Bayesian Vector AutoRegression with Factorised Granger-Causal Graphs

TL;DR

This work tackles the problem of discovering Granger-causal relations from observational multivariate time-series by introducing a Bayesian VAR framework that decouples GC graphs from coefficients via a hierarchical, factorised prior. The Poisson Factorised Granger-Causal Graph (PFGCG) uses a Generalised Bernoulli Poisson Link to generate sparse GC graphs from latent factors, enabling principled uncertainty quantification and automatic factor shrinkage with a tractable Gibbs-sampling inference. Empirical results on synthetic, semi-synthetic, and climate data demonstrate robust performance and uncertainty calibration, especially in low-data regimes, while requiring fewer hyperparameters than competing approaches. The approach provides interpretable GC graphs with posterior edge probabilities and shows potential for climate and other real-world time-series applications where interventions are impractical.

Abstract

We study the problem of automatically discovering Granger causal relations from observational multivariate time-series data.Vector autoregressive (VAR) models have been time-tested for this problem, including Bayesian variants and more recent developments using deep neural networks. Most existing VAR methods for Granger causality use sparsity-inducing penalties/priors or post-hoc thresholds to interpret their coefficients as Granger causal graphs. Instead, we propose a new Bayesian VAR model with a hierarchical factorised prior distribution over binary Granger causal graphs, separately from the VAR coefficients. We develop an efficient algorithm to infer the posterior over binary Granger causal graphs. Comprehensive experiments on synthetic, semi-synthetic, and climate data show that our method is more uncertainty aware, has less hyperparameters, and achieves better performance than competing approaches, especially in low-data regimes where there are less observations.

Figures (14)

• Figure 1: Results on Lorenz 96 and Lotka–Volterra. VAR (FBH) and BVAR(d) failed to learn when $T=100$ on Lorenz 96 and $T=200$ on Lotka–Volterra.
• Figure 2: Results on FMRI.
• Figure 3: Qualitative analysis of PFGCG ($V=1$, $\tau_{\text{max}}=5$) on Lotka–Volterra. Left: Ground-truth GC graph, middle: Bernoulli posterior mean of the discovered GC graphs, right: Matrix of $\{r^{\tau}_k\}_{k=1, \tau}^{K, \tau_{\text{max}}}$. Each rectangle in the figures indicates a value of a matrix and brighter colors indicates larger values.
• Figure 4: Results on JR55. Left: Discovered casual links between indices on JR55 where the weights are from the Bernoulli posterior of the graph (links with weights less than 0.2 are not shown) and thicker links indicate stronger connections. Right: The geographical locations of the indices.
• Figure 5: MSE over iterations of PFGCG ($V=2$). For better visualisation, we show MSE in the iterations between $[a, b]$ where $a$ is the first iteration that MSE goes below 2.0 and $b=500$.

Theorems & Definitions (4)

• Definition 3.1
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