Humans have been shown to adapt to the temporal statistics of timing tasks so as to optimize the accuracy of their responses, in agreement with the predictions of Bayesian integration. This suggests that they build an internal representation of both the experimentally imposed distribution of time intervals (the prior) and of the error (the loss function). The responses of a Bayesian ideal observer depend crucially on these internal representations, which have only been previously studied for simple distributions. To study the nature of these representations we asked subjects to reproduce time intervals drawn from underlying temporal distributions of varying complexity, from uniform to highly skewed or bimodal while also varying the error mapping that determined the performance feedback. Interval reproduction times were affected by both the distribution and feedback, in good agreement with a performance-optimizing Bayesian observer and actor model. Bayesian model comparison highlighted that subjects were integrating the provided feedback and represented the experimental distribution with a smoothed approximation. A nonparametric reconstruction of the subjective priors from the data shows that they are generally in agreement with the true distributions up to third-order moments, but with systematically heavier tails. In particular, higher-order statistical features (kurtosis, multimodality) seem much harder to acquire. Our findings suggest that humans have only minor constraints on learning lower-order statistical properties of unimodal (including peaked and skewed) distributions of time intervals under the guidance of corrective feedback, and that their behavior is well explained by Bayesian decision theory.

Humans have been shown to combine noisy sensory information with previous experience (priors), in qualitative and sometimes quantitative agreement with the statistically-optimal predictions of Bayesian integration. However, when the prior distribution becomes more complex than a simple Gaussian, such as skewed or bimodal, training takes much longer and performance appears suboptimal. It is unclear whether such suboptimality arises from an imprecise internal representation of the complex prior, or from additional constraints in performing probabilistic computations on complex distributions, even when accurately represented. Here we probe the sources of suboptimality in probabilistic inference using a novel estimation task in which subjects are exposed to an explicitly provided distribution, thereby removing the need to remember the prior. Subjects had to estimate the location of a target given a noisy cue and a visual representation of the prior probability density over locations, which changed on each trial. Different classes of priors were examined (Gaussian, unimodal, bimodal). Subjects' performance was in qualitative agreement with the predictions of Bayesian Decision Theory although generally suboptimal. The degree of suboptimality was modulated by statistical features of the priors but was largely independent of the class of the prior and level of noise in the cue, suggesting that suboptimality in dealing with complex statistical features, such as bimodality, may be due to a problem of acquiring the priors rather than computing with them. We performed a factorial model comparison across a large set of Bayesian observer models to identify additional sources of noise and suboptimality. Our analysis rejects several models of stochastic behavior, including probability matching and sample-averaging strategies. Instead we show that subjects' response variability was mainly driven by a combination of a noisy estimation of the parameters of the priors, and by variability in the decision process, which we represent as a noisy or stochastic posterior.

The precision of multisensory perception improves when cues arising from the same cause are integrated, such as visual and vestibular heading cues for an observer moving through a stationary environment. In order to determine how the cues should be processed, the brain must infer the causal relationship underlying the multisensory cues. In heading perception, however, it is unclear whether observers follow the Bayesian strategy, a simpler non-Bayesian heuristic, or even perform causal inference at all. We developed an efficient and robust computational framework to perform Bayesian model comparison of causal inference strategies, which incorporates a number of alternative assumptions about the observers. With this framework, we investigated whether human observers’ performance in an explicit cause attribution and an implicit heading discrimination task can be modeled as a causal inference process. In the explicit causal inference task, all subjects accounted for cue disparity when reporting judgments of common cause, although not necessarily all in a Bayesian fashion. By contrast, but in agreement with previous findings, data from the heading discrimination task only could not rule out that several of the same observers were adopting a forced-fusion strategy, whereby cues are integrated regardless of disparity. Only when we combined evidence from both tasks we were able to rule out forced-fusion in the heading discrimination task. Crucially, findings were robust across a number of variants of models and analyses. Our results demonstrate that our proposed computational framework allows researchers to ask complex questions within a rigorous Bayesian framework that accounts for parameter and model uncertainty.

The precision of multisensory heading perception improves when visual and vestibular cues arising from the same cause, namely motion of the observer through a stationary environment, are integrated. Thus, in order to determine how the cues should be processed, the brain must infer the causal relationship underlying the multisensory cues.In heading perception, however, it is unclear whether observers follow the Bayesian strategy, a simpler non-Bayesian heuristic, or even perform causal inference at all. We developed an efficient and robust computational framework to perform Bayesian model comparison of causal inference strategies, which incorporates a number of alternative assumptions about the observers. With this framework, we investigated whether human observers' performance in an explicit cause attribution and an implicit heading discrimination task can be modeled as a causal inference process. In the explicit inference task, all subjects accounted for cue disparity when reporting judgments of common cause, although not necessarily all in a Bayesian fashion. By contrast, but in agreement with previous findings, data from the heading discrimination task only could not rule out that several of the same observers were adopting a forced-fusion strategy, whereby cues are integrated regardless of disparity. Only when we combined evidence from both tasks we were able to rule out forced-fusion in the heading discrimination task. Crucially, findings were robust across a number of variants of models and analyses. Our results demonstrate that our proposed computational framework allows researchers to ask complex questions within a rigorous Bayesian framework that accounts for parameter and model uncertainty.

