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Estimating the changes of epidemiological parameters, such as instantaneous reproduction number, Rt, is important for understanding the transmission dynamics of infectious diseases. Current estimates of time-varying epidemiological parameters often face problems such as lagging observations, averaging inference, and improper quantification of uncertainties. To address these problems, we propose a Bayesian data assimilation framework for time-varying parameter estimation. Specifically, this framework is applied to estimate the instantaneous reproduction number Rt during emerging epidemics, resulting in the state-of-the-art 'DARt' system. With DARt, time misalignment caused by lagging observations is tackled by incorporating observation delays into the joint inference of infections and Rt; the drawback of averaging is overcome by instantaneously updating upon new observations and developing a model selection mechanism that captures abrupt changes; the uncertainty is quantified and reduced by employing Bayesian smoothing. We validate the performance of DARt and demonstrate its power in describing the transmission dynamics of COVID-19. The proposed approach provides a promising solution for making accurate and timely estimation for transmission dynamics based on reported data., Competing Interests: The authors have declared that no competing interests exist.

This work considers a class of canonical neural networks comprising rate coding models, wherein neural activity and plasticity minimise a common cost function-and plasticity is modulated with a certain delay. We show that such neural networks implicitly perform active inference and learning to minimise the risk associated with future outcomes. Mathematical analyses demonstrate that this biological optimisation can be cast as maximisation of model evidence, or equivalently minimisation of variational free energy, under the well-known form of a partially observed Markov decision process model. This equivalence indicates that the delayed modulation of Hebbian plasticity-accompanied with adaptation of firing thresholds-is a sufficient neuronal substrate to attain Bayes optimal inference and control. We corroborated this proposition using numerical analyses of maze tasks. This theory offers a universal characterisation of canonical neural networks in terms of Bayesian belief updating and provides insight into the neuronal mechanisms underlying planning and adaptive behavioural control., (© 2022. The Author(s).)

Neural rhythms or oscillations are ubiquitous in neuroimaging data. These spectral responses have been linked to several cognitive processes; including working memory, attention, perceptual binding and neuronal coordination. In this paper, we show how Bayesian methods can be used to finesse the ill-posed problem of reconstructing-and explaining-oscillatory responses. We offer an overview of recent developments in this field, focusing on (i) the use of MEG data and Empirical Bayes to build hierarchical models for group analyses-and the identification of important sources of inter-subject variability and (ii) the construction of novel dynamic causal models of intralaminar recordings to explain layer-specific activity. We hope to show that electrophysiological measurements contain much more spatial information than is often thought: on the one hand, the dynamic causal modelling of non-invasive (low spatial resolution) electrophysiology can afford sub-millimetre (hyper-acute) resolution that is limited only by the (spatial) complexity of the underlying (dynamic causal) forward model. On the other hand, invasive microelectrode recordings (that penetrate different cortical layers) can reveal laminar-specific responses and elucidate hierarchical message passing and information processing within and between cortical regions at a macroscopic scale. In short, the careful and biophysically grounded modelling of sparse data enables one to characterise the neuronal architectures generating oscillations in a remarkable detail.

Empirical evidence suggests the incentive value of an option is affected by other options available during choice and by options presented in the past. These contextual effects are hard to reconcile with classical theories and have inspired accounts where contextual influences play a crucial role. However, each account only addresses one or the other of the empirical findings and a unifying perspective has been elusive. Here, we offer a unifying theory of context effects on incentive value attribution and choice based on normative Bayesian principles. This formulation assumes that incentive value corresponds to a precision-weighted prediction error, where predictions are based upon expectations about reward. We show that this scheme explains a wide range of contextual effects, such as those elicited by other options available during choice (or within-choice context effects). These include both conditions in which choice requires an integration of multiple attributes and conditions where a multi-attribute integration is not necessary. Moreover, the same scheme explains context effects elicited by options presented in the past or between-choice context effects. Our formulation encompasses a wide range of contextual influences (comprising both within- and between-choice effects) by calling on Bayesian principles, without invoking ad-hoc assumptions. This helps clarify the contextual nature of incentive value and choice behaviour and may offer insights into psychopathologies characterized by dysfunctional decision-making, such as addiction and pathological gambling.

