Probability Revision Research Articles

Combining Probabilities with Full Beliefs

AbstractOne of the major problems in formal epistemology is the difficulty of combining probabilistic and dichotomous (all-or-nothing) beliefs in one and the same formal framework. Based on the properties of actual human belief systems, a set of ten desiderata is proposed for belief system models that contain both probabilistic and full beliefs. Previously proposed combined belief systems are shown to fall short of these desiderata. A proof of concept is provided to show that they can be satisfied in a formal model. In the proposed model, the amnesia problem of probabilistic updating is solved by introducing infinitesimal probabilities as carriers of memories of beliefs that have been given up. In the resulting system of probability revision, contingent propositions can be transferred in both directions between full beliefs and beliefs held to a lower degree, in a way that mirrors real-life acquisitions and losses of full beliefs. The subsystem consisting of full beliefs has a pattern of change that constitutes a credible system of dichotomous belief change.

Bayesian probability revision and infection prevention behavior in Japan: A quantitative analysis of the first wave of COVID-19

The relationship between cognitive biases and infection prevention behavior remains unexplored in the existing literature. This study uses data from a questionnaire survey conducted in Japan on the first wave of Coronavirus Disease 2019 (COVID-19) from February to May 2020 to empirically investigate the impact of Bayesian probability inference, the influence of cognitive biases of PCR test results on infection prevention behavior, and the discrepancy between infection prevention intentions and behaviors. We used a bivariate ordinal probit model when considering the correlation between behaviors. The results showed that the higher probability responses, implying pessimistic biases, were more likely to indicate that declaring a state of emergency was necessary and effective, and were more health-oriented in ensuring infection prevention behavior even at the expense of the economy. However, the study found that although they wanted to reduce the frequency of their outings and the number of people they met, they did not reduce them in terms of actual behavior change. It also found that those with pessimistic biases had a higher WTP for the vaccine.

A Basis for AGM Revision in Bayesian Probability Revision

In standard Bayesian probability revision, the adoption of full beliefs (propositions with probability 1) is irreversible. Once an agent has full belief in a proposition, no subsequent revision can remove that belief. This is an unrealistic feature, and it also makes probability revision incompatible with belief change theory, which focuses on how the set of full beliefs is modified through both additions and retractions. This problem in probability theory can be solved in a model that (i) lets the codomain of the probability function be a hyperreal-valued rather than the real-valued closed interval [0, 1], and (ii) identifies the full beliefs as the propositions whose probability is either 1 or infinitesimally smaller than 1. In this model, changes in the probability function will result in changes in the set of full beliefs (belief set), which constitutes a submodel that can be conceived as the “tip of the iceberg” within the larger model that also contains beliefs on lower levels of probability. The patterns of change in the set of full beliefs in this modified Bayesian model coincides with the corresponding pattern in a slightly modified version of AGM revision, which is commonly conceived as the gold standard of (dichotomous) belief change. The modification only concerns the marginal case of revision by an inconsistent input sentence. These results show that probability revision and dichotomous belief change can be unified in one and the same framework, or – if we so wish – that belief change theory can be subsumed under a modified version of probability revision that allows for iterated change and for the removal of full beliefs.

Iterated AGM Revision Based on Probability Revision

Close connections between probability theory and the theory of belief change emerge if the codomain of probability functions is extended from the real-valued interval [0, 1] to a hyperreal interval with the same limits. Full beliefs are identified as propositions with a probability at most infinitesimally smaller than 1. Full beliefs can then be given up, and changes in the set of full beliefs follow a pattern very close to that of AGM revision. In this contribution, iterated revision is investigated. The iterated changes in the set of full beliefs generated by repeated revisions of a hyperreal probability function can, semantically, be modelled with the same basic structure as the sphere models of belief change theory. The changes on the set of full beliefs induced by probability revision satisfy the Darwiche–Pearl postulates for iterated belief change.

