Belief Update Research Articles

Two Views of Constrained Differential Privacy: Belief Revision and Update

In this paper, we provide two views of constrained differential private (DP) mechanisms. The first one is as belief revision. A constrained DP mechanism is obtained by standard probabilistic conditioning, and hence can be naturally implemented by Monte Carlo algorithms. The other is as belief update. A constrained DP is defined according to l2-distance minimization postprocessing or projection and hence can be naturally implemented by optimization algorithms. The main advantage of these two perspectives is that we can make full use of the machinery of belief revision and update to show basic properties for constrained differential privacy especially some important new composition properties. Within the framework established in this paper, constrained DP algorithms in the literature can be classified either as belief revision or belief update. At the end of the paper, we demonstrate their differences especially in utility on a couple of scenarios.

De se names

AbstractWe argue that there are names with de se contents and that they are theoretically fruitful. De se names serve to challenge intuitive and otherwise plausible orthodoxies such as Stalnaker's view of communication and Bayesian views of belief update. These implications are also significant for those already sympathetic to the irreducibility of de se content.

Neural Evidence for Boundary Updating as the Source of the Repulsive Bias in Classification.

Binary classification, an act of sorting items into two classes by setting a boundary, is biased by recent history. One common form of such bias is repulsive bias, a tendency to sort an item into the class opposite to its preceding items. Sensory-adaptation and boundary-updating are considered as two contending sources of the repulsive bias, yet no neural support has been provided for either source. Here, we explored human brains of both men and women, using functional magnetic resonance imaging (fMRI), to find such support by relating the brain signals of sensory-adaptation and boundary-updating to human classification behavior. We found that the stimulus-encoding signal in the early visual cortex adapted to previous stimuli, yet its adaptation-related changes were dissociated from current choices. Contrastingly, the boundary-representing signals in the inferior-parietal and superior-temporal cortices shifted to previous stimuli and covaried with current choices. Our exploration points to boundary-updating, rather than sensory-adaptation, as the origin of the repulsive bias in binary classification.SIGNIFICANCE STATEMENT Many animal and human studies on perceptual decision-making have reported an intriguing history effect called "repulsive bias," a tendency to classify an item as the opposite class of its previous item. Regarding the origin of repulsive bias, two contending ideas have been proposed: "bias in stimulus representation because of sensory adaptation" versus "bias in class-boundary setting because of belief updating." By conducting model-based neuroimaging experiments, we verified their predictions about which brain signal should contribute to the trial-to-trial variability in choice behavior. We found that the brain signal of class boundary, but not stimulus representation, contributed to the choice variability associated with repulsive bias. Our study provides the first neural evidence supporting the boundary-based hypothesis of repulsive bias.

Belief Decay or Persistence? A Mixed‐method Study on Belief Movement Over Time

AbstractWhen individuals encounter new information (data), that information is incorporated with their existing beliefs (prior) to form a new belief (posterior) in a process referred to as belief updating. While most studies on rational belief updating in visual data analysis elicit beliefs immediately after data is shown, we posit that there may be critical movement in an individual's beliefs when elicited immediately after data is shown v. after a temporal delay (e.g., due to forgetfulness or weak incorporation of the data). Our paper investigates the hypothesis that posterior beliefs elicited after a time interval will “decay” back towards the prior beliefs compared to the posterior beliefs elicited immediately after new data is presented. In this study, we recruit 101 participants to complete three tasks where beliefs are elicited immediately after seeing new data and again after a brief distractor task. We conduct (1) a quantitative analysis of the results to understand if there are any systematic differences in beliefs elicited immediately after seeing new data or after a distractor task and (2) a qualitative analysis of participants' reflections on the reasons for their belief update. While we find no statistically significant global trends across the participants beliefs elicited immediately v. after the delay, the qualitative analysis provides rich insight into the reasons for an individual's belief movement across 9 prototypical scenarios, which includes (i) decay of beliefs as a result of either forgetting the information shown or strongly held prior beliefs, (ii) strengthening of confidence in updated beliefs by positively integrating the new data and (iii) maintaining a consistently updated belief over time, among others. These results can guide subsequent experiments to disambiguate when and by what mechanism new data is truly incorporated into one's belief system.