The fate of scientific hypotheses often relies on the ability of a computational model to explain the data, quantified in modern statistical approaches by the likelihood function. The log-likelihood is the key element for parameter estimation and model evaluation. However, the log-likelihood of complex models in fields such as computational biology and neuroscience is often intractable to compute analytically or numerically. In those cases, researchers can often only estimate the log-likelihood by comparing observed data with synthetic observations generated by model simulations. Standard techniques to approximate the likelihood via simulation either use summary statistics of the data or are at risk of producing substantial biases in the estimate. Here, we explore another method, inverse binomial sampling (IBS), which can estimate the log-likelihood of an entire data set efficiently and without bias. For each observation, IBS draws samples from the simulator model until one matches the observation. The log-likelihood estimate is then a function of the number of samples drawn. The variance of this estimator is uniformly bounded, achieves the minimum variance for an unbiased estimator, and we can compute calibrated estimates of the variance. We provide theoretical arguments in favor of IBS and an empirical assessment of the method for maximum-likelihood estimation with simulation-based models. As case studies, we take three model-fitting problems of increasing complexity from computational and cognitive neuroscience. In all problems, IBS generally produces lower error in the estimated parameters and maximum log-likelihood values than alternative sampling methods with the same average number of samples. Our results demonstrate the potential of IBS as a practical, robust, and easy to implement method for log-likelihood evaluation when exact techniques are not available.

Optimal sensory decision-making requires the combination of uncertain sensory signals with prior expectations. The effect of prior probability is often described as a shift in the decision criterion. Can observers track sudden changes in probability? To answer this question, we used a change-point detection paradigm that is frequently used to examine behavior in changing environments. In a pair of orientation-categorization tasks, we investigated the effects of changing probabilities on decision-making. In both tasks, category probability was updated using a sample-and-hold procedure. We developed an ideal Bayesian change-point detection model in which the observer marginalizes over both the current run length (i.e., time since last change) and the current category probability. We compared this model to various alternative models that correspond to different strategies -from approximately Bayesian to simple heuristics -that the observers may have adopted to update their beliefs about probabilities. We find that probability is estimated following an exponential averaging model with a bias towards equal priors, consistent with a conservative bias. The mechanism underlying change of decision criterion is a combination of on-line estimation of prior probability and a stable, long-term equal-probability prior, thus operating at two very different timescales. Author summaryWe demonstrate how people learn and adapt to changes to the probability of occurrence of one of two categories on decision-making under uncertainty. The study combined psychophysical behavioral tasks with computational modeling. We used two behavioral tasks: a typical forced-choice categorization task as well as one in which the observer specified the decision criterion to use on each trial before the stimulus was displayed. We formulated an ideal Bayesian change-point detection model and compared it to several alternative models. We found that the data are best fit by a model that estimates category probability based on recently observed exemplars with a bias towards equal probability. Our results suggest that the brain takes multiple relevant time scales into account when setting category expectations. Introduction 1 Sensory decision-making involves making decisions under uncertainty. Furthermore, 2 optimal sensory decision-making requires the combination of uncertain sensory signals 3 PLOS 1/24 with prior expectations. Perceptual models of decision-making often incorporate prior 4 expectations to describe human behavior. In Bayesian models, priors are combined 5 with likelihoods to compute a posterior [1]. In signal detection theory, the effect of 6 unequal probabilities (signal present vs. absent) is a shift of the decision criterion [2]. 7 The effects of prior probability on the decision criterion have been observed in 8 detection [2-4], line tilt [5], numerosity estimation [6, 7], recognition memory [8], and 9 perceptual categorization [9] tasks, among others. These studies generally use explicit 10 priors, assume a fixed effect, and treat lear...

Optimal sensory decision-making requires the combination of uncertain sensory signals with prior expectations. The effect of prior probability is often described as a shift in the decision criterion. Can observers track sudden changes in probability? To answer this question, we used a change-point detection paradigm that is frequently used to examine behavior in changing environments. In a pair of orientation-categorization tasks, we investigated the effects of changing probabilities on decision-making. In both tasks, category probability was updated using a sample-and-hold procedure: probability was held constant for a period of time before jumping to another probability state that was randomly selected from a predetermined set of probability states. We developed an ideal Bayesian change-point detection model in which the observer marginalizes over both the current run length (i.e., time since last change) and the current category probability. We compared this model to various alternative models that correspond to different strategies—from approximately Bayesian to simple heuristics—that the observers may have adopted to update their beliefs about probabilities. While a number of models provided decent fits to the data, model comparison favored a model in which probability is estimated following an exponential averaging model with a bias towards equal priors, consistent with a conservative bias, and a flexible variant of the Bayesian change-point detection model with incorrect beliefs. We interpret the former as a simpler, more biologically plausible explanation suggesting that the mechanism underlying change of decision criterion is a combination of on-line estimation of prior probability and a stable, long-term equal-probability prior, thus operating at two very different timescales.

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