面向相依数据的WAIC：具有线性时间复杂度的协方差校正框架
On WAIC for Dependent Data: A Covariance-Corrected Framework with Linear-Time Complexity
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中文总结 AI 辅助
针对WAIC不适用于相依数据的问题，本文提出线性时间复杂度的CC-WAIC，通过协方差校正与有效样本量修正，在模拟与实际应用中显著优于现有准则，为相依数据贝叶斯模型选择提供可扩展工具。
中文摘要 AI 辅助
广泛适用信息准则（WAIC）是贝叶斯模型选择的基石，但其条件独立性假设使其不适用于序列数据和空间相依数据，这一局限会导致模型复杂度被系统性低估、预测评估结果过于乐观。我们提出CC-WAIC，它是WAIC的原则性推广，明确纳入了对数似然贡献的完整后验协方差结构；当数据满足独立性假设时，CC-WAIC会精确退化为WAIC，因此是自然的扩展而非临时修改。为克服完整协方差矩阵的过高计算成本，我们开发了使用带状协方差近似的线性时间实现，该实现降低了复杂度且具有近似误差的理论保证；我们还引入了有效样本量校正，以缓解MCMC估计中的有限样本偏差。通过对隐马尔可夫模型的大量模拟，以及对老忠实间歇泉数据和标普500波动率建模的实际应用，我们证明CC-WAIC的性能显著优于WAIC、留一交叉验证（LOO-CV）、iWAIC和WAICNF，尤其在强时间相依和样本量有限的情况下表现突出。该准则为现代科学中普遍存在的相依数据场景下的贝叶斯模型选择提供了计算可扩展且理论扎实的工具；其局限性包括依赖指数混合和精确条件似然，文中还讨论了对长记忆过程和近似推断的扩展方向。
英文摘要
The Widely Applicable Information Criterion (WAIC) is a cornerstone of Bayesian model selection, but its conditional independence assumption renders it inappropriate for sequential and spatially correlated data - a limitation that leads to systematically underestimated model complexity and over-optimistic predictive assessments. We introduce CC-WAIC, a principled generalization of WAIC that explicitly incorporates the full posterior covariance structure of log-likelihood contributions. CC-WAIC reduces exactly to WAIC when independence holds, making it a natural extension rather than an ad-hoc modification. To overcome the prohibitive computational cost of the full covariance matrix, we develop a linear-time implementation using a banded covariance approximation that reduces complexity, with theoretical guarantees on the approximation error. We further introduce an effective sample size correction to mitigate finite-sample bias in MCMC estimation. Through extensive simulations on Hidden Markov Models and real-world applications to Old Faithful geyser data and S&P 500 volatility modelling, we demonstrate that CC-WAIC substantially outperforms WAIC, LOO-CV, iWAIC, and WAICNF, particularly under strong temporal dependence and limited sample sizes. The proposed criterion offers a computationally scalable and theoretically grounded tool for Bayesian model selection in the dependent data settings that are ubiquitous across modern science. Limitations include reliance on exponential mixing and exact conditional likelihoods; extensions to long-memory processes and approximate inference are discussed.
