Christoph Mathys

A Bayesian Foundation for Individual Learning Under Uncertainty

Computational learning models are critical for understanding mechanisms of adaptive behavior. However, the two major current frameworks, reinforcement learning (RL) and Bayesian learning, both have certain limitations. For example, many Bayesian models are agnostic of inter-individual variability and involve complicated integrals, making online learning difficult. Here, we introduce a generic hierarchical Bayesian framework for individual learning under multiple forms of uncertainty (e.g., environmental volatility and perceptual uncertainty). The model assumes Gaussian random walks of states at all but the first level, with the step size determined by the next highest level. The coupling between levels is controlled by parameters that shape the influence of uncertainty on learning in a subject-specific fashion. Using variational Bayes under a mean-field approximation and a novel approximation to the posterior energy function, we derive trial-by-trial update equations which (i) are analytical and extremely efficient, enabling real-time learning, (ii) have a natural interpretation in terms of RL, and (iii) contain parameters representing processes which play a key role in current theories of learning, e.g., precision-weighting of prediction error. These parameters allow for the expression of individual differences in learning and may relate to specific neuromodulatory mechanisms in the brain. Our model is very general: it can deal with both discrete and continuous states and equally accounts for deterministic and probabilistic relations between environmental events and perceptual states (i.e., situations with and without perceptual uncertainty). These properties are illustrated by simulations and analyses of empirical time series. Overall, our framework provides a novel foundation for understanding normal and pathological learning that contextualizes RL within a generic Bayesian scheme and thus connects it to principles of optimality from probability theory.

Active inference and epistemic value.

We offer a formal treatment of choice behavior based on the premise that agents minimize the expected free energy of future outcomes. Crucially, the negative free energy or quality of a policy can be decomposed into extrinsic and epistemic (or intrinsic) value. Minimizing expected free energy is therefore equivalent to maximizing extrinsic value or expected utility (defined in terms of prior preferences or goals), while maximizing information gain or intrinsic value (or reducing uncertainty about the causes of valuable outcomes). The resulting scheme resolves the exploration-exploitation dilemma: Epistemic value is maximized until there is no further information gain, after which exploitation is assured through maximization of extrinsic value. This is formally consistent with the Infomax principle, generalizing formulations of active vision based upon salience (Bayesian surprise) and optimal decisions based on expected utility and risk-sensitive (Kullback-Leibler) control. Furthermore, as with previous active inference formulations of discrete (Markovian) problems, ad hoc softmax parameters become the expected (Bayes-optimal) precision of beliefs about, or confidence in, policies. This article focuses on the basic theory, illustrating the ideas with simulations. A key aspect of these simulations is the similarity between precision updates and dopaminergic discharges observed in conditioning paradigms.

Hierarchical Prediction Errors in Midbrain and Basal Forebrain during Sensory Learning

In Bayesian brain theories, hierarchically related prediction errors (PEs) play a central role for predicting sensory inputs and inferring their underlying causes, e.g., the probabilistic structure of the environment and its volatility. Notably, PEs at different hierarchical levels may be encoded by different neuromodulatory transmitters. Here, we tested this possibility in computational fMRI studies of audio-visual learning. Using a hierarchical Bayesian model, we found that low-level PEs about visual stimulus outcome were reflected by widespread activity in visual and supramodal areas but also in the midbrain. In contrast, high-level PEs about stimulus probabilities were encoded by the basal forebrain. These findings were replicated in two groups of healthy volunteers. While our fMRI measures do not reveal the exact neuron types activated in midbrain and basal forebrain, they suggest a dichotomy between neuromodulatory systems, linking dopamine to low-level PEs about stimulus outcome and acetylcholine to more abstract PEs about stimulus probabilities.

Uncertainty in perception and the Hierarchical Gaussian Filter

In its full sense, perception rests on an agents model of how its sensory input comes about and the inferences it draws based on this model. These inferences are necessarily uncertain. Here, we illustrate how the Hierarchical Gaussian Filter (HGF) offers a principled and generic way to deal with the several forms that uncertainty in perception takes. The HGF is a recent derivation of one-step update equations from Bayesian principles that rests on a hierarchical generative model of the environment and its (in)stability. It is computationally highly efficient, allows for online estimates of hidden states, and has found numerous applications to experimental data from human subjects. In this paper, we generalize previous descriptions of the HGF and its account of perceptual uncertainty. First, we explicitly formulate the extension of the HGFs hierarchy to any number of levels; second, we discuss how various forms of uncertainty are accommodated by the minimization of variational free energy as encoded in the update equations; third, we combine the HGF with decision models and demonstrate the inversion of this combination; finally, we report a simulation study that compared four optimization methods for inverting the HGF/decision model combination at different noise levels. These four methods (Nelder–Mead simplex algorithm, Gaussian process-based global optimization, variational Bayes and Markov chain Monte Carlo sampling) all performed well even under considerable noise, with variational Bayes offering the best combination of efficiency and informativeness of inference. Our results demonstrate that the HGF provides a principled, flexible, and efficient—but at the same time intuitive—framework for the resolution of perceptual uncertainty in behaving agents.

