Positive regression dependence on true-null coordinates (PRDS)

Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space, let $m\ge 1$, and let $\mathbf{P}=(P_1,\dots ,P_m):\mathrm{\Omega }\to [0,1{]}^m$ be a random vector whose coordinates are p-value statistics (Valid p-values). Fix a designated true-null index set $I_0\subseteq \{1,\dots ,m\}$. For $i\in I_0$, write ${\mu }_i$ for the law of $P_i$ under $\mathbb{P}$ and define its support relative to $[0,1]$ by

$\mathrm{supp}({\mu }_i):=\{x\in [0,1]:{\mu }_i(U)>0\text{ for every relatively open neighborhood }U\text{ of }x\}\mathrm{.}$

Give each coordinate $[0,1]$ its trace Borel sigma-algebra (The Borel sigma-algebra of a topological space) and give $[0,1{]}^m$ the finite product sigma-algebra (The product sigma-algebra and its finite iterates). In this definition, “Borel set” means a member of that product sigma-algebra; thus “increasing measurable set” means coordinatewise increasing in this stated structure. The unit-interval lemma proves that $[0,1]$ with this trace Borel structure is standard Borel (The unit interval with its Borel structure is standard Borel). The finite-product result then makes $[0,1{]}^m$ with the specified product sigma-algebra a standard-Borel target (Standard Borel spaces, Finite products of standard Borel spaces are standard Borel); its finite-product proof uses only finite choice in ZF. Under the Axiom of Choice (The Axiom of Choice), this target has a regular conditional distribution given $\sigma (P_i)$, and the conditional-law factorization result supplies a kernel $K_i:[0,1]\to [0,1{]}^m$ representing the conditional law of $\mathbf{P}$ given $P_i=x$ (Existence of regular conditional distributions for standard borel targets, Regular conditional kernels factor through a standard borel conditioning variable, Conditional law given a random element).

The law of $\mathbf{P}$ is positive regression dependent on the true-null coordinates (PRDS on $I_0$) if, for each $i\in I_0$, such a conditional-law version can be chosen so that for every coordinatewise increasing Borel set $D$, the function

$x⟼K_i(x,D)$

is nondecreasing on $\mathrm{supp}({\mu }_i)$. If $I_0=\mathrm{\varnothing }$, this condition is vacuous. The monotonicity condition is a full conditional-law requirement; pairwise correlations alone do not verify it.

The regular conditional law is a chosen version. PRDS requires the monotonicity on the stated support, including support points where a conditional version is not determined by an atom of positive mass. The companion examples include a finite counterexample showing that valid p-values can have positive pairwise covariance without satisfying PRDS. The BY source's one-sided Gaussian sufficient conditions do not imply a general guarantee for two-sided Gaussian tests; the PRDS theorem below applies only after its exact conditional monotonicity premise is established.