  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.
