Updating Probabilities

We show that Skilling's method of induction leads to a unique general theory of inductive inference, the method of Maximum relative Entropy (ME). The main tool for updating probabilities is the logarithmic relative entropy; other entropies such as those of Renyi or Tsallis are ruled out. We also show that Bayes updating is a special case of ME updating and thus, that the two are completely compatible.Comment: Presented at MaxEnt 2006, the 26th International Workshop on Bayesian Inference and Maximum Entropy Methods (July 8-13, 2006, Paris, France

Updating Probabilities with Data and Moments

We use the method of Maximum (relative) Entropy to process information in the form of observed data and moment constraints. The generic "canonical" form of the posterior distribution for the problem of simultaneous updating with data and moments is obtained. We discuss the general problem of non-commuting constraints, when they should be processed sequentially and when simultaneously. As an illustration, the multinomial example of die tosses is solved in detail for two superficially similar but actually very different problems.Comment: Presented at the 27th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Saratoga Springs, NY, July 8-13, 2007. 10 pages, 1 figure V2 has a small typo in the end of the appendix that was fixed. aj=mj+1 is now aj=m(k-j)+

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

Cognitive Abilities and Behavioral Biases

We use a simple, three-item test for cognitive abilities to investigate whether established behavioral biases that play a prominent role in behavioral economics and finance are related to cognitive abilities. We find that higher test scores on the Cognitive Reflection Test of Frederick (2005) indeed are correlated with lower incidences of the conjunction fallacy, conservatism in updating probabilities, and overconfidence. Test scores are also significantly related to subjects' time and risk preferences. We find no influence on anchoring. However, even if biases are lower for people with higher cognitive abilities, they still remain substantial.

Keeping Up-to-Date with Bayes (Abstract)

Bayes’ theorem, a rule for updating probabilities as new information is obtained, may be over two centuries old but it has been the driving force behind many of the most significant recent advances in statistics and other sciences

Cognitive Abilities and Behavioral Biases

We use a simple, three-item test for cognitive abilities to investigate whether established behavioral biases that play a prominent role in behavioral economics and finance are related to cognitive abilities. We find that higher test scores on the Cognitive Reflection Test of Frederick (2005) indeed are correlated with lower incidences of the conjunction fallacy, conservatism in updating probabilities, and overconfidence. Test scores are also significantly related to subjects’ time and risk preferences. We find no influence on anchoring. However, even if biases are lower for people with higher cognitive abilities, they still remain substantial.cognitive reflection test, behavioral finance, biases, cognitive abilities