The Gaussian-Linear Hidden Markov model: a Python package

We propose the Gaussian-Linear Hidden Markov model (GLHMM), a generalisation of different types of HMMs commonly used in neuroscience. In short, the GLHMM is a general framework where linear regression is used to flexibly parameterise the Gaussian state distribution, thereby accommodating a wide range of uses -including unsupervised, encoding and decoding models. GLHMM is implemented as a Python toolbox with an emphasis on statistical testing and out-of-sample prediction -i.e. aimed at finding and characterising brain-behaviour associations. The toolbox uses a stochastic variational inference approach, enabling it to handle large data sets at reasonable computational time. Overall, the approach can be applied to several data modalities, including animal recordings or non-brain data, and applied over a broad range of experimental paradigms. For demonstration, we show examples with fMRI, electrocorticography, magnetoencephalo-graphy and pupillometry.

The FMRIB Variational Bayesian Inference Tutorial II: Stochastic Variational Bayes

Bayesian methods have proved powerful in many applications for the inference of model parameters from data. These methods are based on Bayes' theorem, which itself is deceptively simple. However, in practice the computations required are intractable even for simple cases. Hence methods for Bayesian inference have historically either been significantly approximate, e.g., the Laplace approximation, or achieve samples from the exact solution at significant computational expense, e.g., Markov Chain Monte Carlo methods. Since around the year 2000 so-called Variational approaches to Bayesian inference have been increasingly deployed. In its most general form Variational Bayes (VB) involves approximating the true posterior probability distribution via another more 'manageable' distribution, the aim being to achieve as good an approximation as possible. In the original FMRIB Variational Bayes tutorial we documented an approach to VB based that took a 'mean field' approach to forming the approximate posterior, required the conjugacy of prior and likelihood, and exploited the Calculus of Variations, to derive an iterative series of update equations, akin to Expectation Maximisation. In this tutorial we revisit VB, but now take a stochastic approach to the problem that potentially circumvents some of the limitations imposed by the earlier methodology. This new approach bears a lot of similarity to, and has benefited from, computational methods applied to machine learning algorithms. Although, what we document here is still recognisably Bayesian inference in the classic sense, and not an attempt to use machine learning as a black-box to solve the inference problem.

Stochastic Variational Bayesian Inference for a Nonlinear Forward Model

Variational Bayes (VB) has been used to facilitate the calculation of the posterior distribution in the context of Bayesian inference of the parameters of nonlinear models from data. Previously an analytical formulation of VB has been derived for nonlinear model inference on data with additive gaussian noise as an alternative to nonlinear least squares. Here a stochastic solution is derived that avoids some of the approximations required of the analytical formulation, offering a solution that can be more flexibly deployed for nonlinear model inference problems. The stochastic VB solution was used for inference on a biexponential toy case and the algorithmic parameter space explored, before being deployed on real data from a magnetic resonance imaging study of perfusion. The new method was found to achieve comparable parameter recovery to the analytic solution and be competitive in terms of computational speed despite being reliant on sampling.