Using Occam's razor and Bayesian modelling to compare discrete and continuous representations in numerosity judgements.

Previous research has established that numeric estimates are based not just on perceptual data but also past experience, and so may be influenced by the form of this stored information. It remains unclear, however, how such experience is represented: numerical data can be processed by either a continuous analogue number system or a discrete symbolic number system, with each predicting different generalisation effects. The present paper therefore contrasts discrete and continuous prior formats within the domain of numerical estimation using both direct comparisons of computational models of this process using these representations, as well as empirical contrasts exploiting different predicted reactions of these formats to uncertainty via Occam's razor. Both computational and empirical results indicate that numeric estimates commonly rely on a continuous prior format, mirroring the analogue approximate number system, or 'number sense'. This implies a general preference for the use of continuous numerical representations even where both stimuli and responses are discrete, with learners seemingly relying on innate number systems rather than the symbolic forms acquired in later life. There is however remaining uncertainty in these results regarding individual differences in the use of these systems, which we address in recommendations for future work.

Bayesian transfer in a complex spatial localization task.

Prior knowledge can help observers in various situations. Adults can simultaneously learn two location priors and integrate these with sensory information to locate hidden objects. Importantly, observers weight prior and sensory (likelihood) information differently depending on their respective reliabilities, in line with principles of Bayesian inference. Yet, there is limited evidence that observers actually perform Bayesian inference, rather than a heuristic, such as forming a look-up table. To distinguish these possibilities, we ask whether previously learned priors will be immediately integrated with a new, untrained likelihood. If observers use Bayesian principles, they should immediately put less weight on the new, less reliable, likelihood ("Bayesian transfer"). In an initial experiment, observers estimated the position of a hidden target, drawn from one of two distinct distributions, using sensory and prior information. The sensory cue consisted of dots drawn from a Gaussian distribution centered on the true location with either low, medium, or high variance; the latter introduced after block three of five to test for evidence of Bayesian transfer. Observers did not weight the cue (relative to the prior) significantly less in the high compared to medium variance condition, counter to Bayesian predictions. However, when explicitly informed of the different prior variabilities, observers placed less weight on the new high variance likelihood ("Bayesian transfer"), yet, substantially diverged from ideal. Much of this divergence can be captured by a model that weights sensory information, according only to internal noise in using the cue. These results emphasize the limits of Bayesian models in complex tasks.

Prior expectation of objects in space is dependent on the direction of gaze.

Many studies of multisensory spatial localization have shown that observers' responses are well-characterized by Bayesian inference, as localization judgments are influenced not only by the reliability of sensory encoding, but expectations about where things occur in space. Here, we investigate the frame of reference for the prior expectation of objects in space. Using an audiovisual localization task, we systematically manipulate fixation position and evaluate whether this prior is encoded in an eye-centered, head-centered, or hybrid frame of reference. Results show that in a majority of participants, this prior is encoded in an eye-centered frame of reference.

Opposing effects of reward and punishment on human vigor.

The vigor with which humans and animals engage in a task is often a determinant of the likelihood of the task's success. An influential theoretical model suggests that the speed and rate at which responses are made should depend on the availability of rewards and punishments. While vigor facilitates the gathering of rewards in a bountiful environment, there is an incentive to slow down when punishments are forthcoming so as to decrease the rate of punishments, in conflict with the urge to perform fast to escape punishment. Previous experiments confirmed the former, leaving the latter unanswered. We tested the influence of punishment in an experiment involving economic incentives and contrasted this with reward related behavior on the same task. We found that behavior corresponded with the theoretical model; while instantaneous threat of punishment caused subjects to increase the vigor of their response, subjects' response times would slow as the overall rate of punishment increased. We quantitatively show that this is in direct contrast to increases in vigor in the face of increased overall reward rates. These results highlight the opposed effects of rewards and punishments and provide further evidence for their roles in the variety of types of human decisions.

Fast and Accurate Learning When Making Discrete Numerical Estimates.

