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TV-close laws have identical Huber mixtures at the contamination threshold

Statement

Assume the Axiom of Choice. Let $P_{0}, P_{1}$ be probability measures on one measurable space. If $0 \leq ϵ < 1$ and $∥ P_{0} - P_{1} ∥_{T V} \leq ϵ / (1 - ϵ)$ , there are probability measures $Q_{0}, Q_{1}$ for which

(1 - ϵ) P_{0} + ϵ Q_{0} = (1 - ϵ) P_{1} + ϵ Q_{1} .

At $ϵ = 1$ , the same equality is possible without a TV condition.

Facts & Assumptions

Given: The Axiom of Choice, probability measures $P_{0}, P_{1}$ , and the stated total-variation condition.

[A1]

The Axiom of Choice: the proof uses the choice-qualified Radon–Nikodym supplier to obtain nonnegative density representatives.

[F1]

Total variation distance for probability laws: $∥ P_{0} - P_{1} ∥_{T V} = \sup_{A} ∣ P_{0} (A) - P_{1} (A) ∣$ .

[F2]

Probability measures and probability spaces: $P_{0}$ and $P_{1}$ are nonnegative measures of total mass one.

[F3]

The Radon-Nikodym derivative as an almost-everywhere equivalence class: the Radon–Nikodym derivative is an almost-everywhere class of densities satisfying the setwise integral formula, under the hypotheses of its existence theorem.

[F4]

A sigma-finite positive measure has a nonnegative Radon–Nikodym density version: under AC, if $μ$ and $λ$ are sigma-finite positive measures on the same measurable space and $λ ≪ μ$ , then $d λ / d μ$ has a finite-valued nonnegative measurable representative.

[F5]

The Lebesgue integral is linear on $L^{1} (μ)$ and Monotonicity and nonnegative homogeneity of the nonnegative integral: integrable differences have linear integrals, and order bounds pass to their nonnegative integrals.

[F6]

Closure properties of measurable functions used by the integral: sums, differences, and positive and negative parts of measurable real functions are measurable.

[F7]

Extended-real-valued measurable functions: the level sets of a measurable real-valued function are measurable.

[F8]

The indefinite integral of a nonnegative measurable function is a measure: a nonnegative measurable density defines a measure by integration over measurable sets.

[F9]

A convex mixture of probability laws is a probability law: convex mixtures of probability measures are probability measures.

[F10]

Huber contamination model: the one-observation law is the mixture $(1 - ϵ) P_{θ} + ϵ Q$ .

Proof

1.1givenF1choose

If $ϵ = 1$ , set $Q_{0} = Q_{1} = P_{0}$ ; both displayed mixtures equal $P_{0}$ . If $ϵ < 1$ and $∥ P_{0} - P_{1} ∥_{T V} = 0$ , [F1] gives $P_{0} = P_{1}$ , so again set $Q_{0} = Q_{1} = P_{0}$ . It remains to consider $0 < ϵ < 1$ and $δ : = ∥ P_{0} - P_{1} ∥_{T V} > 0$ .

1.2A1F2F3F4choose

Put $ν = P_{0} + P_{1}$ . Countable additivity and nonnegativity of $P_{0}, P_{1}$ show that $ν$ is a positive measure, and $ν (X) = 2$ , so it is finite and sigma-finite. If $ν (A) = 0$ , then $P_{0} (A) + P_{1} (A) = 0$ with both terms nonnegative, so each is zero; hence $P_{i} ≪ ν$ . By [A1], [F3], and [F4], choose nonnegative measurable densities $p_{i}$ with $P_{i} (A) = \int_{A} p_{i} d ν$ for every measurable $A$ . In particular $\int p_{i} d ν = 1$ , so $p_{i} \in L^{1} (ν)$ .

2.1F1F5F6F7step 1.2algebra

Let $g = p_{0} - p_{1}$ , with measurable positive and negative parts $g^{+}, g^{-}$ by [F6]. Since $\int g d ν = 0$ , [F5] gives $\int g^{+} d ν = \int g^{-} d ν$ . For every measurable $A$ , $∣ \int_{A} g d ν ∣ \leq \max (\int g^{+} d ν, \int g^{-} d ν)$ ; the set $A = {g > 0}$ is measurable by [F7] and attains $\int_{A} g d ν = \int g^{+} d ν$ . Therefore [F1] gives $δ = \int g^{+} d ν = \int g^{-} d ν$ .

3.1F8step 2.1algebra

Define $Q_{0} (A) = δ^{- 1} \int_{A} g^{-} d ν$ and $Q_{1} (A) = δ^{- 1} \int_{A} g^{+} d ν$ . By [F8] these are measures; step 2.1 gives total mass one, so they are probability measures.

4.1givenF2F3F4F9step 3.1algebra

Let $ϵ^{'} = δ / (1 + δ)$ . The assumed inequality implies $ϵ^{'} \leq ϵ$ , and $ϵ^{'} / δ = 1 - ϵ^{'}$ . The densities of $(1 - ϵ^{'}) P_{0} + ϵ^{'} Q_{0}$ and $(1 - ϵ^{'}) P_{1} + ϵ^{'} Q_{1}$ are respectively $(1 - ϵ^{'}) (p_{0} + g^{-}) = (1 - ϵ^{'}) \max (p_{0}, p_{1})$ and $(1 - ϵ^{'}) (p_{1} + g^{+}) = (1 - ϵ^{'}) \max (p_{0}, p_{1})$ . Both mixtures are probability measures by [F9], and equal densities give equality of the measures.

5.1F9F10step 4.1algebra

By [F10], the target laws have the Huber mixture form. Since $ϵ > 0$ , set ${\tilde{Q}}_{i} = (ϵ^{'} / ϵ) Q_{i} + (1 - ϵ^{'} / ϵ) P_{i}$ . By [F9] each ${\tilde{Q}}_{i}$ is a probability measure, and $(1 - ϵ) P_{i} + ϵ {\tilde{Q}}_{i} = (1 - ϵ^{'}) P_{i} + ϵ^{'} Q_{i}$ . Step 4.1 therefore gives the required equality at the original fraction $ϵ$ .

6.1step 1.1step 5.1∎

Steps 1.1 and 5.1 cover all endpoint and positive-distance cases, respectively.

Depends on

Used by

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Sources

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber’s Contamination Model, §5, Theorem 5.1 and proof in the supplement (standard reference, not scraped)
Philippe Rigollet, 18.S997: High-Dimensional Statistics, Lecture Notes, Chapter 5, §5.3 (standard reference, not scraped)