Kybernetika 51 no. 3, 469-485, 2015

Modeling biased information seeking with second order probability distributions

Gernot D. KleiterDOI: 10.14736/kyb-2015-3-0469


Updating probabilities by information from only one hypothesis and thereby ignoring alternative hypotheses, is not only biased but leads to progressively imprecise conclusions. In psychology this phenomenon was studied in experiments with the "pseudodiagnosticity task''. In probability logic the phenomenon that additional premises increase the imprecision of a conclusion is known as "degradation''. The present contribution investigates degradation in the context of second order probability distributions. It uses beta distributions as marginals and copulae together with C-vines to represent dependence structures. It demonstrates that in Bayes' theorem the posterior distributions of the lower and upper probabilities approach 0 and 1 as more and more likelihoods belonging to only one hypothesis are included in the analysis.


probability logic, degradation, Bayes' theorem, pseudodiagnosticity task, second order probability distributions


03B48, 49N30, 62F15, 91E10


  1. G. Boole: An Investigation of the Laws of Thought. Macmillan/Dover Publication, New York 1854/1958.   CrossRef
  2. G. Coletti, D. Petturiti and B. Vantaggi: Bayesian inference: the role of coherence to deal with a prior belief function. Statist. Methods Appl., online, 2014.   CrossRef
  3. G. Coletti and R. Scozzafava: Probabilistic Logic in a Coherent Setting. Kluwer, Dordrecht 2002.   CrossRef
  4. M. E. Doherty, C. R. Mynatt, R. D. Tweney and M. D. Schiavo: Pseudodiagnosticity. Acta Psychologica 43 (1979), 111-121.   DOI:10.1016/0001-6918(79)90017-9
  5. A. Gilio: Generalization of inference rules in coherence-based probabilistic default reasoning. Int. J. Approx. Reasoning 53 (2012), 413-434.   DOI:10.1016/j.ijar.2011.08.004
  6. A. Hanea: Dependence modeling. Vine copula handbook. In: Dependence Modeling. Vine Copula Handbook (D. Kurowicka and H. Joe, eds.), chapter Non-parametric Bayesian belief nets versus vines, World Scientific, New Jersey 2011, pp. 281-303.   DOI:10.1142/9789814299886_0014
  7. H. Joe: Dependence Modeling with Copulas. Chapman and Hall/CRC, Boca Raton 2015.   CrossRef
  8. L. Kern and M. E. Doherty: ``Pseudodiagnosticity'' in an idealized medical problem-solving environment. J. Medical Education 57 (1982), 100-104.   CrossRef
  9. G. D. Kleiter: Propagating imprecise probabilities in Bayesian networks. Artificial Intelligence 88 (1996), 143-161.   DOI:10.1016/s0004-3702(96)00021-5
  10. G. D. Kleiter: Ockham's razor in probability logic. In: Synergies of Soft Computing and Statistics for Intelligent Data Analysis (R. Kruse, M.xQ,R. Berthold, C. Moewes, M. A. Gil, P. Grzegorzewski, and O. Hryniewicz, eds.), Advances in Intelligent Systems and Computation 190, Springer, Heidelberg 2012. pp. 409-417.   DOI:10.1007/978-3-642-33042-1_44
  11. D. Kurowicka and R. Cooke: Distribution-free continuous Bayesian belief nets. In: Proc. Fourth International Conference on Mathematical Methods in Reliability Methodology and Practice, Santa Fe 2004.   CrossRef
  12. D. Kurowicka and R. Cooke: Uncertainty Analysis with High Dimension Dependence Modelling. Wiley, Chichester, 2006.   CrossRef
  13. D. Kurowicka and R. Joe: Dependence Modeling: Vine Copula Handbook. World Scientific, Singapure 2011.   CrossRef
  14. J.-F. Mai and M. Scherer: Simulating Copulas. Stochastic Models, Sampling Algorithms, and Applications. Imperial College Press, London 2012.   CrossRef
  15. R. B. Nelsen: An introduction to Copulas. Springer, Berlin 2006.   CrossRef
  16. R Development Core Team, Vienna and Austria: R: A Language and Environment for Statistical Computing, 2014.    CrossRef
  17. U. Schepsmeier, J. Stoeber, E. C. Brechmann and B. Graeler: Statistical inference of vine copulas. Version 1.2 edition, 2013.   CrossRef
  18. T. Seidenfeld and L. Wasserman: Dilation for sets of probabilities. Ann. Statist. 21 (1993), 1139-1154.   DOI:10.1214/aos/1176349254
  19. R. D. Tweney, M. E. Doherty and G. D. Kleiter: The pseudodiagnosticity trap. Should subjects consider alternative hypotheses? Thinking and Reasoning 16 (2010), 332-345.   DOI:10.1080/13546783.2010.525860
  20. C. Wallmann and G. D. Kleiter: Exchangeability in probability logic. In: Communications in Computer and Information Science (S. Greco, B. Bouchon-Meunier, G. Coletti, M. Fedrizzi, B. Matarazzo, and R. R. Yager, eds.), IPMU (4) 300, Springer, Berlin 2012, pp. 157-167.   DOI:10.1007/978-3-642-31724-8_17
  21. C. Wallmann and G. D. Kleiter: Degradation in probability logic: When more information leads to less precise conclusions. Kybernetika 50 (2014), 268-283.   DOI:10.14736/kyb-2014-2-0268
  22. C. Wallmann and G. D. Kleiter: Probability propagation in generalized inference forms. Studia Logica 102 (2014), 913-929.   DOI:10.1007/s11225-013-9513-4
  23. L. A. Wasserman: Prior envelopes based on belief functions. Annals Statist. 18 (1990), 454-464.   CrossRef