An Improved Method for Estimating Ballistic Target Micromotion and Structural Parameters Based on ISAR Image Sequences

Micro-motion and structural parameters extraction is a critical technique for recognizing ballistic missiles. It is generally known that parameter estimation methods based on inverse synthetic aperture radar(ISAR) image sequences can effectively extract target information. However, when the signal-to-noise ratio(SNR) is low, the traditional methods are prone to cross-association problems in the nearest-neighbor association procedure, and the structural parameters obtained may only be the local optimal solution. To solve the above two problems, an improved method for estimating ballistic target micro-motion and structural parameters based on ISAR image sequences is proposed in this paper. A distance association probability revision matrix is used to address the cross-association problem, reducing the risk of association with the incorrect nearest neighbor; at the same time, the impact of noise on the frequency estimation is decreased by utilizing refinement domain search-match algorithm, which can discover a narrower estimation interval with less noise to improve SNR. The global optimization capacity for structural parameters estimation of the traditional particle swarm optimization(PSO) algorithm is enhanced by step-by-step optimization algorithm, which reduces the dimension of the high-dimensional micro-motion projection cost function to ensure optimization direction variety. Simulation results show that when SNR is -5dB, the accuracy of the precession frequency estimation is improved by 0.2Hz compared with the traditional FFT method, and the RMSE values for cone height, precession angle, and nutation angle are 0.02m, 0.2 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$°$</tex-math></inline-formula> , and 0.3 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$°$</tex-math></inline-formula> , respectively, which are better than the traditional PSO algorithm.

Recovering a Prior from a Posterior: Some Parameterizations of Jeffrey Conditioning

Given someone’s fully specified posterior probability distribution q and information about the revision method that they employed to produce q, what can you infer about their prior probabilistic commitments? This question provides an entrée into a thoroughgoing discussion of a class of parameterizations of Jeffrey conditioning in which the parameters furnish information above and beyond that incorporated in \(q\). Our analysis highlights the ubiquity of Bayes factors (ratios of new to old odds) in the study of probability revision.

A Characterization of Probability-based Dichotomous Belief Revision

This article investigates the properties of multistate top revision, a dichotomous (AGM-style) model of belief revision that is based on an underlying model of probability revision. A proposition is included in the belief set if and only if its probability is either 1 or infinitesimally close to 1. Infinitesimal probabilities are used to keep track of propositions that are currently considered to have negligible probability, so that they are available if future information makes them more plausible. Multistate top revision satisfies a slightly modified version of the set of basic and supplementary AGM postulates, except the inclusion and success postulates. This result shows that hyperreal probabilities can provide us with efficient tools for overcoming the well known difficulties in combining dichotomous and probabilistic models of belief change.

Combining Results of Performance-Based and Informant Test Accuracy Studies: Bayes or Boole?

Background/Aims: Since screening and diagnostic tests for dementia do not have perfect accuracy, >1 test is often administered when assessing patients with cognitive complaints. Use of both patient performance tests and informant questionnaires has been recommended. Combination of individual test results may be based on methods originally defined by Thomas Bayes (revision or updating of pretest probabilities to post-test probabilities given the test results) and by George Boole (application of associative “AND” or “OR” operator). This study sought to apply these methods in clinical practice. Methods: Using the dataset of a pragmatic test accuracy study of the Six-Item Cognitive Impairment Test (6CIT) and informant Ascertain Dementia 8 (AD8), post-test probabilities for the combination were calculated using Bayes’ formula and compared to Boolean “AND” combination. Combined test sensitivity and specificity was calculated using either Boolean “AND” or “OR” operator and compared to results using equations based on individual test sensitivity and specificity. Results: Both Bayesian and Boolean methods produced similar improvements from pretest probability (0.288) to combined post-test probability for dementia (≈0.5). Likewise, the 2 different methods for calculating combined sensitivities and specificities gave similar results, with, as anticipated, the “AND” combination improving overall specificity (to ≈0.65) whereas the “OR” combination improved sensitivity (to ≈1.00). Conclusion: Combination of individual screening test results using Bayesian and Boolean methods is relatively straightforward and may add to clinicians’ intuitive judgements when combining test results.

