Bayesian inference of sparsity in stationary, multivariate autoregressive processes

Advances in sensing technology have made it possible to collect large volumes of time-series data on many variables. In a diverse array of fields, a key question is whether such data can be used to learn directed relationships between variables. In other words, whether changes in one variable consistently precede those in another, an idea formalized in the concept of ‘Granger causality’. In this project, we use two motivating examples from neuroscience, where learning the drivers of change could lead to substantial improvements in our understanding of the underpinning physiology. The first concerns abnormal brain activity patterns which typically affect people with epilepsy. The second concerns the 24-hour biological clock, measured by a range of physiological variables. Graphical vector autoregressions (VARs) are a popular tool for learning such lag-lead relationships in multivariate time-series. A VAR of order p expresses the observation at time t as a regression on the preceding p terms. The pattern of zeros in the autoregressive coefficients has a graphical interpretation: absence of an edge from variable i to variable j is tantamount to i being Granger non-causal for j. In the Bayesian inferential framework, this kind of sparsity in model parameters can be accommodated by prior distributions which assign non-zero probability to every pattern of zeros. One limitation with current approaches to fitting graphical VARs is that useful Markov properties of Granger causality graphs rely on the process being stable (e.g. constant mean, variance, covariances), at least locally. For the process to be stable, the autoregressive coefficients must lie in a constrained space with a complex geometry, and so stability is generally assumed without being enforced. This can be problematic when there are not enough data to learn, with certainty, that a process is stable. A second limitation arises when data, though recorded at discrete intervals in time, would be more naturally described through the underlying continuous-time process. In this case, although the time discretisation is often chosen for convenience, the notion of Granger causality is dependent on it. Directly modelling the continuous-time system through the continuous-time analogue of a sparse VAR overcomes this problem but, again, enforcing stability imposes complex constraints on the parameters. Our aims are therefore: Develop prior distributions for VARs and continuous-time linear systems that simultaneously encourage sparsity in the parameters and enforce stability; Develop associated procedures for computational inference; Apply our ideas to the motivating applications from neuroscience. There are two main challenges. The first is that the complex constraints imposed by the stability condition make it difficult to specify a prior distribution with appropriately restricted support. This problem is then compounded by requiring the prior to be sparse, essentially making its dimension one of the unknowns. There is then a second challenge: constructing a Markov chain Monte Carlo (MCMC) sampler over a constrained space of unknown dimension. In our two applications, interpretation of Granger causality networks has huge potential to improve understanding of the relationships between variables, with important implications for the treatment of disease. There are also numerous other fields which rely on network inference in their research and which therefore stand to gain from our methods. This includes genetics, finance and microbial ecology, with potential benefits further down the line from the broader agenda their research serves, be that society, the economy or the environment.