No evidence of biased updating in beliefs about absolute performance: A replication and generalization of Grossman and Owens (2012)

Many studies report that following feedback, individuals do not update their beliefs enough (a conservatism bias), and react more to good news than to bad news (an asymmetry bias), consistent with the idea of motivated beliefs. In the literature on conservatism and asymmetric updating, however, only one prior study focuses on judgments on absolute performance (Grossman & Owens, 2012), which finds that belief updating is well described by the Bayesian benchmark in that case. Here, we set out to test the replicability of these results and their robustness across several experimental manipulations, varying the uncertainty of participants’ priors, the tasks to perform, the format of beliefs and the elicitation rules used to incentivize these beliefs. We also introduce new measures of ego-relevance of these beliefs, and of the credibility of the feedback received by participants. Overall, we confirm across various experimental conditions that individuals exhibit no conservatism and asymmetry bias when they update their beliefs about their absolute performance. As in Grossman & Owens (2012), most observations are well-described by a Bayesian benchmark in our data. These results suggest a limit to the manifestation of motivated beliefs, and call for more research on the conditions under which biases in belief updating occur.

Computational mechanisms of belief updating in relation to psychotic-like experiences.

Psychotic-like experiences (PLEs) may occur due to changes in weighting prior beliefs and new evidence in the belief updating process. It is still unclear whether the acquisition or integration of stable beliefs is altered, and whether such alteration depends on the level of environmental and belief precision, which reflects the associated uncertainty. This motivated us to investigate uncertainty-related dynamics of belief updating in relation to PLEs using an online study design. We selected a sample (n = 300) of participants who performed a belief updating task with sudden change points and provided self-report questionnaires for PLEs. The task required participants to observe bags dropping from a hidden helicopter, infer its position, and dynamically update their belief about the helicopter's position. Participants could optimize performance by adjusting learning rates according to inferred belief uncertainty (inverse prior precision) and the probability of environmental change points. We used a normative learning model to examine the relationship between adherence to specific model parameters and PLEs. PLEs were linked to lower accuracy in tracking the outcome (helicopter location) (β = 0.26 ± 0.11, p = 0.018) and to a smaller increase of belief precision across observations after a change point (β = -0.003 ± 0.0007, p < 0.001). PLEs were related to slower belief updating when participants encountered large prediction errors (β = -0.03 ± 0.009, p = 0.001). Computational modeling suggested that PLEs were associated with reduced overall belief updating in response to prediction errors (βPE = -1.00 ± 0.45, p = 0.028) and reduced modulation of updating at inferred environmental change points (βCPP = -0.84 ± 0.38, p = 0.023). We conclude that PLEs are associated with altered dynamics of belief updating. These findings support the idea that the process of balancing prior belief and new evidence, as a function of environmental uncertainty, is altered in PLEs, which may contribute to the development of delusions. Specifically, slower learning after large prediction errors in people with high PLEs may result in rigid beliefs. Disregarding environmental change points may limit the flexibility to establish new beliefs in the face of contradictory evidence. The present study fosters a deeper understanding of inferential belief updating mechanisms underlying PLEs.