This technical note describes some Bayesian procedures for the analysis of group studies that use nonlinear models at the first (within-subject) level - e.g., dynamic causal models - and linear models at subsequent (between-subject) levels. Its focus is on using Bayesian model reduction to finesse the inversion of multiple models of a single dataset or a single (hierarchical or empirical Bayes) model of multiple datasets. These applications of Bayesian model reduction allow one to consider parametric random effects and make inferences about group effects very efficiently (in a few seconds). We provide the relatively straightforward theoretical background to these procedures and illustrate their application using a worked example. This example uses a simulated mismatch negativity study of schizophrenia. We illustrate the robustness of Bayesian model reduction to violations of the (commonly used) Laplace assumption in dynamic causal modelling and show how its recursive application can facilitate both classical and Bayesian inference about group differences. Finally, we consider the application of these empirical Bayesian procedures to classification and prediction., (Copyright © 2015 The Authors. Published by Elsevier Inc. All rights reserved.)

Dynamic causal modeling (DCM) is a Bayesian framework for inferring effective connectivity among brain regions from neuroimaging data. While the validity of DCM has been investigated in various previous studies, the reliability of DCM parameter estimates across sessions has been examined less systematically. Here, we report results of a software comparison with regard to test-retest reliability of DCM for fMRI, using a challenging scenario where complex models with many parameters were applied to relatively few data points. Specifically, we examined the reliability of different DCM implementations (in terms of the intra-class correlation coefficient, ICC) based on fMRI data from 35 human subjects performing a simple motor task in two separate sessions, one month apart. We constructed DCMs of motor regions with fair to excellent reliability of conventional activation measures. Using classical DCM (cDCM) in SPM5, we found that the test-retest reliability of DCM results was high, both concerning the model evidence (ICC=0.94) and the model parameter estimates (median ICC=0.47). However, when using a more recent DCM version (DCM10 in SPM8), test-retest reliability was reduced notably. Analyses indicated that, in our particular case, the prior distributions played a crucial role in this change in reliability across software versions. Specifically, when using cDCM priors for model inversion in DCM10, this not only restored reliability but yielded even better results than in cDCM. Analyzing each component of the objective function in DCM, we found a selective change in the reliability of posterior mean estimates. This suggests that tighter regularization afforded by cDCM priors reduces the possibility of local extrema in the objective function. We conclude this paper with an outlook to ongoing developments for overcoming the software-dependency of reliability observed in this study, including global optimization and empirical Bayesian procedures., (Copyright © 2014 Elsevier Inc. All rights reserved.)

Deciding how much evidence to accumulate before making a decision is a problem we and other animals often face, but one that is not completely understood. This issue is particularly important because a tendency to sample less information (often known as reflection impulsivity) is a feature in several psychopathologies, such as psychosis. A formal understanding of information sampling may therefore clarify the computational anatomy of psychopathology. In this theoretical letter, we consider evidence accumulation in terms of active (Bayesian) inference using a generic model of Markov decision processes. Here, agents are equipped with beliefs about their own behavior--in this case, that they will make informed decisions. Normative decision making is then modeled using variational Bayes to minimize surprise about choice outcomes. Under this scheme, different facets of belief updating map naturally onto the functional anatomy of the brain (at least at a heuristic level). Of particular interest is the key role played by the expected precision of beliefs about control, which we have previously suggested may be encoded by dopaminergic neurons in the midbrain. We show that manipulating expected precision strongly affects how much information an agent characteristically samples, and thus provides a possible link between impulsivity and dopaminergic dysfunction. Our study therefore represents a step toward understanding evidence accumulation in terms of neurobiologically plausible Bayesian inference and may cast light on why this process is disordered in psychopathology.

