Humans can meaningfully report their confidence in a perceptual or cognitive decision. It is widely believed that these reports reflect the Bayesian probability that the decision is correct, but this hypothesis has not been rigorously tested against non-Bayesian alternatives. We use two perceptual categorization tasks in which Bayesian confidence reporting requires subjects to take sensory uncertainty into account in a specific way. We find that subjects do take sensory uncertainty into account when reporting confidence, suggesting that brain areas involved in reporting confidence can access low-level representations of sensory uncertainty, a prerequisite of Bayesian inference. However, behavior is not fully consistent with the Bayesian hypothesis and is better described by simple heuristic models that use uncertainty in a non-Bayesian way. Both conclusions are robust to changes in the uncertainty manipulation, task, response modality, model comparison metric, and additional flexibility in the Bayesian model. Our results suggest that adhering to a rational account of confidence behavior may require incorporating implementational constraints.
People often have a sense of a level of confidence about their decisions. Such a “feeling of knowing” [1,2] may serve to improve performance in subsequent decisions [3], learning [1], and group decision-making [4]. Much recent work has focused on identifying brain regions and neural mechanisms responsible for the computation of confidence in humans [5–7], nonhuman primates [8–10], and rodents [11]. In the search for the neural correlates of confidence, the leading premise has been that confidence is Bayesian, i. e. , the observer’s estimated probability that a choice is correct [1,12–14]. In human studies, however, naïve subjects can give a meaningful answer when you ask them to rate their confidence about a decision [15]; thus, “confidence” intrinsically means something to people, and it is not a foregone conclusion that this intrinsic sense corresponds to the Bayesian definition. Therefore, we regard the above “definition” as a testable hypothesis about the way the brain computes explicit confidence reports; we use Bayesian decision theory to formalize this hypothesis. Bayesian decision theory provides a general and often quantitatively accurate account of perceptual decisions in a wide variety of tasks [16–18]. According to this theory, the decision-maker combines knowledge about the statistical structure of the world with the present sensory input to compute a posterior probability distribution over possible states of the world. In principle, a confidence report might be derived from the same posterior distribution; this is the hypothesis described above, which we will call the Bayesian confidence hypothesis (BCH). The main goal of this paper is to test that hypothesis. Recent studies have attempted to test the BCH [19,20] but, because of their experimental designs, are unable to meaningfully distinguish the Bayesian model from any other model of confidence. Recent work has proposed possible qualitative signatures of Bayesian confidence [21]. However, the observation (or lack thereof) of these signatures provides an uncertain amount of evidence in favor of (or against) the Bayesian model, and the signatures are therefore not useful for determining which computations underlie confidence reports [22]. To objectively and quantitatively determine whether confidence ratings appear to be Bayesian, we use a formal model comparison approach. We test the predictions of the BCH as we vary the quality of the sensory evidence and the task structure within individuals. We compare Bayesian models against a variety of alternative models, something that is important for the epistemological standing of Bayesian claims [23,24]. We find that the BCH qualitatively describes human behavior but that quantitatively, even the most flexible Bayesian model is outperformed by models that take uncertainty into account in a non-Bayesian way.
