Risk-Sensitive Optimal Feedback Control Accounts for Sensorimotor Behavior under Uncertainty

Many aspects of human motor behavior can be understood using optimality principles such as optimal feedback control. However, these proposed optimal control models are risk-neutral; that is, they are indifferent to the variability of the movement cost. Here, we propose the use of a risk-sensitive optimal controller that incorporates movement cost variance either as an added cost (risk-averse controller) or as an added value (risk-seeking controller) to model human motor behavior in the face of uncertainty. We use a sensorimotor task to test the hypothesis that subjects are risk-sensitive. Subjects controlled a virtual ball undergoing Brownian motion towards a target. Subjects were required to minimize an explicit cost, in points, that was a combination of the final positional error of the ball and the integrated control cost. By testing subjects on different levels of Brownian motion noise and relative weighting of the position and control cost, we could distinguish between risk-sensitive and risk-neutral control. We show that subjects change their movement strategy pessimistically in the face of increased uncertainty in accord with the predictions of a risk-averse optimal controller. Our results suggest that risk-sensitivity is a fundamental attribute that needs to be incorporated into optimal feedback control models.     

Risk-sensitivity and the mean-variance trade-off: decision making in sensorimotor control

Numerous psychophysical studies suggest that the sensorimotor system chooses actions that optimize the average cost associated with a movement. Recently, however, violations of this hypothesis have been reported in line with economic theories of decision-making that not only consider the mean payoff, but are also sensitive to risk, that is the variability of the payoff. Here, we examine the hypothesis that risk-sensitivity in sensorimotor control arises as a mean-variance trade-off in movement costs. We designed a motor task in which participants could choose between a sure motor action that resulted in a fixed amount of effort and a risky motor action that resulted in a variable amount of effort that could be either lower or higher than the fixed effort. By changing the mean effort of the risky action while experimentally fixing its variance, we determined indifference points at which participants chose equiprobably between the sure, fixed amount of effort option and the risky, variable effort option. Depending on whether participants accepted a variable effort with a mean that was higher, lower or equal to the fixed effort, they could be classified as risk-seeking, risk-averse or risk-neutral. Most subjects were risk-sensitive in our task consistent with a mean-variance trade-off in effort, thereby, underlining the importance of risk-sensitivity in computational models of sensorimotor control.     

Continuous evolution of statistical estimators for optimal decision-making

In many everyday situations, humans must make precise decisions in the presence of uncertain sensory information. For example, when asked to combine information from multiple sources we often assign greater weight to the more reliable information. It has been proposed that statistical-optimality often observed in human perception and decision-making requires that humans have access to the uncertainty of both their senses and their decisions. However, the mechanisms underlying the processes of uncertainty estimation remain largely unexplored. In this paper we introduce a novel visual tracking experiment that requires subjects to continuously report their evolving perception of the mean and uncertainty of noisy visual cues over time. We show that subjects accumulate sensory information over the course of a trial to form a continuous estimate of the mean, hindered only by natural kinematic constraints (sensorimotor latency etc.). Furthermore, subjects have access to a measure of their continuous objective uncertainty, rapidly acquired from sensory information available within a trial, but limited by natural kinematic constraints and a conservative margin for error. Our results provide the first direct evidence of the continuous mean and uncertainty estimation mechanisms in humans that may underlie optimal decision making.     

Doubly Bayesian Analysis of Confidence in Perceptual Decision-Making

Humans stand out from other animals in that they are able to explicitly report on the reliability of their internal operations. This ability, which is known as metacognition, is typically studied by asking people to report their confidence in the correctness of some decision. However, the computations underlying confidence reports remain unclear. In this paper, we present a fully Bayesian method for directly comparing models of confidence. Using a visual two-interval forced-choice task, we tested whether confidence reports reflect heuristic computations (e.g. the magnitude of sensory data) or Bayes optimal ones (i.e. how likely a decision is to be correct given the sensory data). In a standard design in which subjects were first asked to make a decision, and only then gave their confidence, subjects were mostly Bayes optimal. In contrast, in a less-commonly used design in which subjects indicated their confidence and decision simultaneously, they were roughly equally likely to use the Bayes optimal strategy or to use a heuristic but suboptimal strategy. Our results suggest that, while people’s confidence reports can reflect Bayes optimal computations, even a small unusual twist or additional element of complexity can prevent optimality.     