All organisms must integrate cognition, emotion, and motivation to guide action toward valuable (goal) states, as described by active inference. Within this framework, cognition, emotion, and motivation interact through the (Bayesian) fusion of exteroceptive, proprioceptive, and interoceptive signals, the precision-weighting of prediction errors, and the "affective tuning" of neuronal representations. Crucially, misregulation of these processes may have profound psychopathological consequences.

This paper considers goal-directed decision-making in terms of embodied or active inference. We associate bounded rationality with approximate Bayesian inference that optimizes a free energy bound on model evidence. Several constructs such as expected utility, exploration or novelty bonuses, softmax choice rules and optimism bias emerge as natural consequences of free energy minimization. Previous accounts of active inference have focused on predictive coding. In this paper, we consider variational Bayes as a scheme that the brain might use for approximate Bayesian inference. This scheme provides formal constraints on the computational anatomy of inference and action, which appear to be remarkably consistent with neuroanatomy. Active inference contextualizes optimal decision theory within embodied inference, where goals become prior beliefs. For example, expected utility theory emerges as a special case of free energy minimization, where the sensitivity or inverse temperature (associated with softmax functions and quantal response equilibria) has a unique and Bayes-optimal solution. Crucially, this sensitivity corresponds to the precision of beliefs about behaviour. The changes in precision during variational updates are remarkably reminiscent of empirical dopaminergic responses-and they may provide a new perspective on the role of dopamine in assimilating reward prediction errors to optimize decision-making.

Inferring the environment's statistical structure and adapting behavior accordingly is a fundamental modus operandi of the brain. A simple form of this faculty based on spatial attentional orienting can be studied with Posner's location-cueing paradigm in which a cue indicates the target location with a known probability. The present study focuses on a more complex version of this task, where probabilistic context (percentage of cue validity) changes unpredictably over time, thereby creating a volatile environment. Saccadic response speed (RS) was recorded in 15 subjects and used to estimate subject-specific parameters of a Bayesian learning scheme modeling the subjects' trial-by-trial updates of beliefs. Different response models-specifying how computational states translate into observable behavior-were compared using Bayesian model selection. Saccadic RS was most plausibly explained as a function of the precision of the belief about the causes of sensory input. This finding is in accordance with current Bayesian theories of brain function, and specifically with the proposal that spatial attention is mediated by a precision-dependent gain modulation of sensory input. Our results provide empirical support for precision-dependent changes in beliefs about saccade target locations and motivate future neuroimaging and neuropharmacological studies of how Bayesian inference may determine spatial attention.

Viewing the brain as an organ of approximate Bayesian inference can help us understand how it represents the self. We suggest that inferred representations of the self have a normative function: to predict and optimise the likely outcomes of social interactions. Technically, we cast this predict-and-optimise as maximising the chance of favourable outcomes through active inference. Here the utility of outcomes can be conceptualised as prior beliefs about final states. Actions based on interpersonal representations can therefore be understood as minimising surprise - under the prior belief that one will end up in states with high utility. Interpersonal representations thus serve to render interactions more predictable, while the affective valence of interpersonal inference renders self-perception evaluative. Distortions of self-representation contribute to major psychiatric disorders such as depression, personality disorder and paranoia. The approach we review may therefore operationalise the study of interpersonal representations in pathological states., (Copyright © 2014 The Authors. Published by Elsevier Inc. All rights reserved.)

This article provides a neurobiological account of symptoms that have been called 'hysterical', 'psychogenic' or 'medically unexplained', which we will call functional motor and sensory symptoms. We use a neurobiologically informed model of hierarchical Bayesian inference in the brain to explain functional motor and sensory symptoms in terms of perception and action arising from inference based on prior beliefs and sensory information. This explanation exploits the key balance between prior beliefs and sensory evidence that is mediated by (body focused) attention, symptom expectations, physical and emotional experiences and beliefs about illness. Crucially, this furnishes an explanation at three different levels: (i) underlying neuromodulatory (synaptic) mechanisms; (ii) cognitive and experiential processes (attention and attribution of agency); and (iii) formal computations that underlie perceptual inference (representation of uncertainty or precision). Our explanation involves primary and secondary failures of inference; the primary failure is the (autonomous) emergence of a percept or belief that is held with undue certainty (precision) following top-down attentional modulation of synaptic gain. This belief can constitute a sensory percept (or its absence) or induce movement (or its absence). The secondary failure of inference is when the ensuing percept (and any somatosensory consequences) is falsely inferred to be a symptom to explain why its content was not predicted by the source of attentional modulation. This account accommodates several fundamental observations about functional motor and sensory symptoms, including: (i) their induction and maintenance by attention; (ii) their modification by expectation, prior experience and cultural beliefs and (iii) their involuntary and symptomatic nature.

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:, (Copyright © 2011 Elsevier Inc. All rights reserved.)

This note describes a Bayesian model selection or optimization procedure for post hoc inferences about reduced versions of a full model. The scheme provides the evidence (marginal likelihood) for any reduced model as a function of the posterior density over the parameters of the full model. It rests upon specifying models through priors on their parameters, under the assumption that the likelihood remains the same for all models considered. This provides a quick and efficient scheme for scoring arbitrarily large numbers of models, after inverting a single (full) model. In turn, this enables the selection among discrete models that are distinguished by the presence or absence of free parameters, where free parameters are effectively removed from the model using very precise shrinkage priors. An alternative application of this post hoc model selection considers continuous model spaces, defined in terms of hyperparameters (sufficient statistics) of the prior density over model parameters. In this instance, the prior (model) can be optimized with respect to its evidence. The expressions for model evidence become remarkably simple under the Laplace (Gaussian) approximation to the posterior density. Special cases of this scheme include Savage-Dickey density ratio tests for reduced models and automatic relevance determination in model optimization. We illustrate the approach using general linear models and a more complicated nonlinear state-space model., (Copyright © 2011 Elsevier Inc. All rights reserved.)

Mathematical models of scientific data can be formally compared using Bayesian model evidence. Previous applications in the biological sciences have mainly focussed on model selection in which one first selects the model with the highest evidence and then makes inferences based on the parameters of that model. This "best model" approach is very useful but can become brittle if there are a large number of models to compare, and if different subjects use different models. To overcome this shortcoming we propose the combination of two further approaches: (i) family level inference and (ii) Bayesian model averaging within families. Family level inference removes uncertainty about aspects of model structure other than the characteristic of interest. For example: What are the inputs to the system? Is processing serial or parallel? Is it linear or nonlinear? Is it mediated by a single, crucial connection? We apply Bayesian model averaging within families to provide inferences about parameters that are independent of further assumptions about model structure. We illustrate the methods using Dynamic Causal Models of brain imaging data.

Bayesian model selection (BMS) is a powerful method for determining the most likely among a set of competing hypotheses about the mechanisms that generated observed data. BMS has recently found widespread application in neuroimaging, particularly in the context of dynamic causal modelling (DCM). However, so far, combining BMS results from several subjects has relied on simple (fixed effects) metrics, e.g. the group Bayes factor (GBF), that do not account for group heterogeneity or outliers. In this paper, we compare the GBF with two random effects methods for BMS at the between-subject or group level. These methods provide inference on model-space using a classical and Bayesian perspective respectively. First, a classical (frequentist) approach uses the log model evidence as a subject-specific summary statistic. This enables one to use analysis of variance to test for differences in log-evidences over models, relative to inter-subject differences. We then consider the same problem in Bayesian terms and describe a novel hierarchical model, which is optimised to furnish a probability density on the models themselves. This new variational Bayes method rests on treating the model as a random variable and estimating the parameters of a Dirichlet distribution which describes the probabilities for all models considered. These probabilities then define a multinomial distribution over model space, allowing one to compute how likely it is that a specific model generated the data of a randomly chosen subject as well as the exceedance probability of one model being more likely than any other model. Using empirical and synthetic data, we show that optimising a conditional density of the model probabilities, given the log-evidences for each model over subjects, is more informative and appropriate than both the GBF and frequentist tests of the log-evidences. In particular, we found that the hierarchical Bayesian approach is considerably more robust than either of the other approaches in the presence of outliers. We expect that this new random effects method will prove useful for a wide range of group studies, not only in the context of DCM, but also for other modelling endeavours, e.g. comparing different source reconstruction methods for EEG/MEG or selecting among competing computational models of learning and decision-making.