Role of the Medial Prefrontal Cortex in Impaired Decision Making in Juvenile Attention-Deficit/Hyperactivity Disorder

IMPORTANCE Attention-deficit/hyperactivity disorder (ADHD) has been associated with deficient decision making and learning. Models of ADHD have suggested that these deficits could be caused by impaired reward prediction errors (RPEs). Reward prediction errors are signals that indicate violations of expectations and are known to be encoded by the dopaminergic system. However, the precise learning and decision-making deficits and their neurobiological correlates in ADHD are not well known. OBJECTIVE To determine the impaired decision-making and learning mechanisms in juvenile ADHD using advanced computational models, as well as the related neural RPE processes using multimodal neuroimaging. DESIGN, SETTING, AND PARTICIPANTS Twenty adolescents with ADHD and 20 healthy adolescents serving as controls (aged 12-16 years) were examined using a probabilistic reversal learning task while simultaneous functional magnetic resonance imaging and electroencephalogram were recorded. MAIN OUTCOMES AND MEASURES Learning and decision making were investigated by contrasting a hierarchical Bayesian model with an advanced reinforcement learning model and by comparing the model parameters. The neural correlates of RPEs were studied in functional magnetic resonance imaging and electroencephalogram. RESULTS Adolescents with ADHD showed more simplistic learning as reflected by the reinforcement learning model (exceedance probability, Px = .92) and had increased exploratory behavior compared with healthy controls (mean [SD] decision steepness parameter β: ADHD, 4.83 [2.97]; controls, 6.04 [2.53]; P = .02). The functional magnetic resonance imaging analysis revealed impaired RPE processing in the medial prefrontal cortex during cue as well as during outcome presentation (P < .05, family-wise error correction). The outcome-related impairment in the medial prefrontal cortex could be attributed to deficient processing at 200 to 400 milliseconds after feedback presentation as reflected by reduced feedback-related negativity (ADHD, 0.61 [3.90] μV; controls, -1.68 [2.52] μV; P = .04). CONCLUSIONS AND RELEVANCE The combination of computational modeling of behavior and multimodal neuroimaging revealed that impaired decision making and learning mechanisms in adolescents with ADHD are driven by impaired RPE processing in the medial prefrontal cortex. This novel, combined approach furthers the understanding of the pathomechanisms in ADHD and may advance treatment strategies.

Pavlovian conditioning-induced hallucinations result from overweighting of perceptual priors

Neural mechanisms for hallucinations Pairing a stimulus in one modality (vision) with a stimulus in another (sound) can lead to task-induced hallucinations in healthy individuals. After many trials, people eventually report perceiving a nonexistent stimulus contingent on the presence of the previously paired stimulus. Powers et al. investigated how different groups of volunteers and patients respond to this conditioning paradigm. They used behavior, neuroimaging, and computational modeling to dissect the effect of perceptual priors versus sensory evidence on such induced hallucinations. People who are more prone to hear voices were more susceptible to the induced auditory hallucinations. The network of brain regions that was active during the conditioned hallucinations resembled the network observed during clinical symptom capture in individuals who hallucinate while in a brain scanner. Science, this issue p. 596 Perceptual beliefs, stimulus associations, and belief volatility drive task-induced hallucinations in voice-hearers. Some people hear voices that others do not, but only some of those people seek treatment. Using a Pavlovian learning task, we induced conditioned hallucinations in four groups of people who differed orthogonally in their voice-hearing and treatment-seeking statuses. People who hear voices were significantly more susceptible to the effect. Using functional neuroimaging and computational modeling of perception, we identified processes that differentiated voice-hearers from non–voice-hearers and treatment-seekers from non–treatment-seekers and characterized a brain circuit that mediated the conditioned hallucinations. These data demonstrate the profound and sometimes pathological impact of top-down cognitive processes on perception and may represent an objective means to discern people with a need for treatment from those without.