During each session, each subject completed two orientation categorization tasks, Tasks A and B. On each trial, a category C was selected randomly (both categories were equally probable), and a stimulus s was drawn from the corresponding stimulus distribution and displayed. The subject categorized the stimulus and simultaneously reported their confidence on a 4-point scale, with a single button press (Fig 1a). Using a single button press for choice and confidence may prevent post-choice influences on the confidence judgment ([25], but see [26]) and emphasized that confidence should reflect the observer’s perception rather than a preceding motor response. The categories were defined by normal distributions on orientation, which differed by task (Fig 1b). In Task A, the distributions had different means (±μC) and the same standard deviation (σC); leftward-tilting stimuli were more likely to be from category 1. Variants of Task A are common in decision-making studies [27]. In Task B, the distributions had the same mean (0°) and different standard deviations (σ1, σ2); stimuli around the horizontal were more likely to be from category 1. Variants of Task B are less common [28–30] but have some properties of perceptual organization tasks; for example, a subject may have to detect when a stimulus belongs to a narrow category (e. g. , in which two line segments are collinear) that is embedded in a a broader category (e. g. , in which two line segments are unrelated). Subjects were highly trained on the categories; during training, we only used highest-reliability stimuli, and we provided trial-to-trial category correctness feedback. Subjects were then tested with 6 different reliability levels, which were chosen randomly on each trial. During testing, correctness feedback was withheld to avoid the possibility that confidence simply reflects a learned mapping between stimulus orientation and reliability and the probability of being correct [30–33]. Because we are interested in subjects’ intrinsic computation of confidence, we did not instruct or incentivize them to assign probability ranges to each button (e. g. , by using a scoring rule [34–36]). If we had, we would have essentially been training subjects to use a specific model of confidence. To ensure that our results were independent of stimulus type, we used two kinds of stimuli. Some subjects saw oriented drifting Gabors; for these subjects, stimulus reliability was manipulated through contrast. Other subjects saw oriented ellipses; for these subjects, stimulus reliability was manipulated through ellipse elongation (Fig 1c). We found no major differences in model rankings between Gabor and ellipse subjects, therefore we will make no distinctions between the groups. For modeling purposes, we assume that the observer’s internal representation of the stimulus is a noisy measurement x, drawn from a Gaussian distribution with mean s and s. d. σ (Fig 1d and 1e). In the model, σ (i. e. , uncertainty) is a fitted function of stimulus reliability. A Bayes-optimal observer uses knowledge of the generative model to make a decision that maximizes the probability of being correct. Here, when the measurement on a given trial is x, this strategy amounts to choosing the category C for which the posterior probability p (C ∣ x) is highest. This is equivalent to reporting category 1 when the log posterior ratio, d = log p (C = 1 ∣ x) p (C = 2 ∣ x), is positive. In Task A, d is d A = 2 x μ C σ 2 + σ C 2. Therefore, the ideal observer reports category 1 when x is positive; this is the structure of many psychophysical tasks [37]. In Task B, however, d is d B = 1 2 log σ 2 + σ 2 2 σ 2 + σ 1 2 - σ 2 2 - σ 1 2 2 (σ 2 + σ 1 2) (σ 2 + σ 2 2) x 2; the observer needs both x and σ in order to make an optimal decision. From the point of view of the observer, σ is the trial-to-trial level of sensory uncertainty associated with the measurement [38]. In a minor variation of the optimal observer, we allow for the possibility that the observer’s prior belief over category, p (C), is different from the true value of (0. 5,0. 