Effect of Teaching Bayesian Methods Using Learning by Concept vs Learning by Example on Medical Students’ Ability to Estimate Probability of a Diagnosis

Clinicians use probability estimates to make a diagnosis. Teaching students to make more accurate probability estimates could improve the diagnostic process and, ultimately, the quality of medical care. To test whether novice clinicians can be taught to make more accurate bayesian revisions of diagnostic probabilities using teaching methods that apply either explicit conceptual instruction or repeated examples. A randomized clinical trial of 2 methods for teaching bayesian updating and diagnostic reasoning was performed. A web-based platform was used for consent, randomization, intervention, and testing of the effect of the intervention. Participants included 61 medical students at McMaster University and Eastern Virginia Medical School recruited from May 1 to September 30, 2018. Students were randomized to (1) receive explicit conceptual instruction regarding diagnostic testing and bayesian revision (concept group), (2) exposure to repeated examples of cases with feedback regarding posttest probability (experience group), or (3) a control condition with no conceptual instruction or repeated examples. Students in all 3 groups were tested on their ability to update the probability of a diagnosis based on either negative or positive test results. Their probability revisions were compared with posttest probability revisions that were calculated using the Bayes rule and known test sensitivity and specificity. Of the 61 participants, 22 were assigned to the concept group, 20 to the experience group, and 19 to the control group. Approximate age was 25 years. Two participants were first-year; 37, second-year; 12, third-year; and 10, fourth-year students. Mean (SE) probability estimates of students in the concept group were statistically significantly closer to calculated bayesian probability than the other 2 groups (concept, 0.4%; [0.7%]; experience, 3.5% [0.7%]; control, 4.3% [0.7%]; P < .001). Although statistically significant, the differences between groups were relatively modest, and students in all groups performed better than expected, based on prior reports in the literature. The study showed a modest advantage for students who received theoretical instruction on bayesian concepts. All participants' probability estimates were, on average, close to the bayesian calculation. These findings have implications for how to teach diagnostic reasoning to novice clinicians. ClinicalTrials.gov identifier: NCT04130607.

Open-Minded Orthodox Bayesianism by Epsilon-Conditionalization

Abstract Orthodox Bayesianism endorses revising by conditionalization. This paper investigates the zero-raising problem, or equivalently the certainty-dropping problem of orthodox Bayesianism: previously neglected possibilities remain neglected, although the new evidence might suggest otherwise. Yet, one may want to model open-minded agents, that is, agents capable of raising previously neglected possibilities. Different reasons can be given for open-mindedness, one of which is fallibilism. The paper proposes a family of open-minded propositional revisions depending on a parameter ϵ. The basic idea is this: first extend the prior to the newly suggested possibilities by mixing the prior with the uniform probability on these possibilities, then conditionalize. This may put the agent back on the right track when her beliefs or evidence happen to be false. The paper justifies this family of equivocal epsilon-conditionalizations as minimal non-biased open-minded modifications of conditionalization. Several variations are discussed, such as mixing with an ad hoc or silent prior instead of the uniform prior, and a generalization to probabilistic information is given. The approach is compared to other accounts, such as Jeffrey’s Bayesianism, Gärdenfors’s probabilistic revision, maximizing entropy, and minimal revision. 1Introduction2Downsides of Certainty Conservation3Why Raise Zeros? 3.1Accuracy defects3.2Fallibilism and dissolutions3.3Weak fallibilism and partial solutions4Epsilon-Conditionalization 4.1Definitions4.2Revision types and information assumptions4.3Epsilon interpretation and flavours5Justifications 5.1Open-minded orthodox Bayesianism5.2Minimal revision5.3Transitivity6Conclusion Appendix

