用贝叶斯方法实现高阶功能方差分析，提升可解释性与计算效率
Bayesian Neural Networks for Functional ANOVA model
- 基于贝叶斯推断的TPNN框架，自动检测高阶变量交互
- 相比原方法减少计算开销，支持更高阶成分建模
- 适合需要模型可解释性的复杂系统分析场景
随着机器学习对可解释性需求上升，功能方差分析（functional ANOVA）作为分解高维函数为低维分量的理论工具重新受到关注。近期提出的张量积神经网络（TPNN）被用作功能方差模型的基函数，形成ANOVA-TPNN模型。但该方法需预先指定待估计成分，导致高阶TPNN因计算和内存限制难以应用。本文提出贝叶斯-TPNN，一种基于贝叶斯推断的功能方差模型框架，采用TPNN基函数，能自动发现高阶成分且计算成本更低。我们设计了高效的马尔可夫链蒙特卡洛（MCMC）算法，并在多个基准数据集上验证其性能。理论上证明了贝叶斯-TPNN后验一致性。
原文摘要 · Abstract (English)
With the increasing demand for interpretability in machine learning, functional ANOVA decomposition has gained renewed attention as a principled tool for breaking down high-dimensional function into low-dimensional components that reveal the contributions of different variable groups. Recently, Tensor Product Neural Network (TPNN) has been developed and applied as basis functions in the functional ANOVA model, referred to as ANOVA-TPNN. A disadvantage of ANOVA-TPNN, however, is that the components to be estimated must be specified in advance, which makes it difficult to incorporate higher-order TPNNs into the functional ANOVA model due to computational and memory constraints. In this work, we propose Bayesian-TPNN, a Bayesian inference procedure for the functional ANOVA model with TPNN basis functions, enabling the detection of higher-order components with reduced computational cost compared to ANOVA-TPNN. We develop an efficient MCMC algorithm and demonstrate that Bayesian-TPNN performs well by analyzing multiple benchmark datasets. Theoretically, we prove that the posterior of Bayesian-TPNN is consistent.
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