Asymmetrical Update of Beliefs About Future Outcomes is Driven by Outcome Valence and Social Group Membership

People are eager to update their beliefs, such as a perceived risk, if they receive information that is better than expected but are reluctant to do so when the evidence is unfavourable. When estimating the likelihood of future outcomes, this phenomenon of asymmetrical belief update helps generate and maintain personal optimism bias. In this study, we investigated whether asymmetrical belief update also extends to estimating the future of other individuals. Specifically, we prompted respondents to assess the perceived likelihood of three social targets experiencing future positive and negative events: An in-group, a mild out-group, and an extreme out-group. We then provided the respondents with feedback about the base rates of those events in the general population and prompted them to re-assess their initial estimates for all social targets. Respondents expected more positive than negative outcomes for the in-group and the mild out-group, but more negative outcomes for the extreme out-group. We also found an asymmetrical update of beliefs contingent on the valence of the future event and the social target. For negative outcomes, respondents updated more following good news than bad news, particularly for the mild out-group. For positive outcomes, respondents equally updated their beliefs following good news and bad news for the in-group and the mild out-group. However, they updated their beliefs significantly more following bad news than good news for the extreme out-group member. Our data thus reveal the strong influence of social stereotypes on future expectancies for others.

Assumed shared belief about conspiracy theories in social networks protects paranoid individuals against distress

Paranoia is the belief that others intend you harm. It is related to conspiracy theories, wherein those others represent an organized faction, coordinating the harm against self and others, and violating societal norms. Current psychological studies of paranoid conspiracy theorizing focus either on the individual or their broader social network. Likewise, theories of belief formation and updating often contain individual level processes as well as broader interpersonal and organizational factors. Here we examine paranoia and conspiracy theorizing in terms of individual behavioral predictors (performance on a probabilistic reversal learning task which assays belief updating) as well as social sensing: we ask participants to report the features of their social network, including whether their friends and acquaintances share their paranoid conspiratorial beliefs. We find that people who believe paranoid conspiracy theories expect more volatility during the task. They also assume that members of their social network share their paranoid beliefs. Critically, those participants with larger social networks and greater assumed shared belief tend to harbor their conspiratorial beliefs with less emotional distress and expect less volatility in the task. This is evidence that, like political and religious beliefs, conspiracy theories may flourish under a sacred canopy of belief consensus. These data suggest that friends and acquaintances may serve as sources of credulity and moving between them may sustain conspiracy beliefs when there is detraction. This hybrid individual/social account may shed light on clinical paranoia and persecutory delusion, wherein disability is defined normatively, and social supports are fewer.

Incorporating learning into direct policy search for flood risk management.

Direct policy search (DPS) is a method for identifying optimal policies (i.e., rules) for managing a system in response to changing conditions. In this article, we introduce a new adaptive way to incorporate learning into DPS. The standard DPS approach identifies "robust" policies by optimizing their average performance over a large ensemble of future states of the world (SOW). Our approach exploits information gained over time, updating prior beliefs about the kind of SOW being experienced. We first run the standard DPS approach multiple times, but with varying sets of weights applied to the SOWs when calculating average performance. Adaptive "metapolicies" then further improve performance by specifying how control of the system should switch between policies identified using different weight sets, depending on our updated beliefs about the relative likelihood of being in certain SOWs. We outline the general method and illustrate it using a case study of efficient dike heightening that simultaneously minimizes protection system costs and flood damage resulting from rising sea levels and storm surge. The solutions identified by our adaptive algorithm dominate the standard DPS on these two objectives, with an average marginal damage reduction of 35.1% for policies with similar costs; improvements are largest in SOWs with relatively lower sea level rise. We also evaluate how performance varies under different ways of implementing the algorithm, such as changing the frequency with which beliefs are updated.

A differential privacy based multi-stage network fingerprinting deception game method

As an active deception defense mechanism, network fingerprint obfuscation plays an important role in deceiving the attacker's fingerprinting actions. However, the existing methods show increasing limitations in coping with strategic multi-stage fingerprinting behavior in resource constrained situations. In order to solve this problem, we propose a Differential Privacy based Multi-stage fingerprinting deception Game method (DPMG). DPMG uses the multi-stage game model and the belief update mechanism to give a better description on the continuous fingerprinting interaction process. Besides, the equilibrium existence of such a game model is analyzed. Then a defender strategy optimization mechanism under the resource constraints is designed and a Recursive Algorithm Solving Equilibrium algorithm (RASE) to solve the equilibrium strategy is implemented. Experiment results show that DPMG can deceive the network fingerprinting behavior effectively under the defense resources constraints, and scale well to different network size.