5); this adds a constant to dA and dB. We introduce the Bayesian confidence hypothesis (BCH), stating that confidence reports depend on the internal representation of the stimulus (here x) only via d. In the BCH, the observer chooses a response by comparing d to a set of category and confidence boundaries. For example, whenever d falls within a certain range, the observer presses the “medium-low confidence, category 2” button. The BCH is thus an extension of the choice model described above, wherein the value of d is used to compute confidence as well as chosen category. There is another way of thinking about this. Bayesian models assume that subjects compute d in order to make an optimal choice. Assuming people compute d at all, are they able to use it to report confidence as well? We refer to the Bayesian model here as simply “Bayes. ” We also tested several more constrained versions of this model. The observer’s decision can be summarized as a mapping from a combination of a measurement and an uncertainty level (x, σ) to a response that indicates both category and confidence. We can visualize this mapping as in Fig 2, first column. It is clear that the pattern of decision boundaries in the BCH is qualitatively very different between Task A and Task B. In Task A, the decision boundaries are quadratic functions of uncertainty; confidence decreases monotonically with uncertainty and increases with the distance of the measurement from 0. In Task B, the decision boundaries are neither linear nor quadratic. At first glance, it seems obvious that sensory uncertainty is relevant to the computation of confidence. However, this is by no means a given; in fact, a prominent proposal is that confidence is based on the distance between the measurement and the decision boundary, without any role for sensory uncertainty [10,11,39]. Therefore, we tested a model (Fixed) in which the response is a function of the measurement alone (equivalent to a maximum likelihood estimate of the stimulus orientation), and not of the uncertainty of that measurement (Fig 2, second column). We also tested heuristic models in which the subject uses their knowledge of their sensory uncertainty but does not compute a posterior distribution over category. We have previously classified such models as probabilistic non-Bayesian [40]. In the Orientation Estimation model, subjects base their response on a maximum a posteriori estimate of orientation (rather than category), using the mixture of the two stimulus distributions as a prior distribution. In the Linear Neural model, subjects base their response on a linear function of the output of a hypothetical population of neurons. We derived two additional probabilistic non-Bayesian models, Lin and Quad, from the observation that the Bayesian decision criteria are an approximately linear function of uncertainty in some measurement regimes and approximately quadratic in others. These models are able to produce approximately Bayesian behavior without actually performing any computation of the posterior. In Lin and Quad, subjects base their response on a linear or a quadratic function of x and σ, respectively. A comparison of the Lin and Quad columns to the Bayes column in Fig 2 demonstrates that Lin and Quad can approximate the Bayesian mapping from (x, σ) to response despite not being based on the Bayesian decision variable. All of the models we tested were variants of the six models described so far (Bayes, Fixed, Orientation Estimation, Linear Neural, Lin, Quad). Each trial consists of the experimentally determined orientation and reliability level and the subject’s category and confidence response (an integer between 1 and 8). This is a very rich data set, which we summarize in Fig 3. We find the following effects: performance and confidence increase as a function of reliability (Fig 3a, 3b, 3h and 3i), and high-confidence reports are less frequent than low-confidence reports (Fig 3e and 3f). Note Fig 3c and 3d especially; this is the projection of the data that we will use to demonstrate model fits for the rest of this paper. We use this projection because the vertical axis (mean button press) most closely approximates the form of the raw data. Additionally, because our models are differentiated by how they use uncertainty, it is informative to plot how response changes as a function of reliability, in addition to category and task. Recently, a measure of the degree of association between accuracy and confidence, meta-d′, has been developed [41,42]. While it can be useful for characterizing individual differences, we do not include it in our analyses or display it in Fig 3. That is because one strength of our experimental design is that we parametrically vary stimulus strength and stimulus reliability; this differs from papers in which meta-d′ plays a central role because, in those papers, the stimulus is often only a binary category. We used Markov Chain Monte Carlo (MCMC) sampling to fit models to raw individual-subject data. To account for overfitting, we compared models using leave-one-out cross-validated log likelihood scores (LOO) computed with the full posteriors obtained through MCMC [43]. A model recovery analysis ensured that our models are meaningfully distinguishable (Methods). Unless otherwise noted, models were fit jointly to Task A and B category and confidence responses.
