Updating beliefs with incomplete observations

Currently, there is renewed interest in the problem, raised by Shafer in 1985, of updating probabilities when observations are incomplete. This is a fundamental problem in general, and of particular interest for Bayesian networks. Recently, Grunwald and Halpern have shown that commonly used updating strategies fail in this case, except under very special assumptions. In this paper we propose a new method for updating probabilities with incomplete observations. Our approach is deliberately conservative: we make no assumptions about the so-called incompleteness mechanism that associates complete with incomplete observations. We model our ignorance about this mechanism by a vacuous lower prevision, a tool from the theory of imprecise probabilities, and we use only coherence arguments to turn prior into posterior probabilities. In general, this new approach to updating produces lower and upper posterior probabilities and expectations, as well as partially determinate decisions. This is a logical consequence of the existing ignorance about the incompleteness mechanism. We apply the new approach to the problem of classification of new evidence in probabilistic expert systems, where it leads to a new, so-called conservative updating rule. In the special case of Bayesian networks constructed using expert knowledge, we provide an exact algorithm for classification based on our updating rule, which has linear-time complexity for a class of networks wider than polytrees. This result is then extended to the more general framework of credal networks, where computations are often much harder than with Bayesian nets. Using an example, we show that our rule appears to provide a solid basis for reliable updating with incomplete observations, when no strong assumptions about the incompleteness mechanism are justified.Comment: Replaced with extended versio

Bayesian Updating Rules in Continuous Opinion Dynamics Models

In this article, I investigate the use of Bayesian updating rules applied to modeling social agents in the case of continuos opinions models. Given another agent statement about the continuous value of a variable $x$, we will see that interesting dynamics emerge when an agent assigns a likelihood to that value that is a mixture of a Gaussian and a Uniform distribution. This represents the idea the other agent might have no idea about what he is talking about. The effect of updating only the first moments of the distribution will be studied. and we will see that this generates results similar to those of the Bounded Confidence models. By also updating the second moment, several different opinions always survive in the long run. However, depending on the probability of error and initial uncertainty, those opinions might be clustered around a central value.Comment: 14 pages, 5 figures, presented at SigmaPhi200

Assessment of a method to detect signals for updating systematic reviews.

BackgroundSystematic reviews are a cornerstone of evidence-based medicine but are useful only if up-to-date. Methods for detecting signals of when a systematic review needs updating have face validity, but no proposed method has had an assessment of predictive validity performed.MethodsThe AHRQ Comparative Effectiveness Review program had produced 13 comparative effectiveness reviews (CERs), a subcategory of systematic reviews, by 2009, 11 of which were assessed in 2009 using a surveillance system to determine the degree to which individual conclusions were out of date and to assign a priority for updating each report. Four CERs were judged to be a high priority for updating, four CERs were judged to be medium priority for updating, and three CERs were judged to be low priority for updating. AHRQ then commissioned full update reviews for 9 of these 11 CERs. Where possible, we matched the original conclusions with their corresponding conclusions in the update reports, and compared the congruence between these pairs with our original predictions about which conclusions in each CER remained valid. We then classified the concordance of each pair as good, fair, or poor. We also made a summary determination of the priority for updating each CER based on the actual changes in conclusions in the updated report, and compared these determinations with the earlier assessments of priority.ResultsThe 9 CERs included 149 individual conclusions, 84% with matches in the update reports. Across reports, 83% of matched conclusions had good concordance, and 99% had good or fair concordance. The one instance of poor concordance was partially attributable to the publication of new evidence after the surveillance signal searches had been done. Both CERs originally judged as being low priority for updating had no substantive changes to their conclusions in the actual updated report. The agreement on overall priority for updating between prediction and actual changes to conclusions was Kappa = 0.74.ConclusionsThese results provide some support for the validity of a surveillance system for detecting signals indicating when a systematic review needs updating

Behavioral Social Learning

We revisit the economic models of social learning by assuming that individuals update their beliefs in a non-Bayesian way. Individuals either overweigh or underweigh (in Bayesian terms) their private information relative to the public information revealed by the decisions of others and each individual's updating rule is private information. First, we consider a setting with perfectly rational individuals with a commonly known distribution of updating rules. We show that introducing heterogeneous updating rules in a simple social learning environment reconciles equilibrium predictions with laboratory evidence. Additionally, a model of social learning with bounded private beliefs and sufficiently rich updating rules corresponds to a model of social learning with unbounded private beliefs. A straightforward implication is that heterogeneity in updating rules is efficiency-enhancing in most social learning environments. Second, we investigate the implications of heterogeneous updating rules in social learning environments where individuals only understand the relation between the aggregate distribution of decisions and the state of the world. Unlike in rational social learning, heterogeneous updating rules do not lead to a substantial improvement of the societal welfare and there is always a non-negligible likelihood that individuals become extremely and wrongly conï¬dent about the state of the world.Social learning, Non-Bayesian updating, Herding, Informational cascades

Non-Bayesian Updating : A Theoretical Framework

This paper models an agent in an infinite horizon setting who does not update according to Bayes' Rule, and who is self-aware and anticipates her updating behavior when formulating plans. Choice-theoretic axiomatic foundations are provided. Then the model is specialized axiomatically to capture updating biases that reflect excessive weight given to (i) prior beliefs, or alternatively, (ii) the realized sample. Finally, the paper describes a counterpart of the exchangeable Bayesian model, where the agent tries to learn about parameters, and some answers are provided to the question "what does a non-Bayesian updater learn?"non-Bayesian updating, overreaction, underreaction, confirmatory bias, law of small numbers, gambler's fallacy, hot hand fallacy, temptation, self-control, learning, menus

Bayesian Estimation of the Discrepancy with Misspecified Parametric Models

We study a Bayesian model where we have made specific requests about the parameter values to be estimated. The aim is to find the parameter of a parametric family which minimizes a distance to the data generating density and then to estimate the discrepancy using nonparametric methods. We illustrate how coherent updating can proceed given that the standard Bayesian posterior from an unidentifiable model is inappropriate. Our updating is performed using Markov Chain Monte Carlo methods and in particular a novel method for dealing with intractable normalizing constants is required. Illustrations using synthetic data are provided.European Research Council (ERC) through StG "N-BNP" 306406Regione PiemonteMathematic

Foundations of Bayesian Theory

This paper states necessary and sufficient conditions for the existence, uniqueness, and updating according to Bayes?rule, of subjective probabilities representing individuals?beliefs. The approach is preference based, and the result is an axiomatic subjective expected utility model of Bayesian decision making under uncertainty with statedependent preferences. The theory provides foundations for the existence of prior probabilities representing decision makers?beliefs about the likely realization of events and for the updating of these probabilities according to Bayes?rule.

Logics of Imprecise Comparative Probability

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures