用一个神经算子统一处理多种概率条件，实现高效贝叶斯推断。
One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators
- 设计单一神经算子，将任意联合密度映射为条件密度。
- 理论证明该算子可被神经网络逼近至任意精度。
- 适用于构建通用贝叶斯推理基础模型，尤其适合复杂分布建模。
概率条件化旨在确定给定随机变量 $Y$ 时随机变量 $X$ 的分布，是科学与工程中不确定性建模的核心。传统机器学习方法通常针对固定联合分布直接学习条件分布。本文提出新视角：通过识别一个能将任意联合密度映射为条件密度的单一算子来解决该问题，从而在联合-条件对间实现参数共享与计算摊销。我们证明，在合适密度类上，该条件化算子具有连续性，因此可由神经算子以任意精度逼近。进一步地，我们使用神经算子学习了一类高斯混合分布的条件化映射，验证了框架的有效性。本工作为通用、可摊销的概率条件化方法（如贝叶斯推断的基础模型）提供了理论支撑。
原文摘要 · Abstract (English)
Probabilistic conditioning is concerned with the identification of a distribution of a random variable $X$ given a random variable $Y$. It is a cornerstone of scientific and engineering applications where modeling uncertainty is key. This problem has traditionally been addressed in machine learning by directly learning the conditional distribution of a fixed joint distribution. This paper introduces a novel perspective: we propose to solve the conditioning problem by identifying a single operator that maps any joint density to its conditional, thus amortizing over joint-conditional pairs. We establish that the conditioning operator can be approximated to arbitrary accuracy by neural operators. Our proof relies on new results establishing continuity of the conditioning operator over suitable classes of densities. Finally, we learn the conditioning map for a class of Gaussian mixtures using neural operators, illustrating the promise of our framework. This work provides the theoretical underpinnings for general-purpose, amortized methods for probabilistic conditioning, such as foundation models for Bayesian inference.
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