Computational mechanisms of sensorimotor control

In order to generate skilled and efficient actions, the motor system must find solutions to several problems inherent in sensorimotor control, including nonlinearity, nonstationarity, delays, redundancy, uncertainty, and noise. We review these problems and five computational mechanisms that the brain may use to limit their deleterious effects: optimal feedback control, impedance control, predictive control, Bayesian decision theory, and sensorimotor learning. Together, these computational mechanisms allow skilled and fluent sensorimotor behavior.     

Inferring visuomotor priors for sensorimotor learning

Sensorimotor learning has been shown to depend on both prior expectations and sensory evidence in a way that is consistent with Bayesian integration. Thus, prior beliefs play a key role during the learning process, especially when only ambiguous sensory information is available. Here we develop a novel technique to estimate the covariance structure of the prior over visuomotor transformations--the mapping between actual and visual location of the hand--during a learning task. Subjects performed reaching movements under multiple visuomotor transformations in which they received visual feedback of their hand position only at the end of the movement. After experiencing a particular transformation for one reach, subjects have insufficient information to determine the exact transformation, and so their second reach reflects a combination of their prior over visuomotor transformations and the sensory evidence from the first reach. We developed a Bayesian observer model in order to infer the covariance structure of the subjects' prior, which was found to give high probability to parameter settings consistent with visuomotor rotations. Therefore, although the set of visuomotor transformations experienced had little structure, the subjects had a strong tendency to interpret ambiguous sensory evidence as arising from rotation-like transformations. We then exposed the same subjects to a highly-structured set of visuomotor transformations, designed to be very different from the set of visuomotor rotations. During this exposure the prior was found to have changed significantly to have a covariance structure that no longer favored rotation-like transformations. In summary, we have developed a technique which can estimate the full covariance structure of a prior in a sensorimotor task and have shown that the prior over visuomotor transformations favor a rotation-like structure. Moreover, through experience of a novel task structure, participants can appropriately alter the covariance structure of their prior.     

Biophysics of risk aversion based on neurotransmitter receptor theory

Decision under risk and uncertainty has been attracting attention in neuroeconomics and neuroendocrinology of decision-making. This paper demonstrated that the neurotransmitter receptor theory-based value (utility) function can account for human and animal risk-taking behavior. The theory predicts that (i) when dopaminergic neuronal response is efficiently coupled to the formation of ligand-receptor complex, subjects are risk-aversive (irrespective of their satisfaction level) and (ii) when the coupling is inefficient, subjects are risk-seeking at low satisfaction levels, consistent with risk-sensitive foraging theory in ecology. It is further suggested that some anomalies in decision under risk are due to inefficiency of the coupling between dopamine receptor activation and neuronal response. Future directions in the application of the model to studies in neuroeconomics of addiction and neuroendocrine modulation of risk-taking behavior are discussed.     

Representations of uncertainty in sensorimotor control

Uncertainty is ubiquitous in our sensorimotor interactions, arising from factors such as sensory and motor noise and ambiguity about the environment. Setting it apart from previous theories, a quintessential property of the Bayesian framework for making inference about the state of world so as to select actions, is the requirement to represent the uncertainty associated with inferences in the form of probability distributions. In the context of sensorimotor control and learning, the Bayesian framework suggests that to respond optimally to environmental stimuli the central nervous system needs to construct estimates of the sensorimotor transformations, in the form of internal models, as well as represent the structure of the uncertainty in the inputs, outputs and in the transformations themselves. Here we review Bayesian inference and learning models that have been successful in demonstrating the sensitivity of the sensorimotor system to different forms of uncertainty as well as recent studies aimed at characterizing the representation of the uncertainty at different computational levels.     

Decision-Making under Ambiguity Is Modulated by Visual Framing,but Not by Motor vs. Non-Motor Context. Experiments and an Information-Theoretic Ambiguity Model

A number of recent studies have investigated differences in human choice behavior depending on task framing, especially comparing economic decision-making to choice behavior in equivalent sensorimotor tasks. Here we test whether decision-making under ambiguity exhibits effects of task framing in motor vs. non-motor context. In a first experiment, we designed an experience-based urn task with varying degrees of ambiguity and an equivalent motor task where subjects chose between hitting partially occluded targets. In a second experiment, we controlled for the different stimulus design in the two tasks by introducing an urn task with bar stimuli matching those in the motor task. We found ambiguity attitudes to be mainly influenced by stimulus design. In particular, we found that the same subjects tended to be ambiguity-preferring when choosing between ambiguous bar stimuli, but ambiguity-avoiding when choosing between ambiguous urn sample stimuli. In contrast, subjects’ choice pattern was not affected by changing from a target hitting task to a non-motor context when keeping the stimulus design unchanged. In both tasks subjects’ choice behavior was continuously modulated by the degree of ambiguity. We show that this modulation of behavior can be explained by an information-theoretic model of ambiguity that generalizes Bayes-optimal decision-making by combining Bayesian inference with robust decision-making under model uncertainty. Our results demonstrate the benefits of information-theoretic models of decision-making under varying degrees of ambiguity for a given context, but also demonstrate the sensitivity of ambiguity attitudes across contexts that theoretical models struggle to explain.     

