TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Equal-tailed posterior quantile intervals are credible

Statement

Let $Π_{y}$ be a posterior probability kernel on $R$ and let $0 < α < 1$ . Define $C_{α} (y) = [q_{α / 2} (y), q_{1 - α / 2} (y)]$ using the quantiles from Posterior quantiles are measurable and bracket their probability level. Then $C_{α}$ is graph-measurable and $Π_{y} (C_{α} (y)) \geq 1 - α$ for every $y$ . Thus equal-tailed posterior quantile intervals are one kind of credible set.

Facts & Assumptions

Given: The posterior probability kernel, the quantile convention, and $0 < α < 1$ in the Statement.

[F1]

A credible set is a graph-measurable set-valued rule whose posterior mass is at least its stated level. Posterior credible set

[F2]

By the quantile definition $q_{u} (y) = \inf {r \in Q : F_{y} (r) \geq u}$ , the defining sets are nested oppositely to $u$ , so $u \leq v$ implies $q_{u} (y) \leq q_{v} (y)$ . Each $q_{u}$ is measurable and satisfies $Π_{y} ((- i n f t y, q_{u} (y))) \leq u \leq Π_{y} ((- i n f t y, q_{u} (y)])$ . Posterior quantiles are measurable and bracket their probability level

[F3]

The product sigma-algebra is generated by measurable rectangles. The product sigma-algebra and its finite iterates

[F4]

There is a rational strictly between any two distinct real numbers. Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable

[F5]

If $A \subseteq B$ and $Π_{y} (A) < \infty$ , then $Π_{y} (B ∖ A) = Π_{y} (B) - Π_{y} (A)$ . Measure of a set difference when the smaller set has finite measure

Proof

Given: The hypotheses in the Statement.

1.1givenF2

Since $α / 2 < 1 - α / 2$ , the defining rational sets for the two quantiles are nested in the direction that gives $q_{α / 2} (y) \leq q_{1 - α / 2} (y)$ . Both endpoint functions are measurable by [F2].

1.2givenF2F5algebra

Put $l = q_{α / 2} (y)$ and $u = q_{1 - α / 2} (y)$ . The disjoint difference $(- \infty, u] ∖ (- \infty, l)$ is $[l, u]$ , so $Π_{y} ([l, u]) = Π_{y} ((- \infty, u]) - Π_{y} ((- \infty, l)) .$ By [F2], the first term is at least $1 - α / 2$ and the second is at most $α / 2$ , giving posterior mass at least $1 - α$ .

2.1F2F3F4step 1.1

The complement of the interval membership graph consists of pairs with $x < q_{α / 2} (y)$ or $x > q_{1 - α / 2} (y)$ . By [F4], the first relation is the countable union over $r \in Q$ of ${y : q_{α / 2} (y) > r} \times (- \infty, r),$ and the second is the countable union over $r \in Q$ of ${y : q_{1 - α / 2} (y) < r} \times (r, \infty) .$ Each is a measurable rectangle by endpoint measurability, so [F3] makes the graph measurable.

3.1F1step 2.1step 1.2∎

The graph measurability from step 2.1 and the mass bound from step 1.2 satisfy [F1], so $C_{α}$ is a posterior credible set.

Depends on

Used by

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Sources

Andrew Gelman et al., Bayesian Data Analysis, 3rd ed., §2.5 (standard reference, not scraped)
Alicia A. Johnson, Miles Q. Ott, and Mine Dogucu, Bayes Rules!, Chapter 8 §8.1 (standard reference, not scraped)