Our study has several limitations. For instance, because of our short presentation time, we cannot say much about how our results generalize to tasks that require integration of evidence over time [8,53–55]. Additionally, because our stimuli are very low-level, we cannot say much about high-level stimuli like faces [56]. Also, we only considered explicit confidence ratings, which differ from the implicit confidence that can be gathered from humans (e. g. , by presenting two tasks and asking the subject to choose which one they feel more confident about completing correctly [57,58]) or from nonhuman animals [13] (e. g. , by measuring how frequently they decline to make a difficult choice [8], or how long they will wait for a reward [11]). It is possible that implicit confidence might be more Bayesian than explicit confidence; Barthelmé and Mamassian [58] conduct an implicit confidence experiment and rule out some heuristic models. However, their experimental task is substantially different from the one presented here. In their experiment, the stimulus feature of interest (orientation) only takes on two values rather than varying parametrically, so it requires a different class of heuristic models. Future studies of the difference between implicit and explicit confidence should use experiments that are able to distinguish the models presented here, which has not been done. Like the present study, Aitchison et al. [19] found evidence that confidence reports may emerge from heuristic computations. However, they sampled stimuli from only a small region of their two-dimensional space, where model predictions may not vary greatly. Therefore, their stimulus set did not allow for the models to be strongly distinguished. Furthermore, although they tested for Bayesian computation, they did not test for probabilistic computation (whether observers take sensory uncertainty into account on a trial-to-trial basis [40]) as we do here. Such a test requires that the experimenter vary the reliability, not only the value, of the stimulus feature of interest. Navajas et al. [49] suggested that confidence reports are best described as a weighted average of precision and the probability of being correct. However, their model uses the estimated probability of being correct under a non-Bayesian decision rule [22]. They did not show the fit of a Bayesian model, and therefore their study does not constitute a true test of whether confidence is Bayesian. Here, we tested and rejected the hypothesis that confidence is a weighted average of precision and the posterior probability of being correct under a Bayesian decision rule. Sanders et al. [20] reported that confidence has a “statistical” nature. However, their experiment was unable to determine whether confidence is Bayesian or not [17], because the stimuli varied along only one dimension. Aitchison et al. [19] note that, to distinguish models of confidence, the experimenter must use stimuli that are characterized by two dimensions (e. g. , contrast and orientation as in this experiment, or contrast and crowding as in Barthelmé and Mamassian [58]). This is because, when fitting models that map from an internal variable to an integer confidence rating, it is impossible to distinguish between two internal variables that are monotonically related (in the case of Sanders et al. [20], the measurement and the posterior probability of being correct). Therefore, the only alternative model proposed by Sanders et al. [20] is based on reaction time, rather than on the presented stimuli. In detection and coarse discrimination tasks, Lau, Rahnev, and colleagues report that subjects overestimate their confidence in the periphery and for unattended stimuli. The authors have proposed a signal detection theory model in which high eccentricity or lower attention induces higher noise, and the confidence criterion may not change at all [39,59–63]. As a result, more probability mass will “spill over” the criterion to the high-confidence regime. How do these findings relate to ours? At a qualitative level, they are consistent in that confidence does not seem Bayesian. However, in detection and coarse discrimination tasks, it is not possible to distinguish between fixed-criterion and probabilistic models [64], and their data cannot be used to infer that the criterion is fixed. The paradigms in the present paper are able to distinguish such models because of the parametric manipulation of orientation, the stimulus feature of interest; indeed, we find strong evidence against the Fixed criterion model. It remains to be seen whether claims that confidence can be systematically dissociated from perceptual performance [1,51,65–69] are consistent with the account presented here, in which the brain adjusts confidence criteria based on uncertainty but in a non-Bayesian manner. Another form of non-Bayesian confidence ratings is the recent proposal that, in confidence judgments, only the “positive evidence” in favor of the chosen option matters, instead of the “balance of evidence” between two options [31,53,56,70,71]. In our tasks, this form of suboptimality would entail that confidence is derived from the (log) likelihood of the chosen category, instead of from the (log) likelihood ratio. This does not seem consistent with our data; for example, the likelihood of category 2 decreases as the absolute value of the stimulus and, correspondingly, the measurement, increases (Fig 7b). However, confidence for a category 2 decision steadily increases with the absolute value of the stimulus (Fig 7a). More work is needed to understand whether alternative models could explain the “positive evidence” data, and if not, what causes the difference with our results. What do our findings tell us about the neural basis of confidence? Previous studies have found that neural activity in some