The neurophysiology of eye movements has been studied extensively, and several computational models have been proposed for decision-making processes that underlie the generation of eye movements towards a visual stimulus in a situation of uncertainty. One class of models, known as linear rise-to-threshold models, provides an economical, yet broadly applicable, explanation for the observed variability in the latency between the onset of a peripheral visual target and the saccade towards it. So far, however, these models do not account for the dynamics of learning across a sequence of stimuli, and they do not apply to situations in which subjects are exposed to events with conditional probabilities. In this methodological paper, we extend the class of linear rise-to-threshold models to address these limitations. Specifically, we reformulate previous models in terms of a generative, hierarchical model, by combining two separate sub-models that account for the interplay between learning of target locations across trials and the decision-making process within trials. We derive a maximum-likelihood scheme for parameter estimation as well as model comparison on the basis of log likelihood ratios. The utility of the integrated model is demonstrated by applying it to empirical saccade data acquired from three healthy subjects. Model comparison is used (i) to show that eye movements do not only reflect marginal but also conditional probabilities of target locations, and (ii) to reveal subject-specific learning profiles over trials. These individual learning profiles are sufficiently distinct that test samples can be successfully mapped onto the correct subject by a naïve Bayes classifier. Altogether, our approach extends the class of linear rise-to-threshold models of saccadic decision making, overcomes some of their previous limitations, and enables statistical inference both about learning of target locations across trials and the decision-making process within trials.

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.

We describe an extension of our empirical Bayes approach to magnetoencephalography/electroencephalography (MEG/EEG) source reconstruction that covers both evoked and induced responses. The estimation scheme is based on classical covariance component estimation using restricted maximum likelihood (ReML). We have focused previously on the estimation of spatial covariance components under simple assumptions about the temporal correlations. Here we extend the scheme, using temporal basis functions to place constraints on the temporal form of the responses. We show how the same scheme can estimate evoked responses that are phase-locked to the stimulus and induced responses that are not. For a single trial the model is exactly the same. In the context of multiple trials, however, the inherent distinction between evoked and induced responses calls for different treatments of the underlying hierarchical multitrial model. We derive the respective models and show how they can be estimated efficiently using ReML. This enables the Bayesian estimation of evoked and induced changes in power or, more generally, the energy of wavelet coefficients., ((c) 2006 Wiley-Liss, Inc.)

Distributed linear solutions of the EEG source localisation problem are used routinely. In contrast to discrete dipole equivalent models, distributed linear solutions do not assume a fixed number of active sources and rest on a discretised fully 3D representation of the electrical activity of the brain. The ensuing inverse problem is underdetermined and constraints or priors are required to ensure the uniqueness of the solution. In a Bayesian framework, the conditional expectation of the source distribution, given the data, is attained by carefully balancing the minimisation of the residuals induced by noise and the improbability of the estimates as determined by their priors. This balance is specified by hyperparameters that control the relative importance of fitting and conforming to various constraints. Here we formulate the conventional "Weighted Minimum Norm" (WMN) solution in terms of hierarchical linear models. An "Expectation-Maximisation" (EM) algorithm is used to obtain a "Restricted Maximum Likelihood" (ReML) estimate of the hyperparameters, before estimating the "Maximum a Posteriori" solution itself. This procedure can be considered a generalisation of previous work that encompasses multiple constraints. Our approach was compared with the "classic" WMN and Maximum Smoothness solutions, using a simplified 2D source model with synthetic noisy data. The ReML solution was assessed with four types of source location priors: no priors, accurate priors, inaccurate priors, and both accurate and inaccurate priors. The ReML approach proved useful as: (1) The regularisation (or influence of the a priori source covariance) increased as the noise level increased. (2) The localisation error (LE) was negligible when accurate location priors were used. (3) When accurate and inaccurate location priors were used simultaneously, the solution was not influenced by the inaccurate priors. The ReML solution was then applied to real somatosensory-evoked responses to illustrate the application in an empirical setting.

