Inversion of hierarchical Bayesian models using Gaussian processes

Over the past decade, computational approaches to neuroimaging have increasingly made use of hierarchical Bayesian models (HBMs), either for inferring on physiological mechanisms underlying fMRI data (e.g., dynamic causal modelling, DCM) or for deriving computational trajectories (from behavioural data) which serve as regressors in general linear models. However, an unresolved problem is that standard methods for inverting the hierarchical Bayesian model are either very slow, e.g. Markov Chain Monte Carlo Methods (MCMC), or are vulnerable to local minima in non-convex optimisation problems, such as variational Bayes (VB). This article considers Gaussian process optimisation (GPO) as an alternative approach for global optimisation of sufficiently smooth and efficiently evaluable objective functions. GPO avoids being trapped in local extrema and can be computationally much more efficient than MCMC. Here, we examine the benefits of GPO for inverting HBMs commonly used in neuroimaging, including DCM for fMRI and the Hierarchical Gaussian Filter (HGF). Importantly, to achieve computational efficiency despite high-dimensional optimisation problems, we introduce a novel combination of GPO and local gradient-based search methods. The utility of this GPO implementation for DCM and HGF is evaluated against MCMC and VB, using both synthetic data from simulations and empirical data. Our results demonstrate that GPO provides parameter estimates with equivalent or better accuracy than the other techniques, but at a fraction of the computational cost required for MCMC. We anticipate that GPO will prove useful for robust and efficient inversion of high-dimensional and nonlinear models of neuroimaging data.