Many everyday estimation tasks have an inherently discrete nature, whether the task is counting objects (e.g., a number of paint buckets) or estimating discretized continuous variables (e.g., the number of paint buckets needed to paint a room). While Bayesian inference is often used for modeling estimates made along continuous scales, discrete numerical estimates have not received as much attention, despite their common everyday occurrence. Using two tasks, a numerosity task and an area estimation task, we invoke Bayesian decision theory to characterize how people learn discrete numerical distributions and make numerical estimates. Across three experiments with novel stimulus distributions we found that participants fell between two common decision functions for converting their uncertain representation into a response: drawing a sample from their posterior distribution and taking the maximum of their posterior distribution. While this was consistent with the decision function found in previous work using continuous estimation tasks, surprisingly the prior distributions learned by participants in our experiments were much more adaptive: When making continuous estimates, participants have required thousands of trials to learn bimodal priors, but in our tasks participants learned discrete bimodal and even discrete quadrimodal priors within a few hundred trials. This makes discrete numerical estimation tasks good testbeds for investigating how people learn and make estimates.

Vigor in the face of fluctuating rates of reward: an experimental examination.

Two fundamental questions underlie the expression of behavior, namely what to do and how vigorously to do it. The former is the topic of an overwhelming wealth of theoretical and empirical work particularly in the fields of reinforcement learning and decision-making, with various forms of affective prediction error playing key roles. Although vigor concerns motivation, and so is the subject of many empirical studies in diverse fields, it has suffered a dearth of computational models. Recently, Niv et al. [Niv, Y., Daw, N. D., Joel, D., & Dayan, P. Tonic dopamine: Opportunity costs and the control of response vigor. Psychopharmacology (Berlin), 191, 507-520, 2007] suggested that vigor should be controlled by the opportunity cost of time, which is itself determined by the average rate of reward. This coupling of reward rate and vigor can be shown to be optimal under the theory of average return reinforcement learning for a particular class of tasks but may also be a more general, perhaps hard-wired, characteristic of the architecture of control. We, therefore, tested the hypothesis that healthy human participants would adjust their RTs on the basis of the average rate of reward. We measured RTs in an odd-ball discrimination task for rewards whose magnitudes varied slowly but systematically. Linear regression on the subjects' individual RTs using the time varying average rate of reward as the regressor of interest, and including nuisance regressors such as the immediate reward in a round and in the preceding round, showed that a significant fraction of the variance in subjects' RTs could indeed be explained by the rate of experienced reward. This validates one of the key proposals associated with the model, illuminating an apparently mandatory form of coupling that may involve tonic levels of dopamine.

Separate encoding of model-based and model-free valuations in the human brain.

Behavioral studies have long shown that humans solve problems in two ways, one intuitive and fast (System 1, model-free), and the other reflective and slow (System 2, model-based). The neurobiological basis of dual process problem solving remains unknown due to challenges of separating activation in concurrent systems. We present a novel neuroeconomic task that predicts distinct subjective valuation and updating signals corresponding to these two systems. We found two concurrent value signals in human prefrontal cortex: a System 1 model-free reinforcement signal and a System 2 model-based Bayesian signal. We also found a System 1 updating signal in striatal areas and a System 2 updating signal in lateral prefrontal cortex. Further, signals in prefrontal cortex preceded choices that are optimal according to either updating principle, while signals in anterior cingulate cortex and globus pallidus preceded deviations from optimal choice for reinforcement learning. These deviations tended to occur when uncertainty regarding optimal values was highest, suggesting that disagreement between dual systems is mediated by uncertainty rather than conflict, confirming recent theoretical proposals.

The human prefrontal cortex mediates integration of potential causes behind observed outcomes.

Prefrontal cortex has long been implicated in tasks involving higher order inference in which decisions must be rendered, not only about which stimulus is currently rewarded, but also which stimulus dimensions are currently relevant. However, the precise computational mechanisms used to solve such tasks have remained unclear. We scanned human participants with functional MRI, while they performed a hierarchical intradimensional/extradimensional shift task to investigate what strategy subjects use while solving higher order decision problems. By using a computational model-based analysis, we found behavioral and neural evidence that humans solve such problems not by occasionally shifting focus from one to the other dimension, but by considering multiple explanations simultaneously. Activity in human prefrontal cortex was better accounted for by a model that integrates over all available evidences than by a model in which attention is selectively gated. Importantly, our model provides an explanation for how the brain determines integration weights, according to which it could distribute its attention. Our results demonstrate that, at the point of choice, the human brain and the prefrontal cortex in particular are capable of a weighted integration of information across multiple evidences.

Pavlovian-instrumental interaction in 'observing behavior'.

