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Marginals and positive-mass conditionals of a finite joint law are probability vectors

Statement

Let $U, V$ be finite nonempty alphabets and let $p_{U V} (u, v) \geq 0$ satisfy $\sum_{u, v} p_{U V} (u, v) = 1$ . Define $p_{U} (u) = \sum_{v} p_{U V} (u, v)$ and $p_{V} (v) = \sum_{u} p_{U V} (u, v)$ . Then both marginals are probability vectors. For each $v$ with $p_{V} (v) > 0$ , the conditional vector $p_{U ∣ v} (u) : = p_{U V} (u, v) / p_{V} (v)$ is a probability vector; likewise for conditioning on positive-mass $u$ . Consequently, if $p_{U V W}$ is a finite joint pmf, then for every $w$ with $p_{W} (w) > 0$ , $p_{U V ∣ w}$ and its two marginals are probability vectors.

Facts & Assumptions

Given: Finite nonempty alphabets $U, V$ and a joint pmf $p_{U V}$ .

[F1]

Finite distributions, entropy, divergence, and stochastic matrices: a finite probability vector has nonnegative coordinates whose sum is $1$ .

[F2]

Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule: for finite sets $S, T$ and a real array $a_{s, t}$ , the sum over $S \times T$ equals either iterated sum, and finite sums are invariant under bijective reindexing.

Proof

1.1givenF1F2algebra

Every marginal coordinate is nonnegative because it is a finite sum of nonnegative entries of $p_{U V}$ . By [F2], $\sum_{u} p_{U} (u) = \sum_{u} \sum_{v} p_{U V} (u, v) = \sum_{(u, v)} p_{U V} (u, v) = 1$ , and the same calculation with the order reversed gives $\sum_{v} p_{V} (v) = 1$ . Thus both marginals are probability vectors by [F1].

2.1givenF1step 1.1algebra∎

Fix $v$ with $p_{V} (v) > 0$ . Each ratio $p_{U V} (u, v) / p_{V} (v)$ is nonnegative, and $\sum_{u} p_{U ∣ v} (u) = (\sum_{u} p_{U V} (u, v)) / p_{V} (v) = p_{V} (v) / p_{V} (v) = 1$ . Hence $p_{U ∣ v}$ is a probability vector by [F1]. The same argument with $u$ fixed gives $p_{V ∣ u}$ . If $p_{U V W}$ is a joint pmf and $p_{W} (w) > 0$ , first apply this argument to the pair of finite alphabets $U \times V$ and $W$ ; then apply step 1.1 to the resulting conditional pmf on $U \times V$ . This gives the last assertion.

Depends on

Used by

Finite conditional entropy Definition
Finite conditional mutual information Definition
Finite mutual information Definition
Conditional block law factors across IID coordinates Lemma
Conditional type-class cardinality bound Lemma
Continuity of finite conditional entropy in the joint law Lemma
Finite conditional mutual information identities and equality Theorem
Finite entropy and mutual-information chain rules Theorem

Dependency tree · two levels

12 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

Robert M. Gray, Entropy and Information Theory, corrected first edition (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author prepublication (standard reference, not scraped)