Peer Disagreement: A Call for the Revision of Prior Probabilities

The current debate about peer disagreement has so far mainly focused on the question of whether peer disagreements provide genuine counterevidence to which we should respond by revising our credences. By contrast, comparatively little attention has been devoted to the question by which process, if any, such revision should be brought about. The standard assumption is that we update our credences by conditionalizing on the evidence that peer disagreements provide. In this paper, we argue that non-dogmatist views have good reasons to reject conditionalization. Instead, peer disagreements should be understood to call for a revision of our prior conditional probabilities: rather than merely adding to our original evidence, they pose a challenge to the thought that we have properly assessed the probative force of our original evidence.

Updating a progic

This paper presents a progic, or probabilistic logic, in the sense of Haenni et al. [8]. The progic presented here is based on Bayesianism, as the progic discussed by Williamson [15]. However, the underlying generalised Bayesianism differs from the objective Bayesianism used by Williamson, in the calibration norm, and the liberalisation and interpretation of the reference probability in the norm of equivocation. As a consequence, the updating dynamics of both Bayesianisms differ essentially. Whereas objective Bayesianism is based on a probabilistic re-evaluation, orthodox Bayesianism is based on a probabilistic revision. I formulate a generalised and iterable orthodox Bayesian revision dynamics. This allows to define an updating procedure for the generalised Bayesian progic. The paper compares the generalised Bayesian progic and Williamson's objective Bayesian progic in strength, update dynamics and with respect to language (in)sensitivity.

Individual differences in managerial accounting judgments and decision making

Managers are influenced in their decisions by the information provided by managerial accountants. Two typical examples from textbooks are the irrelevance of sunk costs and, more recently, the affect of knowing the outcome of a decision or revised budget forecast. Individual differences in the cognitive ability of decision makers to use information can lead to systematic differences in judgments. We identify and label one of these individual cognitive differences comprehensive thinking ability: the ability to think about multiple paths, branches or alternatives. Significant comprehensive thinking ability is likely to mitigate systematic differences in judgment in many contexts. We report the results of a series of studies using a variation on the investment trap (sunk cost or irrelevant cost) problem and a probability revision task. The findings suggest that comprehensive thinking ability may also explain other common systematic differences in judgment.

Trial-by-trial modeling of electrophysiological signals during inverse Bayesian inference

Empirical support for the Bayesian brain hypothesis, although of major theoretical importance for cognitive neuroscience, is surprisingly scarce. The literature still lacks definitive functional neuroimaging evidence that neural activities code and compute Bayesian probabilities. Here, we introduce a new experimental design to relate electrophysiological measures to Bayesian inference. Specifically, an urns-and-balls paradigm was used to study neural underpinnings of probabilistic inverse inference. Event-related potentials (ERPs) were recorded from human participants who performed the urns-and-balls paradigm, and computational modeling was conducted on trial-by-trial electrophysiological signals. Five computational models were compared with respect to their capacity to predict electrophysiological measures. One Bayesian model (BAY) was compared with another Bayesian model which takes potential effects of non-linear probability weighting into account (BAYS). A predictive surprise model (TOPS) of sequential probability revisions was derived from the Bayesian models. A comparison was made with two published models of surprise (DIF [1] and OST [2]). Subsets of the trial-by-trial electrophysiological signals were differentially sensitive to model predictors: The anteriorly distributed N250 was best fit by the DIF model, the BAYS model provided the best fit to the anteriorly distributed P3a, whereas the posteriorly distributed P3b and Slow Wave were best fit by the TOPS model.

The solitary pulmonary nodule-deciding when to act?

