Split conformal prediction set

Let $m\ge 0$ calibration examples $Z_i=(X_i,Y_i)$ be scored by one fixed, measurable score rule $s_H$ obtained from the training statistic $H$, and let $0<\alpha <1$. Set $n=m+1,k=\lceil (m+1)(1-\alpha )\rceil \mathrm{.}$ Compute $R_i=s_H(Z_i)$ for $1\le i\le m$. Sort calibration observations by the ordered keys $(R_i,i)$, so ties are broken by the fixed index order. If $k\le m$, let $q$ be the score component of the $k$th ordered pair; if $k=m+1$, set $q=+\mathrm{\infty }$. For a feature value $x$, return $C(x)=\{y\in \mathcal{Y}:s_H(x,y)\le q\}\mathrm{.}$

The optional tie-randomized version uses one independent $U$ with Lebesgue probability law on $[0,1]$, shared across the candidate scan, and returns $C_U(x)=\{y\in \mathcal{Y}:p_U(y)>\alpha \},$ where $p_U$ is the score-rank formula in Conformal scores and rank p-values. The candidate score is included in its tie group. The score maps must be jointly measurable in the training, calibration, feature, and response variables; then membership in each returned set is a measurable event by Finite order statistics and rank maps are measurable.