From local match/mismatch signals to updating of task-relevant beliefs: The temporo-parietal junction and its embedment in cortical networks: Comment on "Left and right temporal-parietal junctions (TPJs) as "match/mismatch" hedonic machines: A unifying account of TPJ function" by Doricchi et al.

From local match/mismatch signals to updating of task-relevant beliefs: The temporo-parietal junction and its embedment in cortical networks: Comment on "Left and right temporal-parietal junctions (TPJs) as "match/mismatch" hedonic machines: A unifying account of TPJ function" by Doricchi et al.

Bayesian Investor Belief Updating Speed and Market Underreaction to Earnings Announcements

AbstractBuilding on the Bayesian Theorem, we propose a multi‐period market microstructure model to understand how Bayesian investors underact new information and the duration of market underreaction. Applying the model to post‐earnings‐announcement drifts, our simulation and regression analyses show that the duration of the post‐announcement price adjustment process and the post‐announcement drifts can be explained by the new measure of belief updating speed that quantifies the uncertainties faced by Bayesian investors when incorporating new information into prices. Our study highlights the importance of incorporating the belief uncertainties of uninformed investors in explaining market underreaction in the Bayesian framework.

Condition-based pavement management systems accounting for model uncertainty and facility heterogeneity with belief updates

Due to the tighter budget for pavement management, schedules of inspection activities should be jointly optimized with the maintenance and reconstruction (M&R) plans for pavement systems. Conducting inspections every year is unnecessary and will decrease the budget for M&R activities, while infrequent inspections may lead to suboptimal M&R planning due to the lack of accurate information. This paper presents a methodology for jointly optimizing the inspection scheduling and M&R planning for pavement systems, considering model uncertainty and facility-specific heterogeneity. The problem is defined as a Partially Observable Markov Decision Process (POMDP) model, accounting for the tradeoff between the information value and inspection costs. Moreover, a statistical learning method is used to update the prediction of pavement conditions using the collected inspection data and, eventually, improve the condition-based decisions. This “belief update” process can gradually reduce the model uncertainty as the dataset size increases. We demonstrate the proposed stochastic optimization framework through a numerical example with a system of fifty heterogenous pavement facilities under a combined budget for inspection and M&R activities. Several managerial insights and implications are discussed. For example, the optimal inspection frequencies are less sensitive to the budget; and the agency should perform fewer reconstructions and more rehabilitations when the budget is limited.

Changing minds about climate change: a pervasive role for domain-general metacognition

Updating one’s beliefs about the causes and effects of climate change is crucial for altering attitudes and behaviours. Importantly, metacognitive abilities - insight into the (in)correctness of one’s beliefs- play a key role in the formation of polarised beliefs. We here aimed at investigated the role of metacognition in changing beliefs about climate change. To that end, we focused on the role of domain-general and domain-specific metacognition in updating prior beliefs about climate change across the spectrum of climate change scepticism. We also considered the role of how climate science is communicated in the form of textual or visuo-textual presentations. We asked two large US samples to perform a perceptual decision-making task (to assess domain-general decision-making and metacognitive abilities. They next performed a belief-updating task, where they were exposed to good and bad news about climate change and we asked them about their beliefs and their updating. Lastly, they completed a series of questionnaires probing their attitudes to climate change. We show that climate change scepticism is associated with differences in domain-general as well as domain-specific metacognitive abilities. Moreover, domain-general metacognitive sensitivity influenced belief updating in an asymmetric way: lower domain-general metacognition decreased the updating of prior beliefs, especially in the face of negative evidence. Our findings highlight the role of metacognitive failures in revising erroneous beliefs about climate change and point to their adverse social effects.

Modulation of Motor Vigor by Expectation of Reward Probability Trial-by-Trial Is Preserved in Healthy Ageing and Parkinson's Disease Patients.