Solitary pulmonary nodules (SPNs) are commonly encountered in pulmonary practice. Their management is complex, and multiple clinical factors must be considered. The three common management strategies applied to solitary pulmonary nodules are careful observation, diagnostic testing, and surgery. Fundamental concepts derived from decision analysis can be used to help clinicians choose optimal management strategies for individual patients with SPNs. This process begins with estimating the pretest probability of cancer. Then the consequences of treatment are considered-including the benefit of surgery if the patient has cancer and the harm of treatment if the patient does not have cancer. Patient comorbidities and competing risks affect the consequences of treatment. Knowledge of the benefits and harms of treatment allows clinicians to determine the treatment threshold and then rationally develop the optimal management plan. Probability revision using the pretest probability, test characteristics, and Bayes theorem is used to refine the probability of cancer until a decision threshold is reached and definitive treatment can be determined. Patients with very low pretest probability of cancer are managed with a strategy of careful observation by serial computed tomography (CT). Patients who have a high pretest probability of cancer merit surgical diagnosis. Patients with an intermediate pretest probability of cancer go on to further diagnostic testing, primarily with CT-guided fine needle aspiration or positron-emission tomography. Patient preferences are considered throughout the process because the absolute difference in outcome between some strategies may be small.

OP96 Survey of use and Application of test Accuracy Measures for Decision Making in Primary Care

BackgroundIncrease in test use over recent decades has occurred despite disappointing results from test accuracy evaluations. Difficulties with understanding and application of test accuracy information are purported to be important...

Updating: A psychologically basic situation of probability revision

The Bayesian model has been used in psychology as the standard reference for the study of probability revision. In the first part of this paper we show that this traditional choice restricts the scope of the experimental investigation of revision to a stable universe. This is the case of a situation that, technically, is known as focusing. We argue that it is essential for a better understanding of human probability revision to consider another situation called updating (Katsuno & Mendelzon, 1992), in which the universe is evolving. In that case the structure of the universe has definitely been transformed and the revision message conveys information on the resulting universe. The second part of the paper presents four experiments based on the Monty Hall puzzle that aim to show that updating is a natural frame for individuals to revise their beliefs.

The Smith-Walley Interpretation of Subjective Probability: An Appreciation

The right interpretation of subjective probability is implicit in the theories of upper and lower odds, and upper and lower previsions, developed, respectively, by Cedric Smith (1961) and Peter Walley (1991). On this interpretation you are free to assign contingent events the probability 1 (and thus to employ conditionalization as a method of probability revision) without becoming vulnerable to a weak Dutch book.

Cognitive processes underlying lottery and sports gambling decisions: The role of stated probabilities and background knowledge

Two studies compared choice and underlying cognitive processes in equivalent decision tasks involving risk and uncertainty (lotteries versus sports gambles including displayed expert probability judgements). In sports gambles, background knowledge was triggered via information on team location, home or away. Otherwise, displayed risk information (stake, winnings, odds, and outcome probabilities) was controlled across gamble type. In a choice study, home win bets were chosen significantly more frequently than draws or away wins, compared to lottery equivalents. In a parallel study eliciting concurrent verbal protocols, participants made fewer evaluations of odds and probabilities, and more statements involving background knowledge in sports gambles. Furthermore, some sports gamble protocols indicated modifications of stated probabilities and decision strategies contingent on domain knowledge. It was concluded that stated probability revision and knowledge-based reasoning are key cognitive processes in sports gambling not normally applied in the lottery paradigms often employed in decision research.

On the revision of probabilistic beliefs using uncertain evidence

We revisit the problem of revising probabilistic beliefs using uncertain evidence, and report results on several major issues relating to this problem: how should one specify uncertain evidence? How should one revise a probability distribution? How should one interpret informal evidential statements? Should, and do, iterated belief revisions commute? And what guarantees can be offered on the amount of belief change induced by a particular revision? Our discussion is focused on two main methods for probabilistic revision: Jeffrey's rule of probability kinematics and Pearl's method of virtual evidence, where we analyze and unify these methods from the perspective of the questions posed above.