brain areas (e. g. , human medial temporal lobe [7] and prefrontal cortex [72], monkey lateral intraparietal cortex [8] and pulvinar [10], rodent orbitofrontal cortex [11]) is associated with behavioral indicators of confidence, and/or with the distance of a stimulus to a decision boundary. However, such studies mostly used stimuli that vary along a single dimension (e. g. , net retinal dot motion energy, mixture of two odors). Because measurement is indistinguishable from the probability of being correct in these classes of tasks, neural activity associated with confidence may represent either the measurement or the probability of being correct [19]. In addition to the recommendation of Aitchison et al. [19] to distinguish between these possibilities by varying stimuli along two dimensions, we recommend fitting both Bayesian and non-Bayesian probabilistic models to behavior. In view of the relatively poor performance of the Bayesian models in the present study, the proposal [12] to correlate behavior and neural activity with predictions of the Bayesian confidence model should be viewed with skepticism. Our results raise general issues about the status of Bayesian models as descriptions of behavior. First, because it is impossible to exhaustively test all models that might be considered “Bayesian, ” we cannot rule out the entire class of models. However, we have tried to alleviate this issue as much as possible by testing a large number of Bayesian models—far more than the number of Bayesian and non-Bayesian models tested in other studies of confidence. Second, Bayesian models are often held in favor for their generalizability; one can determine the performance-maximizing strategy for any task. Although generalizability indeed makes Bayesian models attractive and powerful, we do not believe that this property should override a bad fit. One could take two different views of our heuristic model results. The first view is that the heuristics should be taken seriously as principled models [73]; here, the challenge is to demonstrate that they describe behavior in a variety of tasks and can be motivated based on underlying principles. The second view is that these are descriptive models simply meant to demonstrate that a simple model can provide a good fit to the data; here, the heuristics are benchmarks for more principled models, and the challenge is to find a principled model that fits the data as well as the heuristics. We lean towards the second view and interpret our results as demonstrating that the purest form of the Bayesian confidence hypothesis does not describe human confidence reports particularly well. However, one might still conclude, after examining the fits of the Bayesian model, that the behavior is “approximately Bayesian” rather than “non-Bayesian. ” As written, this is a semantic distinction because it relies on one’s definition of “approximate. ” However, it can be turned into a more meaningful question: Are the differences between human behavior and Bayesian models accounted for by an unknown principle, such as an ecologically relevant objective function that includes both task performance and biological constraints? Although there are benefits associated with veridical explicit representations of confidence [74–76], there are also neural constraints that may give rise to non-Bayesian behavior [23,24]. Such constraints include the kinds of operations that neurons can perform, the high energy cost of spiking [77,78], and the cost of neural wiring length [79,80]. A search for ecologically rational constraints on Bayesian computation benefits from a positive characterization of the deviations from Bayesian computation, in the form of heuristic models such as Lin and Quad. Specifically, one could define neural networks with various combinations of constraints, and train them as if they were psychophysical subjects in our tasks. After training, one could fit behavioral models to them; this approach has already shown that the output from such neural networks is sometimes best described by heuristic models [81]. Using model ranking as a measure of similarity, one could determine which network architecture and training procedure produces confidence behavior that is most similar to that of humans. This could reveal which constraints are responsible for the specific deviations from Bayesian computation that we have observed.
The experiments were approved by the University Committee on Activities Involving Human Subjects of New York University. Informed consent was given by each subject before the experiment. This control experiment was identical to experiment 1 except for the following modifications: This experiment was identical to experiment 1 except for the following modifications:
Humans are able to report a sense of confidence in decisions that we make. It is widely hypothesized that confidence reflects the computed probability that a decision is accurate; however, this hypothesis has not been fully explored. We use several human behavioral experiments to test a variety of models that may be considered to be distinct hypotheses about the computational underpinnings of confidence. We find that reported confidence does not appear to reflect the probability that a decision is correct, but instead emerges from a heuristic approximation of this probability.
statistical noise
ellipses
decision making
statistics
social sciences
geometry
neuroscience
cognitive psychology
mathematics
probability distribution
cognition
research and analysis methods
gaussian noise
animal cells
behavior
research assessment
probability theory
psychology
cellular neuroscience
cell biology
normal distribution
research validity
neurons
biology and life sciences
cellular types
physical sciences
cognitive science
Comparing Bayesian and non-Bayesian accounts of human confidence reports