Motor improvements, such as faster movement times or increased velocity, have been associated with reward magnitude in deterministic contexts. Yet whether individual inferences on reward probability influence motor vigor dynamically remains undetermined. We investigated how dynamically inferring volatile action-reward contingencies modulated motor performance trial-by-trial. We conducted three studies that coupled a reversal learning paradigm with a motor sequence task and used a validated hierarchical Bayesian model to fit trial-by-trial data. In Study 1, we tested healthy younger [HYA; 37 (24 females)] and older adults [HOA; 37 (17 females)], and medicated Parkinson's disease (PD) patients [20 (7 females)]. We showed that stronger predictions about the tendency of the action-reward contingency led to faster performance tempo, commensurate with movement time, on a trial-by-trial basis without robustly modulating reaction time (RT). Using Bayesian linear mixed models, we demonstrated a similar invigoration effect on performance tempo in HYA, HOA, and PD, despite HOA and PD being slower than HYA. In Study 2 [HYA, 39 (29 females)], we additionally showed that retrospective subjective inference about credit assignment did not contribute to differences in motor vigor effects. Last, Study 3 [HYA, 33 (27 females)] revealed that explicit beliefs about the reward tendency (confidence ratings) modulated performance tempo trial-by-trial. Our study is the first to reveal that the dynamic updating of beliefs about volatile action-reward contingencies positively biases motor performance through faster tempo. We also provide robust evidence for a preserved sensitivity of motor vigor to inferences about the action-reward mapping in aging and medicated PD.SIGNIFICANCE STATEMENT Navigating a world rich in uncertainty relies on updating beliefs about the probability that our actions lead to reward. Here, we investigated how inferring the action-reward contingencies in a volatile environment modulated motor vigor trial-by-trial in healthy younger and older adults, and in Parkinson's disease (PD) patients on medication. We found an association between trial-by-trial predictions about the tendency of the action-reward contingency and performance tempo, with stronger expectations speeding the movement. We additionally provided evidence for a similar sensitivity of performance tempo to the strength of these predictions in all groups. Thus, dynamic beliefs about the changing relationship between actions and their outcome enhanced motor vigor. This positive bias was not compromised by age or Parkinson's disease.

Seeing the Error in My "Bayes": A Quantified Degree of Belief Change Correlates with Children's Pupillary Surprise Responses Following Explicit Predictions.

Bayesian models allow us to investigate children's belief revision alongside physiological states, such as "surprise". Recent work finds that pupil dilation (or the "pupillary surprise response") following expectancy violations is predictive of belief revision. How can probabilistic models inform the interpretations of "surprise"? Shannon Information considers the likelihood of an observed event, given prior beliefs, and suggests stronger surprise occurs following unlikely events. In contrast, Kullback-Leibler divergence considers the dissimilarity between prior beliefs and updated beliefs following observations-with greater surprise indicating more change between belief states to accommodate information. To assess these accounts under different learning contexts, we use Bayesian models that compare these computational measures of "surprise" to contexts where children are asked to either predict or evaluate the same evidence during a water displacement task. We find correlations between the computed Kullback-Leibler divergence and the children's pupillometric responses only when the children actively make predictions, and no correlation between Shannon Information and pupillometry. This suggests that when children attend to their beliefs and make predictions, pupillary responses may signal the degree of divergence between a child's current beliefs and the updated, more accommodating beliefs.

A generalized Bayes framework for probabilistic clustering

Summary Loss-based clustering methods, such as k-means clustering and its variants, are standard tools for finding groups in data. However, the lack of quantification of uncertainty in the estimated clusters is a disadvantage. Model-based clustering based on mixture models provides an alternative approach, but such methods face computational problems and are highly sensitive to the choice of kernel. In this article we propose a generalized Bayes framework that bridges between these paradigms through the use of Gibbs posteriors. In conducting Bayesian updating, the loglikelihood is replaced by a loss function for clustering, leading to a rich family of clustering methods. The Gibbs posterior represents a coherent updating of Bayesian beliefs without needing to specify a likelihood for the data, and can be used for characterizing uncertainty in clustering. We consider losses based on Bregman divergence and pairwise similarities, and develop efficient deterministic algorithms for point estimation along with sampling algorithms for uncertainty quantification. Several existing clustering algorithms, including k-means, can be interpreted as generalized Bayes estimators in our framework, and thus we provide a method of uncertainty quantification for these approaches, allowing, for example, calculation of the probability that a data point is well clustered.