Observing the observer (I): meta-bayesian models of learning and decision-making

In this paper, we present a generic approach that can be used to infer how subjects make optimal decisions under uncertainty. This approach induces a distinction between a subject's perceptual model, which underlies the representation of a hidden "state of affairs" and a response model, which predicts the ensuing behavioural (or neurophysiological) responses to those inputs. We start with the premise that subjects continuously update a probabilistic representation of the causes of their sensory inputs to optimise their behaviour. In addition, subjects have preferences or goals that guide decisions about actions given the above uncertain representation of these hidden causes or state of affairs. From a Bayesian decision theoretic perspective, uncertain representations are so-called "posterior" beliefs, which are influenced by subjective "prior" beliefs. Preferences and goals are encoded through a "loss" (or "utility") function, which measures the cost incurred by making any admissible decision for any given (hidden) state of affair. By assuming that subjects make optimal decisions on the basis of updated (posterior) beliefs and utility (loss) functions, one can evaluate the likelihood of observed behaviour. Critically, this enables one to "observe the observer", i.e. identify (context- or subject-dependent) prior beliefs and utility-functions using psychophysical or neurophysiological measures. In this paper, we describe the main theoretical components of this meta-Bayesian approach (i.e. a Bayesian treatment of Bayesian decision theoretic predictions). In a companion paper ('Observing the observer (II): deciding when to decide'), we describe a concrete implementation of it and demonstrate its utility by applying it to simulated and real reaction time data from an associative learning task.     

Observing the observer (II): deciding when to decide

In a companion paper [1], we have presented a generic approach for inferring how subjects make optimal decisions under uncertainty. From a Bayesian decision theoretic perspective, uncertain representations correspond to "posterior" beliefs, which result from integrating (sensory) information with subjective "prior" beliefs. Preferences and goals are encoded through a "loss" (or "utility") function, which measures the cost incurred by making any admissible decision for any given (hidden or unknown) state of the world. By assuming that subjects make optimal decisions on the basis of updated (posterior) beliefs and utility (loss) functions, one can evaluate the likelihood of observed behaviour. In this paper, we describe a concrete implementation of this meta-Bayesian approach (i.e. a Bayesian treatment of Bayesian decision theoretic predictions) and demonstrate its utility by applying it to both simulated and empirical reaction time data from an associative learning task. Here, inter-trial variability in reaction times is modelled as reflecting the dynamics of the subjects' internal recognition process, i.e. the updating of representations (posterior densities) of hidden states over trials while subjects learn probabilistic audio-visual associations. We use this paradigm to demonstrate that our meta-Bayesian framework allows for (i) probabilistic inference on the dynamics of the subject's representation of environmental states, and for (ii) model selection to disambiguate between alternative preferences (loss functions) human subjects could employ when dealing with trade-offs, such as between speed and accuracy. Finally, we illustrate how our approach can be used to quantify subjective beliefs and preferences that underlie inter-individual differences in behaviour.     

An evaluation of new parsimony‐based versus parametric inference methods in biogeography: a case study using the globally distributed plant family Sapindaceae

