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A convex mixture of probability laws is a probability law

Statement

Let $(X, A)$ be a measurable space, let $P, Q$ be probability measures on it, and let $t \in [0, 1]$ . The set function $M_{t} (A) = (1 - t) P (A) + t Q (A)$ is a probability measure.

Facts & Assumptions

Given: A measurable space $(X, A)$ , probability measures $P, Q$ on it, and $t \in [0, 1]$ .

[F1]

Probability measures and probability spaces: a probability measure is a measure of total mass one.

Proof

1.1givenF1algebra

For each $A \in A$ , both terms in $M_{t} (A)$ are nonnegative and finite. Also $M_{t} (\emptyset) = 0$ and $M_{t} (X) = (1 - t) + t = 1$ , using [F1].

2.1givenF1step 1.1algebra∎

If $(A_{j})_{j \geq 1}$ are pairwise disjoint, countable additivity of $P, Q$ gives $M_{t} (⋃_{j} A_{j}) = (1 - t) \sum_{j} P (A_{j}) + t \sum_{j} Q (A_{j}) = \sum_{j} M_{t} (A_{j})$ . Together with step 1.1, this is a probability measure.

Depends on

Probability measures and probability spaces

Used by

Gross-error contamination path Definition
Huber contamination model Definition
Exact Bernoulli Huber modulus and indistinguishable mixtures Example
TV-close laws have identical Huber mixtures at the contamination threshold Lemma

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Mengjie Chen, Chao Gao and Zhao Ren, Robust Covariance and Scatter Matrix Estimation under Huber’s Contamination Model, §1 (standard reference, not scraped)