proofs
Provedcertified-residual-attentioncontrol-variatesimportance-samplingsecond-moment

Residual importance sampling: exact mean, exact variance identity, and residual second-moment bound

For all J∈Type⁡\htmlData{mx=1}{J \in \htmlData{mx=0}{\operatorname{Type}}} and π,r,v,g:J→R\htmlData{mx=3}{\pi, r, v, g : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}, if ∀j, 0≤π(j)\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}}, ∑jπ(j)=1\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{\pi}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1, ∀j, 0<r(j)\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}} and ∑jr(j)=1\htmlData{mx=3}{\sum_{j} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}} \mathrel{\htmlData{mx=4}{=}} 1 (given Fintype⁡ J\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}), then

let⁡(b)(:= Σ j, π j ∗ g j let o : R := Σ j, π j ∗ v j let E : J → R := fun j => b + (π j / r j) ∗ (v j − g j) (Σ j, r j ∗ E j = o) ∧ (Σ j, r j ∗ (E j − o)  ˆ2 =   (Σ j, π j  ˆ2 ∗ (v j − g j)  ˆ2 / r j) − (Σ j, π j ∗ (v j − g j))  ˆ2) ∧ (Σ j, r j ∗ (E j − o)  ˆ2 ≤ Σ j, π j  ˆ2 ∗ (v j − g j)  ˆ2 / r j) ∧ ((∀ j, v j = g j) → ∀ j, E j = o)))\htmlData{mx=4}{\htmlData{mx=3}{\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{let}}(\htmlData{mx=1}{b})}}(\mathtt{:=\ \Sigma \ j,\ {\pi}\ j\ *\ g\ j\ let\ o\ :\ \text{ℝ}\ :=\ \Sigma \ j,\ {\pi}\ j\ *\ v\ j\ let\ E\ :\ J\ \to \ \text{ℝ}\ :=\ fun\ j\ =>\ b\ +\ ({\pi}\ j\ /\ r\ j)\ *\ (v\ j\ -\ g\ j)\ (\Sigma \ j,\ r\ j\ *\ E\ j\ =\ o)\ \text{∧}\ (\Sigma \ j,\ r\ j\ *\ (E\ j\ -\ o)\ \^\ 2\ =\ \ \ (\Sigma \ j,\ {\pi}\ j\ \^\ 2\ *\ (v\ j\ -\ g\ j)\ \^\ 2\ /\ r\ j)\ -\ (\Sigma \ j,\ {\pi}\ j\ *\ (v\ j\ -\ g\ j))\ \^\ 2)\ \text{∧}\ (\Sigma \ j,\ r\ j\ *\ (E\ j\ -\ o)\ \^\ 2\ \leq \ \Sigma \ j,\ {\pi}\ j\ \^\ 2\ *\ (v\ j\ -\ g\ j)\ \^\ 2\ /\ r\ j)\ \text{∧}\ ((\forall \ j,\ v\ j\ =\ g\ j)\ \to \ \forall \ j,\ E\ j\ =\ o))})}
What the notation means · or point at any symbol
Type
Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
Fintype
Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application
Fintype applied to J. Lean writes f x; papers write f(x). (Fintype J)
⟹Implies
If J, then ℝ. (Between types, A → B is the type of functions from A to B.) (J → ℝ)
x : TVariables
π, r, v, g range over J → ℝ: the statement must hold for every choice. ((π r v g : J → ℝ))
∀For all
The statement after the comma holds for every j. (∀ j, 0 ≤ π j)
∑Summation
Add up π(j) for every j. (∑ j, π j)
=Equality
∑ j, π(j) and 1 are the same. (∑ j, π j = 1)
(x : T)As real numbers
let(b) read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((let b : ℝ)
let
let: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (let)
RReal numbers
The real numbers: every point on the number line. (ℝ)
≤At most
0 is less than or equal to π(j). (0 ≤ π j)
<Less than
0 is strictly smaller than r(j). (0 < r j)
πPi
The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)

With an exactly supplied control-variate mean b=Σπg and proposal r, the one-sample residual estimator b+(π/r)(v-g) is unbiased. Its exact centered second moment is Σπ²(v-g)²/r minus the squared residual mean, hence bounded by the uncentered residual second moment. If g=v pointwise, the estimator is identically exact.

How it was proved

Proved
5 October 2026
By
Ajax
Checked by
Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms
Classical.choice, Quot.sound, propext
Proof
61 lines of Lean
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