Estimating time-varying graphical models are of paramount importance in various social, financial, biological, and engineering systems, since the evolution of such networks can be utilized for example to spot trends, detect anomalies, predict vulnerability, and evaluate the impact of interventions. Existing methods require extensive tuning of parameters that control the graph sparsity and temporal smoothness. Furthermore, these methods are computationally burdensome with time complexity $O(NP^3)$ for $P$ variables and $N$ time points. As a remedy, we propose a low-complexity tuning-free Bayesian approach, named BASS. Specifically, we impose temporally-dependent spike-and-slab priors on the graphs such that they are sparse and varying smoothly across time. A variational inference algorithm is then derived to learn the graph structures from the data automatically. Owning to the pseudo-likelihood and the mean-field approximation, the time complexity of BASS is only $O(NP^2)$. Additionally, by identifying the frequency-domain resemblance to the time-varying graphical models, we show that BASS can be extended to learning frequency-varying inverse spectral density matrices, and yields graphical models for multivariate stationary time series. Numerical results on both synthetic and real data show that that BASS can better recover the underlying true graphs, while being more efficient than the existing methods, especially for high-dimensional cases.

翻译：估计时变图模型在各类社会、金融、生物和工程系统中具有至关重要的作用，因为此类网络的演变可用于识别趋势、检测异常、预测脆弱性以及评估干预措施的影响。现有方法需要大量调参来控制图的稀疏性和时间平滑性。此外，这些方法计算负担沉重，对于$P$个变量和$N$个时间点，时间复杂度为$O(NP^3)$。为解决这一问题，我们提出一种低复杂度、免调参的贝叶斯方法，命名为BASS。具体而言，我们在图上施加时间相关的尖峰-板先验（spike-and-slab prior），从而保证图的稀疏性及随时间平滑变化。随后推导出一种变分推理算法，从数据自动学习图结构。得益于伪似然（pseudo-likelihood）与平均场近似，BASS的时间复杂度仅为$O(NP^2)$。此外，通过识别时变图模型与频率域的相似性，我们证明BASS可扩展至学习时变逆谱密度矩阵，从而为多元平稳时间序列提供图模型。在合成数据和真实数据上的数值结果表明，BASS能够更准确地恢复真实底层图结构，同时相比现有方法效率更高，尤其适用于高维情形。