The Dopaminergic Midbrain Encodes the Expected Certainty about Desired Outcomes

Dopamine plays a key role in learning; however, its exact function in decision making and choice remains unclear. Recently, we proposed a generic model based on active (Bayesian) inference wherein dopamine encodes the precision of beliefs about optimal policies. Put simply, dopamine discharges reflect the confidence that a chosen policy will lead to desired outcomes. We designed a novel task to test this hypothesis, where subjects played a “limited offer” game in a functional magnetic resonance imaging experiment. Subjects had to decide how long to wait for a high offer before accepting a low offer, with the risk of losing everything if they waited too long. Bayesian model comparison showed that behavior strongly supported active inference, based on surprise minimization, over classical utility maximization schemes. Furthermore, midbrain activity, encompassing dopamine projection neurons, was accurately predicted by trial-by-trial variations in model-based estimates of precision. Our findings demonstrate that human subjects infer both optimal policies and the precision of those inferences, and thus support the notion that humans perform hierarchical probabilistic Bayesian inference. In other words, subjects have to infer both what they should do as well as how confident they are in their choices, where confidence may be encoded by dopaminergic firing.

Computations of uncertainty mediate acute stress responses in humans.

The effects of stress are frequently studied, yet its proximal causes remain unclear. Here we demonstrate that subjective estimates of uncertainty predict the dynamics of subjective and physiological stress responses. Subjects learned a probabilistic mapping between visual stimuli and electric shocks. Salivary cortisol confirmed that our stressor elicited changes in endocrine activity. Using a hierarchical Bayesian learning model, we quantified the relationship between the different forms of subjective task uncertainty and acute stress responses. Subjective stress, pupil diameter and skin conductance all tracked the evolution of irreducible uncertainty. We observed a coupling between emotional and somatic state, with subjective and physiological tuning to uncertainty tightly correlated. Furthermore, the uncertainty tuning of subjective and physiological stress predicted individual task performance, consistent with an adaptive role for stress in learning under uncertain threat. Our finding that stress responses are tuned to environmental uncertainty provides new insight into their generation and likely adaptive function.

Allostatic Self-efficacy: A Metacognitive Theory of Dyshomeostasis-Induced Fatigue and Depression

This paper outlines a hierarchical Bayesian framework for interoception, homeostatic/allostatic control, and meta-cognition that connects fatigue and depression to the experience of chronic dyshomeostasis. Specifically, viewing interoception as the inversion of a generative model of viscerosensory inputs allows for a formal definition of dyshomeostasis (as chronically enhanced surprise about bodily signals, or, equivalently, low evidence for the brains model of bodily states) and allostasis (as a change in prior beliefs or predictions which define setpoints for homeostatic reflex arcs). Critically, we propose that the performance of interoceptive-allostatic circuitry is monitored by a metacognitive layer that updates beliefs about the brains capacity to successfully regulate bodily states (allostatic self-efficacy). In this framework, fatigue and depression can be understood as sequential responses to the interoceptive experience of dyshomeostasis and the ensuing metacognitive diagnosis of low allostatic self-efficacy. While fatigue might represent an early response with adaptive value (cf. sickness behavior), the experience of chronic dyshomeostasis may trigger a generalized belief of low self-efficacy and lack of control (cf. learned helplessness), resulting in depression. This perspective implies alternative pathophysiological mechanisms that are reflected by differential abnormalities in the effective connectivity of circuits for interoception and allostasis. We discuss suitably extended models of effective connectivity that could distinguish these connectivity patterns in individual patients and may help inform differential diagnosis of fatigue and depression in the future.

Scene Construction, Visual Foraging, and Active Inference

This paper describes an active inference scheme for visual searches and the perceptual synthesis entailed by scene construction. Active inference assumes that perception and action minimize variational free energy, where actions are selected to minimize the free energy expected in the future. This assumption generalizes risk-sensitive control and expected utility theory to include epistemic value; namely, the value (or salience) of information inherent in resolving uncertainty about the causes of ambiguous cues or outcomes. Here, we apply active inference to saccadic searches of a visual scene. We consider the (difficult) problem of categorizing a scene, based on the spatial relationship among visual objects where, crucially, visual cues are sampled myopically through a sequence of saccadic eye movements. This means that evidence for competing hypotheses about the scene has to be accumulated sequentially, calling upon both prediction (planning) and postdiction (memory). Our aim is to highlight some simple but fundamental aspects of the requisite functional anatomy; namely, the link between approximate Bayesian inference under mean field assumptions and functional segregation in the visual cortex. This link rests upon the (neurobiologically plausible) process theory that accompanies the normative formulation of active inference for Markov decision processes. In future work, we hope to use this scheme to model empirical saccadic searches and identify the prior beliefs that underwrite intersubject variability in the way people forage for information in visual scenes (e.g., in schizophrenia).