionalInequalities/KLS/LogConcave/Density/GaussianSmoothing/Moments.lean#L255-L271 (repository human-ai-mathematics/kls-lean, commit 408915f, module `FunctionalInequalities.KLS.LogConcave.Density.GaussianSmoothing.Moments`, declaration `FunctionalInequalities.KLS.LogConcave.GaussianSmoothing.smoothedLaw_variance`; the repository is private at upload time).

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