This article is about how the brain data mines its sensory inputs. There are several architectural principles of functional brain anatomy that have emerged from careful anatomic and physiologic studies over the past century. These principles are considered in the light of representational learning to see if they could have been predicted a priori on the basis of purely theoretical considerations. We first review the organisation of hierarchical sensory cortices, paying special attention to the distinction between forward and backward connections. We then review various approaches to representational learning as special cases of generative models, starting with supervised learning and ending with learning based upon empirical Bayes. The latter predicts many features, such as a hierarchical cortical system, prevalent top-down backward influences and functional asymmetries between forward and backward connections that are seen in the real brain. The key points made in article are: (i). hierarchical generative models enable the learning of empirical priors and eschew prior assumptions about the causes of sensory input that are inherent in non-hierarchical models. These assumptions are necessary for learning schemes based on information theory and efficient or sparse coding, but are not necessary in a hierarchical context. Critically, the anatomical infrastructure that may implement generative models in the brain is hierarchical. Furthermore, learning based on empirical Bayes can proceed in a biologically plausible way. (ii). The second point is that backward connections are essential if the processes generating inputs cannot be inverted, or the inversion cannot be parameterised. Because these processes involve many-to-one mappings, are non-linear and dynamic in nature, they are generally non-invertible. This enforces an explicit parameterisation of generative models (i.e. backward connections) to afford recognition and suggests that forward architectures, on their own, are not sufficient for perception. (iii). Finally, non-linearities in generative models, mediated by backward connections, require these connections to be modulatory, so that representations in higher cortical levels can interact to predict responses in lower levels. This is important in relation to functional asymmetries in forward and backward connections that have been demonstrated empirically.

Mechanistic insight is achieved only when experiments are employed to test formal or computational models. Furthermore, in analogy to lesion studies, phantom perception may serve as a vehicle to understand the fundamental processing principles underlying healthy auditory perception. With a special focus on tinnitus-as the prime example of auditory phantom perception-we review recent work at the intersection of artificial intelligence, psychology and neuroscience. In particular, we discuss why everyone with tinnitus suffers from (at least hidden) hearing loss, but not everyone with hearing loss suffers from tinnitus. We argue that intrinsic neural noise is generated and amplified along the auditory pathway as a compensatory mechanism to restore normal hearing based on adaptive stochastic resonance. The neural noise increase can then be misinterpreted as auditory input and perceived as tinnitus. This mechanism can be formalized in the Bayesian brain framework, where the percept (posterior) assimilates a prior prediction (brain's expectations) and likelihood (bottom-up neural signal). A higher mean and lower variance (i.e. enhanced precision) of the likelihood shifts the posterior, evincing a misinterpretation of sensory evidence, which may be further confounded by plastic changes in the brain that underwrite prior predictions. Hence, two fundamental processing principles provide the most explanatory power for the emergence of auditory phantom perceptions: predictive coding as a top-down and adaptive stochastic resonance as a complementary bottom-up mechanism. We conclude that both principles also play a crucial role in healthy auditory perception. Finally, in the context of neuroscience-inspired artificial intelligence, both processing principles may serve to improve contemporary machine learning techniques.