We introduce NCP (Neural Conditional Probability), a novel operator-theoretic approach for learning conditional distributions with a particular focus on inference tasks. NCP can be used to build conditional confidence regions and extract important statistics like conditional quantiles, mean, and covariance. It offers streamlined learning through a single unconditional training phase, facilitating efficient inference without the need for retraining even when conditioning changes. By tapping into the powerful approximation capabilities of neural networks, our method efficiently handles a wide variety of complex probability distributions, effectively dealing with nonlinear relationships between input and output variables. Theoretical guarantees ensure both optimization consistency and statistical accuracy of the NCP method. Our experiments show that our approach matches or beats leading methods using a simple Multi-Layer Perceptron (MLP) with two hidden layers and GELU activations. This demonstrates that a minimalistic architecture with a theoretically grounded loss function can achieve competitive results without sacrificing performance, even in the face of more complex architectures.

翻译：我们提出NCP（神经条件概率），一种新颖的算子理论方法，用于学习条件分布，特别关注推理任务。NCP可用于构建条件置信区域并提取重要统计量，如条件分位数、均值和协方差。它通过单一的无条件训练阶段实现简化的学习，即使在条件发生变化时也能促进高效推理而无需重新训练。通过利用神经网络的强大逼近能力，我们的方法能高效处理各种复杂的概率分布，有效应对输入与输出变量之间的非线性关系。理论保证确保了NCP方法的优化一致性与统计准确性。实验表明，我们的方法使用简单的两层隐藏层多层感知器（MLP）与GELU激活函数，即可匹配或超越主流方法。这证明，基于理论基础的损失函数配合极简架构，即使面对更复杂的模型结构，也能在不牺牲性能的前提下取得有竞争力的结果。