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A general framework for updating belief distributions.

ABSTRACT: We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data-generating mechanism. For instance, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our framework uses loss functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.

SUBMITTER: Bissiri PG

PROVIDER: S-EPMC5082587 | biostudies-literature | 2016 Nov

REPOSITORIES: biostudies-literature

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A general framework for updating belief distributions.

Bissiri P G PG Holmes C C CC Walker S G SG

Journal of the Royal Statistical Society. Series B, Statistical methodology 20160223 5

We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered as a special case. Modern application areas make it increasingly challenging for Bayesians to attempt to model the true data-generating mechanism. For instance, when the object of interest is low dimensional, such a ...[more]

PMID: 27840585

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Project description:People tend to update beliefs about their future outcomes in a valence-dependent way: they are likely to incorporate good news and to neglect bad news. However, belief formation is a complex process which depends not only on motivational factors such as the desire for favorable conclusions, but also on multiple cognitive variables such as prior beliefs, knowledge about personal vulnerabilities and resources, and the size of the probabilities and estimation errors. Thus, we applied computational modeling in order to test for valence-induced biases in updating while formally controlling for relevant cognitive factors. We compared biased and unbiased Bayesian models of belief updating, and specified alternative models based on reinforcement learning. The experiment consisted of 80 trials with 80 different adverse future life events. In each trial, participants estimated the base rate of one of these events and estimated their own risk of experiencing the event before and after being confronted with the actual base rate. Belief updates corresponded to the difference between the two self-risk estimates. Valence-dependent updating was assessed by comparing trials with good news (better-than-expected base rates) with trials with bad news (worse-than-expected base rates). After receiving bad relative to good news, participants' updates were smaller and deviated more strongly from rational Bayesian predictions, indicating a valence-induced bias. Model comparison revealed that the biased (i.e., optimistic) Bayesian model of belief updating better accounted for data than the unbiased (i.e., rational) Bayesian model, confirming that the valence of the new information influenced the amount of updating. Moreover, alternative computational modeling based on reinforcement learning demonstrated higher learning rates for good than for bad news, as well as a moderating role of personal knowledge. Finally, in this specific experimental context, the approach based on reinforcement learning was superior to the Bayesian approach. The computational validation of valence-dependent belief updating represents a novel support for a genuine optimism bias in human belief formation. Moreover, the precise control of relevant cognitive variables justifies the conclusion that the motivation to adopt the most favorable self-referential conclusions biases human judgments.

Project description:Individuals prone to anxiety and depression often report beliefs and make judgements about themselves that are more negative than those reported by others. We use computational modeling of a richly naturalistic task to disentangle the role of negative priors versus negatively biased belief updating and to investigate their association with different dimensions of Internalizing psychopathology. Undergraduate participants first provided profiles for a hypothetical tech internship. They then viewed pairs of other profiles and selected the individual they would prefer to work alongside out of each pair. In a subsequent phase of the experiment, participants made judgments about their relative popularity as hypothetical internship partners both before any feedback and after each of 20 items of feedback revealing whether or not they had been selected as the preferred teammate from a given pairing. Scores on latent factors of general negative affect, anxiety-specific affect and depression-specific affect were estimated using participants’ self-report scores on standardized measures of anxiety and depression together with factor loadings from a bifactor analysis conducted previously. Higher scores on the depression-specific factor were linked to more negative prior beliefs but were not associated with differences in belief updating. In contrast, higher scores on the anxiety-specific factor were associated with a negative bias in belief updating but no difference in prior beliefs. These findings indicate that, to at least some extent, distinct processes may impact the formation of belief priors and in-the-moment belief updating and that these processes may be differentially disrupted in depression and anxiety. Future directions for enquiry include examination of the possibility that prior beliefs biases in depression might reflect generalization from prior experiences or global schema whereas belief updating biases in anxiety might be more situationally specific. Author summary Pessimistic beliefs are at the core of both depression and anxiety. How do these beliefs form and is this process similar or different for depression and anxiety? Here, we investigate these questions in a naturalistic experimental setting, in which undergraduate participants competed against one another to be selected by their peers in a hypothetical tech internship and were asked to judge their popularity relative to other participants. Importantly, participants were asked to make these judgements before any feedback and after receiving a series of pieces of feedback about whether or not they had been selected over other specific students. This design, along with computational modelling of the formation of participants’ beliefs, allowed us to establish that anxiety is associated with pessimistic beliefs that arise through negatively biased belief updating–i.e., the incorporation of negative information more readily into one’s beliefs than positive information–while depression is associated with pessimistic prior beliefs–that is, beliefs that are formed before receiving explicit feedback and likely on the basis of prior experiences or pre-existing schema.

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A general framework for updating belief distributions.

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A general framework for updating belief distributions.

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