Aim Recently developed parametric methods in historical biogeography allow researchers to integrate temporal and palaeogeographical information into the reconstruction of biogeographical scenarios, thus overcoming a known bias of parsimony‐based approaches. Here, we compare a parametric method, dispersal–extinction–cladogenesis (DEC), against a parsimony‐based method, dispersal–vicariance analysis (DIVA), which does not incorporate branch lengths but accounts for phylogenetic uncertainty through a Bayesian empirical approach (Bayes‐DIVA). We analyse the benefits and limitations of each method using the cosmopolitan plant family Sapindaceae as a case study. Location World‐wide. Methods Phylogenetic relationships were estimated by Bayesian inference on a large dataset representing generic diversity within Sapindaceae. Lineage divergence times were estimated by penalized likelihood over a sample of trees from the posterior distribution of the phylogeny to account for dating uncertainty in biogeographical reconstructions. We compared biogeographical scenarios between Bayes‐DIVA and two different DEC models: one with no geological constraints and another that employed a stratified palaeogeographical model in which dispersal rates were scaled according to area connectivity across four time slices, reflecting the changing continental configuration over the last 110 million years. Results Despite differences in the underlying biogeographical model, Bayes‐DIVA and DEC inferred similar biogeographical scenarios. The main differences were: (1) in the timing of dispersal events – which in Bayes‐DIVA sometimes conflicts with palaeogeographical information, and (2) in the lower frequency of terminal dispersal events inferred by DEC. Uncertainty in divergence time estimations influenced both the inference of ancestral ranges and the decisiveness with which an area can be assigned to a node. Main conclusions By considering lineage divergence times, the DEC method gives more accurate reconstructions that are in agreement with palaeogeographical evidence. In contrast, Bayes‐DIVA showed the highest decisiveness in unequivocally reconstructing ancestral ranges, probably reflecting its ability to integrate phylogenetic uncertainty. Care should be taken in defining the palaeogeographical model in DEC because of the possibility of overestimating the frequency of extinction events, or of inferring ancestral ranges that are outside the extant species ranges, owing to dispersal constraints enforced by the model. The wide‐spanning spatial and temporal model proposed here could prove useful for testing large‐scale biogeographical patterns in plants.     

Effects of branch length uncertainty on Bayesian posterior probabilities for phylogenetic hypotheses

In Bayesian phylogenetics, confidence in evolutionary relationships is expressed as posterior probability--the probability that a tree or clade is true given the data, evolutionary model, and prior assumptions about model parameters. Model parameters, such as branch lengths, are never known in advance; Bayesian methods incorporate this uncertainty by integrating over a range of plausible values given an assumed prior probability distribution for each parameter. Little is known about the effects of integrating over branch length uncertainty on posterior probabilities when different priors are assumed. Here, we show that integrating over uncertainty using a wide range of typical prior assumptions strongly affects posterior probabilities, causing them to deviate from those that would be inferred if branch lengths were known in advance; only when there is no uncertainty to integrate over does the average posterior probability of a group of trees accurately predict the proportion of correct trees in the group. The pattern of branch lengths on the true tree determines whether integrating over uncertainty pushes posterior probabilities upward or downward. The magnitude of the effect depends on the specific prior distributions used and the length of the sequences analyzed. Under realistic conditions, however, even extraordinarily long sequences are not enough to prevent frequent inference of incorrect clades with strong support. We found that across a range of conditions, diffuse priors--either flat or exponential distributions with moderate to large means--provide more reliable inferences than small-mean exponential priors. An empirical Bayes approach that fixes branch lengths at their maximum likelihood estimates yields posterior probabilities that more closely match those that would be inferred if the true branch lengths were known in advance and reduces the rate of strongly supported false inferences compared with fully Bayesian integration.     

Estimation of quantitative trait locus effects with epistasis by variational Bayes algorithms

Bayesian hierarchical shrinkage methods have been widely used for quantitative trait locus mapping. From the computational perspective, the application of the Markov chain Monte Carlo (MCMC) method is not optimal for high-dimensional problems such as the ones arising in epistatic analysis. Maximum a posteriori (MAP) estimation can be a faster alternative, but it usually produces only point estimates without providing any measures of uncertainty (i.e., interval estimates). The variational Bayes method, stemming from the mean field theory in theoretical physics, is regarded as a compromise between MAP and MCMC estimation, which can be efficiently computed and produces the uncertainty measures of the estimates. Furthermore, variational Bayes methods can be regarded as the extension of traditional expectation-maximization (EM) algorithms and can be applied to a broader class of Bayesian models. Thus, the use of variational Bayes algorithms based on three hierarchical shrinkage models including Bayesian adaptive shrinkage, Bayesian LASSO, and extended Bayesian LASSO is proposed here. These methods performed generally well and were found to be highly competitive with their MCMC counterparts in our example analyses. The use of posterior credible intervals and permutation tests are considered for decision making between quantitative trait loci (QTL) and non-QTL. The performance of the presented models is also compared with R/qtlbim and R/BhGLM packages, using a previously studied simulated public epistatic data set.     

