Conformal prediction constructs prediction sets with finite-sample coverage guarantees, but its calibration stage is structurally constrained to a scalar score function and a single threshold variable - forcing shapes of prediction sets to be fixed before calibration, typically through data splitting. We introduce multi-variable conformal prediction (MCP), a framework that extends conformal prediction to vector-valued score functions with multiple simultaneous calibration variables. Building on scenario theory as a principled framework for certifying data-driven decisions, MCP unifies prediction set design and calibration into a single optimization problem, eliminating data splitting without sacrificing coverage guarantees. We propose two computationally efficient variants: RemMCP, grounded in constrained optimization with constraint removal, which admits a clean generalization of split conformal prediction; and RelMCP, based on iterative optimization with constraint relaxation, which supports non-convex score functions at the cost of possibly greater conservatism. Through numerical experiments on ellipsoidal and multi-modal prediction sets, we demonstrate that RemMCP and RelMCP consistently meet the target coverage with prediction set sizes smaller than or comparable to those of baselines with data split, while considerably reducing variance across calibration runs - a direct consequence of using all available data for shape optimization and calibration simultaneously.

翻译：共形预测构建具有有限样本覆盖保证的预测集，但其校准阶段在结构上受限于标量评分函数和单一阈值变量——这迫使预测集的形状在校准前固定，通常通过数据分割实现。我们提出多变量共形预测（MCP），该框架将共形预测扩展至具有多个同时校准变量的向量值评分函数。基于场景理论（一种用于认证数据驱动决策的原则性框架），MCP将预测集设计与校准统一为单一优化问题，在无需数据分割的前提下保持覆盖保证。我们提出两种计算高效的变体：基于带约束移除的约束优化的RemMCP，其实现了分割共形预测的简洁泛化；以及基于带约束松弛的迭代优化的RelMCP，其支持非凸评分函数，但可能以更高的保守性为代价。通过在椭球形和多模态预测集上的数值实验，我们证明RemMCP和RelMCP能一致达到目标覆盖，其预测集尺寸小于或等于基线方法（采用数据分割）的尺寸，同时显著降低跨校准运行的方差——这是同时使用全部可用数据进行形状优化与校准的直接结果。