Finite nested decision families and bounded risk

Let $n,M\ge 1$, $B>0$, and $\alpha \in [0,B)$. Fix a deterministic strictly ordered grid $\mathrm{\Lambda }=\{{\lambda }_1<\cdots <{\lambda }_M\}\subset \mathbb{R}$ and a probability space $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$. For $i=1,\dots ,n+1$, let $W_i$ be a random element of $[0,B{]}^M$ with its Borel sigma-algebra, and write $L_i({\lambda }_j)=(W_i{)}_j$. The rows $W_1,\dots ,W_{n+1}$ are exchangeable when their joint law is invariant under every permutation of the $n+1$ row indices. The coordinates $L_i({\lambda }_j)$ are the losses of the finite decision family at observation $i$; in a prediction problem they may be obtained by applying each fixed prediction rule to that observation and evaluating a specified loss.

The family is loss-nested when, with probability one, $L_i({\lambda }_1)\ge \cdots \ge L_i({\lambda }_M)$ for every $i$. It is endpoint-safe at level $\alpha$ when, with probability one, $L_i({\lambda }_M)\le \alpha$ for every $i$. For any measurable selector $\widehat{\lambda }$ taking values in $\mathrm{\Lambda }$ and determined by the calibration rows $W_1,\dots ,W_n$, its marginal expected test risk is $\mathbb{E}[L_{n+1}(\widehat{\lambda })]$, with expectation over all $n+1$ rows. This expectation is finite because every loss lies in $[0,B]$. Here “nested” means nonincreasing loss on the specified grid; literal inclusion of prediction sets is not required.

Exchangeable random elements: exchangeability means invariance of the joint law under every coordinate permutation.

Random elements and real random variables: a random element is a measurable map from a probability space into a measurable space.

Expectation of a nonnegative or integrable random variable: the expectation of a bounded nonnegative loss is its Lebesgue integral against the probability measure.

Angelopoulos et al., Conformal Risk Control, §1.1

Angelopoulos, Barber, and Bates, Theoretical Foundations of Conformal Prediction, Chapter 2 §§2.1–2.2