This article details a scheme for approximate Bayesian inference, which has underpinned thousands of neuroimaging studies since its introduction 15 years ago. Variational Laplace (VL) provides a generic approach for fitting linear or non-linear models, which may be static or dynamic, returning a posterior probability density over the model parameters and an approximation of log model evidence, which enables Bayesian model comparison. VL applies variational Bayesian inference in conjunction with quadratic or Laplace approximations of the evidence lower bound (free energy). Importantly, update equations do not need to be derived for each model under consideration, providing a general method for fitting a broad class of models. This primer is intended for experimenters and modellers who may wish to fit models to data using variational Bayesian methods, without assuming previous experience of variational Bayes or machine learning. Accompanying code demonstrates how to fit different kinds of model using the reference implementation of the VL scheme in the open-source Statistical Parametric Mapping (SPM) software package. In addition, we provide a standalone software function that does not require SPM, in order to ease translation to other fields, together with detailed pseudocode. Finally, the supplementary materials provide worked derivations of the key equations.
During learning, neural responses decrease over repeated exposure to identical stimuli. This repetition suppression is thought to reflect a progressive optimization of neuronal responses elicited by the task. Functional magnetic resonance imaging was used to study the neural basis of associative learning of visual objects and their locations. As expected, activation in specialized cortical areas decreased with time. However, with path analysis it was shown that, in parallel to this adaptation, increases in effective connectivity occurred between distinct cortical systems specialized for spatial and object processing. The time course of these plastic changes was highly correlated with individual learning performance, suggesting that interactions between brain areas underlie associative learning.
The slight perversion of the original title of this piece (The Future of the Bayesian Brain) reflects my attempt to write prospectively about 'Science and Stories' over the past 20 years. I will meet this challenge by dealing with the future and then turning to its history. The future of the Bayesian brain (in neuroimaging) is clear: it is the application of dynamic causal modeling to understand how the brain conforms to the free energy principle. In this context, the Bayesian brain is a corollary of the free energy principle, which says that any self organizing system (like a brain or neuroimaging community) must maximize the evidence for its own existence, which means it must minimize its free energy using a model of its world. Dynamic causal modeling involves finding models of the brain that have the greatest evidence or the lowest free energy. In short, the future of imaging neuroscience is to refine models of the brain to minimize free energy, where the brain refines models of the world to minimize free energy. This endeavor itself minimizes free energy because our community is itself a self organizing system. I cannot imagine an alternative future that has the same beautiful self consistency as mine. Having dispensed with the future, we can now focus on the past, which is much more interesting:
This note derives the variational free energy under the Laplace approximation, with a focus on accounting for additional model complexity induced by increasing the number of model parameters. This is relevant when using the free energy as an approximation to the log-evidence in Bayesian model averaging and selection. By setting restricted maximum likelihood (ReML) in the larger context of variational learning and expectation maximisation (EM), we show how the ReML objective function can be adjusted to provide an approximation to the log-evidence for a particular model. This means ReML can be used for model selection, specifically to select or compare models with different covariance components. This is useful in the context of hierarchical models because it enables a principled selection of priors that, under simple hyperpriors, can be used for automatic model selection and relevance determination (ARD). Deriving the ReML objective function, from basic variational principles, discloses the simple relationships among Variational Bayes, EM and ReML. Furthermore, we show that EM is formally identical to a full variational treatment when the precisions are linear in the hyperparameters. Finally, we also consider, briefly, dynamic models and how these inform the regularisation of free energy ascent schemes, like EM and ReML.
This paper offers an active inference account of choice behaviour and learning. It focuses on the distinction between goal-directed and habitual behaviour and how they contextualise each other. We show that habits emerge naturally (and autodidactically) from sequential policy optimisation when agents are equipped with state-action policies. In active inference, behaviour has explorative (epistemic) and exploitative (pragmatic) aspects that are sensitive to ambiguity and risk respectively, where epistemic (ambiguity-resolving) behaviour enables pragmatic (reward-seeking) behaviour and the subsequent emergence of habits. Although goal-directed and habitual policies are usually associated with model-based and model-free schemes, we find the more important distinction is between belief-free and belief-based schemes. The underlying (variational) belief updating provides a comprehensive (if metaphorical) process theory for several phenomena, including the transfer of dopamine responses, reversal learning, habit formation and devaluation. Finally, we show that active inference reduces to a classical (Bellman) scheme, in the absence of ambiguity.
At the inception of human brain mapping, two principles of functional anatomy underwrote most conceptions - and analyses - of distributed brain responses: namely functional segregation and integration. There are currently two main approaches to characterising functional integration. The first is a mechanistic modelling of connectomics in terms of directed effective connectivity that mediates neuronal message passing and dynamics on neuronal circuits. The second phenomenological approach usually characterises undirected functional connectivity (i.e., measurable correlations), in terms of intrinsic brain networks, self-organised criticality, dynamical instability, etc. This paper describes a treatment of effective connectivity that speaks to the emergence of intrinsic brain networks and critical dynamics. It is predicated on the notion of Markov blankets that play a fundamental role in the self-organisation of far from equilibrium systems. Using the apparatus of the renormalisation group, we show that much of the phenomenology found in network neuroscience is an emergent property of a particular partition of neuronal states, over progressively larger scales. As such, it offers a way of linking dynamics on directed graphs to the phenomenology of intrinsic brain networks.
Functional segregation requires convergence and divergence of neuroanatomical connections. Furthermore, the nature of functional segregation suggests that (1) signals in convergent afferents are correlated and (2) signals in divergent efferents are uncorrelated. The aim of this article is to show that this arrangement can be predicted mathematically, using information theory and an idealized model of cortical processing. In theory, the existence of bifurcating axons limits the number of independent output channels from any small cortical region, relative to the number of inputs. An information theoretic analysis of this special (high input: output ratio) constraint indicates that the maximal transfer of information between inputs, to a cortical region, and its outputs will occur when (1) extrinsic connectivity to the area is organized such that the entropy of neural activity in afferents is optimally low and (2) connectivity intrinsic to the region is arranged to maximize the entropy measured at the initial segments of projection neurons. Under the constraints of the model, a low entropy is synonymous with high correlations between axonal firing rates (and vice versa). Consequently this antisymmetric arrangement of functional activity in convergent and divergent connections underlying functional segregation is exactly that predicted by the principle of maximum preservation of information, considered in the context of axonal bifurcation. The hypothesis that firing in convergent afferents is correlated (has low entropy) and spatially coherent was tested using positron emission tomographic measurements of cortical synaptic function in man. This hypothesis was confirmed.