Adaptive Sampling of Information in Perceptual Decision-Making

In many perceptual and cognitive decision-making problems, humans sample multiple noisy information sources serially, and integrate the sampled information to make an overall decision. We derive the optimal decision procedure for two-alternative choice tasks in which the different options are sampled one at a time, sources vary in the quality of the information they provide, and the available time is fixed. To maximize accuracy, the optimal observer allocates time to sampling different information sources in proportion to their noise levels. We tested human observers in a corresponding perceptual decision-making task. Observers compared the direction of two random dot motion patterns that were triggered only when fixated. Observers allocated more time to the noisier pattern, in a manner that correlated with their sensory uncertainty about the direction of the patterns. There were several differences between the optimal observer predictions and human behaviour. These differences point to a number of other factors, beyond the quality of the currently available sources of information, that influences the sampling strategy.     

Tactile Gap Detection Deteriorates during Bimanual Symmetrical Movements under Mirror Visual Feedback

It has been suggested that incongruence between signals for motor intention and sensory input can cause pain and other sensory abnormalities. This claim is supported by reports that moving in an environment of induced sensorimotor conflict leads to elevated pain and sensory symptoms in those with certain painful conditions. Similar procedures can lead to reports of anomalous sensations in healthy volunteers too. In the present study, we used mirror visual feedback to investigate the effects of sensorimotor incongruence on responses to stimuli that arise from sources external to the body, in particular, touch. Incongruence between the sensory and motor signals for the right arm was manipulated by having the participants make symmetrical or asymmetrical movements while watching a reflection of their left arm in a parasagittal mirror, or the left hand surface of a similarly positioned opaque board. In contrast to our prediction, sensitivity to the presence of gaps in tactile stimulation of the right forearm was not reduced when participants made asymmetrical movements during mirror visual feedback, as compared to when they made symmetrical or asymmetrical movements with no visual feedback. Instead, sensitivity was reduced when participants made symmetrical movements during mirror visual feedback relative to the other three conditions. We suggest that small discrepancies between sensory and motor information, as they occur during mirror visual feedback with symmetrical movements, can impair tactile processing. In contrast, asymmetrical movements with mirror visual feedback may not impact tactile processing because the larger discrepancies between sensory and motor information may prevent the integration of these sources of information. These results contrast with previous reports of anomalous sensations during exposure to both low and high sensorimotor conflict, but are nevertheless in agreement with a forward model interpretation of perceptual modulations during goal directed movement.     

Uncertainty, neuromodulation, and attention

Uncertainty in various forms plagues our interactions with the environment. In a Bayesian statistical framework, optimal inference and prediction, based on unreliable observations in changing contexts, require the representation and manipulation of different forms of uncertainty. We propose that the neuromodulators acetylcholine and norepinephrine play a major role in the brain's implementation of these uncertainty computations. Acetylcholine signals expected uncertainty, coming from known unreliability of predictive cues within a context. Norepinephrine signals unexpected uncertainty, as when unsignaled context switches produce strongly unexpected observations. These uncertainty signals interact to enable optimal inference and learning in noisy and changeable environments. This formulation is consistent with a wealth of physiological, pharmacological, and behavioral data implicating acetylcholine and norepinephrine in specific aspects of a range of cognitive processes. Moreover, the model suggests a class of attentional cueing tasks that involve both neuromodulators and shows how their interactions may be part-antagonistic, part-synergistic.     

Bayesian phylogenetic model selection using reversible jump Markov chain Monte Carlo

A common problem in molecular phylogenetics is choosing a model of DNA substitution that does a good job of explaining the DNA sequence alignment without introducing superfluous parameters. A number of methods have been used to choose among a small set of candidate substitution models, such as the likelihood ratio test, the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), and Bayes factors. Current implementations of any of these criteria suffer from the limitation that only a small set of models are examined, or that the test does not allow easy comparison of non-nested models. In this article, we expand the pool of candidate substitution models to include all possible time-reversible models. This set includes seven models that have already been described. We show how Bayes factors can be calculated for these models using reversible jump Markov chain Monte Carlo, and apply the method to 16 DNA sequence alignments. For each data set, we compare the model with the best Bayes factor to the best models chosen using AIC and BIC. We find that the best model under any of these criteria is not necessarily the most complicated one; models with an intermediate number of substitution types typically do best. Moreover, almost all of the models that are chosen as best do not constrain a transition rate to be the same as a transversion rate, suggesting that it is the transition/transversion rate bias that plays the largest role in determining which models are selected. Importantly, the reversible jump Markov chain Monte Carlo algorithm described here allows estimation of phylogeny (and other phylogenetic model parameters) to be performed while accounting for uncertainty in the model of DNA substitution.