Subjects typically choose to be presented with stimuli that predict the existence of future reinforcements. This so-called 'observing behavior' is evident in many species under various experimental conditions, including if the choice is expensive, or if there is nothing that subjects can do to improve their lot with the information gained. A recent study showed that the activities of putative midbrain dopamine neurons reflect this preference for observation in a way that appears to challenge the common prediction-error interpretation of these neurons. In this paper, we provide an alternative account according to which observing behavior arises from a small, possibly Pavlovian, bias associated with the operation of working memory.

Causal inference in perception.

Until recently, the question of how the brain performs causal inference has been studied primarily in the context of cognitive reasoning. However, this problem is at least equally crucial in perceptual processing. At any given moment, the perceptual system receives multiple sensory signals within and across modalities and, for example, has to determine the source of each of these signals. Recently, a growing number of studies from various fields of cognitive science have started to address this question and have converged to very similar computational models. Therefore, it seems that a common computational strategy, which is highly consistent with a normative model of causal inference, is exploited by the perceptual system in a variety of domains.

Probability matching as a computational strategy used in perception.

The question of which strategy is employed in human decision making has been studied extensively in the context of cognitive tasks; however, this question has not been investigated systematically in the context of perceptual tasks. The goal of this study was to gain insight into the decision-making strategy used by human observers in a low-level perceptual task. Data from more than 100 individuals who participated in an auditory-visual spatial localization task was evaluated to examine which of three plausible strategies could account for each observer's behavior the best. This task is very suitable for exploring this question because it involves an implicit inference about whether the auditory and visual stimuli were caused by the same object or independent objects, and provides different strategies of how using the inference about causes can lead to distinctly different spatial estimates and response patterns. For example, employing the commonly used cost function of minimizing the mean squared error of spatial estimates would result in a weighted averaging of estimates corresponding to different causal structures. A strategy that would minimize the error in the inferred causal structure would result in the selection of the most likely causal structure and sticking with it in the subsequent inference of location-"model selection." A third strategy is one that selects a causal structure in proportion to its probability, thus attempting to match the probability of the inferred causal structure. This type of probability matching strategy has been reported to be used by participants predominantly in cognitive tasks. Comparing these three strategies, the behavior of the vast majority of observers in this perceptual task was most consistent with probability matching. While this appears to be a suboptimal strategy and hence a surprising choice for the perceptual system to adopt, we discuss potential advantages of such a strategy for perception.

Bayesian priors are encoded independently from likelihoods in human multisensory perception.

It has been shown that human combination of crossmodal information is highly consistent with an optimal Bayesian model performing causal inference. These findings have shed light on the computational principles governing crossmodal integration/segregation. Intuitively, in a Bayesian framework priors represent a priori information about the environment, i.e., information available prior to encountering the given stimuli, and are thus not dependent on the current stimuli. While this interpretation is considered as a defining characteristic of Bayesian computation by many, the Bayes rule per se does not require that priors remain constant despite significant changes in the stimulus, and therefore, the demonstration of Bayes-optimality of a task does not imply the invariance of priors to varying likelihoods. This issue has not been addressed before, but here we empirically investigated the independence of the priors from the likelihoods by strongly manipulating the presumed likelihoods (by using two drastically different sets of stimuli) and examining whether the estimated priors change or remain the same. The results suggest that the estimated prior probabilities are indeed independent of the immediate input and hence, likelihood.

Human trimodal perception follows optimal statistical inference.

Our nervous system typically processes signals from multiple sensory modalities at any given moment and is therefore posed with two important problems: which of the signals are caused by a common event, and how to combine those signals. We investigated human perception in the presence of auditory, visual, and tactile stimulation in a numerosity judgment task. Observers were presented with stimuli in one, two, or three modalities simultaneously and were asked to report their percepts in each modality. The degree of congruency between the modalities varied across trials. For example, a single flash was paired in some trials with two beeps and two taps. Cross-modal illusions were observed in most conditions in which there was incongruence among the two or three stimuli, revealing robust interactions among the three modalities in all directions. The observers' bimodal and trimodal percepts were remarkably consistent with a Bayes-optimal strategy of combining the evidence in each modality with the prior probability of the events. These findings provide evidence that the combination of sensory information among three modalities follows optimal statistical inference for the entire spectrum of conditions.