Small steps for mankind: Modeling the emergence of cumulative culture from joint active inference communication.

Although the increase in the use of dynamical modeling in the literature on cultural evolution makes current models more mathematically sophisticated, these models have yet to be tested or validated. This paper provides a testable deep active inference formulation of social behavior and accompanying simulations of cumulative culture in two steps: First, we cast cultural transmission as a bi-directional process of communication that induces a generalized synchrony (operationalized as a particular convergence) between the belief states of interlocutors. Second, we cast social or cultural exchange as a process of active inference by equipping agents with the choice of who to engage in communication with. This induces trade-offs between confirmation of current beliefs and exploration of the social environment. We find that cumulative culture emerges from belief updating (i.e., active inference and learning) in the form of a joint minimization of uncertainty. The emergent cultural equilibria are characterized by a segregation into groups, whose belief systems are actively sustained by selective, uncertainty minimizing, dyadic exchanges. The nature of these equilibria depends sensitively on the precision afforded by various probabilistic mappings in each individual's generative model of their encultured niche.

Learning from Shared News: When Abundant Information Leads to Belief Polarization

AbstractWe study learning via shared news. Each period agents receive the same quantity and quality of firsthand information and can share it with friends. Some friends (possibly few) share selectively, generating heterogeneous news diets across agents. Agents are aware of selective sharing and update beliefs by Bayes’s rule. Contrary to standard learning results, we show that beliefs can diverge in this environment, leading to polarization. This requires that (i) agents hold misperceptions (even minor) about friends’ sharing and (ii) information quality is sufficiently low. Polarization can worsen when agents’ friend networks expand. When the quantity of firsthand information becomes large, agents can hold opposite extreme beliefs, resulting in severe polarization. We find that news aggregators can curb polarization caused by news sharing. Our results hold without media bias or fake news, so eliminating these is not sufficient to reduce polarization. When fake news is included, it can lead to polarization but only through misperceived selective sharing. We apply our theory to shed light on the polarization of public opinion about climate change in the United States.

Impaired belief updating and devaluation in adult women with bulimia nervosa

Recent models of bulimia nervosa (BN) propose that binge-purge episodes ultimately become automatic in response to cues and insensitive to negative outcomes. Here, we examined whether women with BN show alterations in instrumental learning and devaluation sensitivity using traditional and computational modeling analyses of behavioral data. Adult women with BN (n = 30) and group-matched healthy controls (n = 31) completed a task in which they first learned stimulus-response-outcome associations. Then, participants were required to repeatedly adjust their responses in a “baseline test”, when different sets of stimuli were explicitly devalued, and in a “slips-of-action test”, when outcomes instead of stimuli were devalued. The BN group showed intact behavioral sensitivity to outcome devaluation during the slips-of-action test, but showed difficulty overriding previously learned stimulus-response associations on the baseline test. Results from a Bayesian learner model indicated that this impaired performance could be accounted for by a slower pace of belief updating when a new set of previously learned responses had to be inhibited (p = 0.036). Worse performance and a slower belief update in the baseline test were each associated with more frequent binge eating (p = 0.012) and purging (p = 0.002). Our findings suggest that BN diagnosis and severity are associated with deficits in flexibly updating beliefs to withhold previously learned responses to cues. Additional research is needed to determine whether this impaired ability to adjust behavior is responsible for maintaining automatic and persistent binge eating and purging in response to internal and environmental cues.