Using probability trees to compute marginals with imprecise probabilities

This paper presents an approximate algorithm to obtain a posteriori intervals of probability, when available information is also given with intervals. The algorithm uses probability trees as a means of representing and computing with the convex sets of probabilities associated to the intervals.

[1]  J. López.  Representación y combinación de la información con incertidumbre mediante convexos de probabilidades , 1998 .

[2]  David Poole,et al.  Probabilistic Partial Evaluation: Exploiting Rule Structure in Probabilistic Inference , 1997, IJCAI.

[3]  A. Salmerón,et al.  Importance sampling in Bayesian networks using probability trees , 2000 .

[4]  Glenn Shafer,et al.  A Mathematical Theory of Evidence , 2020, A Mathematical Theory of Evidence.

[5]  Hung T. Nguyen,et al.  Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference , 1994 .

[6]  Bjørnar Tessem,et al.  Interval probability propagation , 1992, Int. J. Approx. Reason..

[7]  Peter Walley,et al.  Towards a unified theory of imprecise probability , 2000, Int. J. Approx. Reason..

[8]  Brent Boerlage.  Link Strength in Bayesian Networks , 1994 .

[9]  L. Wasserman,et al.  Inferences from multinomial data: Learning about a bag of marbles - Discussion , 1996 .

[10]  Peter Walley,et al.  Measures of Uncertainty in Expert Systems , 1996, Artif. Intell..

[11]  Luis M. de Campos,et al.  Updating Uncertain Information , 1990, IPMU.

[12]  Luis M. de Campos,et al.  Independence Concepts for Convex Sets of Probabilities , 1995, UAI.

[13]  David Heckerman,et al.  Troubleshooting Under Uncertainty , 1994 .

[14]  Rina Dechter,et al.  Bucket elimination: A unifying framework for probabilistic inference , 1996, UAI.

[15]  J. Hartigan,et al.  Bayesian Inference Using Intervals of Measures , 1981 .

[16]  Inés Couso,et al.  Examples of Independence for Imprecise Probabilities , 1999, ISIPTA.

[17]  P. Walley.  Statistical Reasoning with Imprecise Probabilities , 1990 .

[18]  Nir Friedman,et al.  Learning Bayesian Networks with Local Structure , 1996, UAI.

[19]  Eric Beattie,et al.  Alarm monitoring system , 2001 .

[20]  L. Zadeh.  Fuzzy sets as a basis for a theory of possibility , 1999 .

[21]  James O. Berger,et al.  An overview of robust Bayesian analysis , 1994 .

[22]  J. Kacprzyk,et al.  Aggregation and Fusion of Imperfect Information , 2001 .

[23]  Peter Walley.  A Bounded Derivative Model for Prior Ignorance about a Real-valued Parameter , 1997 .

[24]  G. Klir,et al.  Fuzzy Measure Theory , 1993 .

[25]  P. Walley.  Inferences from Multinomial Data: Learning About a Bag of Marbles , 1996 .

[26]  菅野 道夫,et al.  Theory of fuzzy integrals and its applications , 1975 .

[27]  Ronald R. Yager,et al.  Uncertainty in Intelligent Systems , 1993 .

[28]  Daphne Koller,et al.  Nonuniform Dynamic Discretization in Hybrid Networks , 1997, UAI.

[29]  Smets Ph.,et al.  Belief functions, Non-standard logics for automated reasoning , 1988 .

[30]  Eugene Santos,et al.  Exploiting case-based independence for approximating marginal probabilities , 1996, Int. J. Approx. Reason..

[31]  Robert Nau,et al.  The Aggregation of Imprecise Probabilities , 2002, ISIPTA.

[32]  Serafín Moral,et al.  Aggregation of Imprecise Probabilities , 1998 .

[33]  George J. Klir,et al.  Fuzzy sets, uncertainty and information , 1988 .

[34]  I. Levi.  On Indeterminate Probabilities , 1974 .

[35]  C. Howson.  Theories of Probability , 1995 .

[36]  Werner Kießling,et al.  Towards Precision of Probabilistic Bounds Propagation , 1992, UAI.

[37]  J. E. Cano,et al.  Simulation Algorithms for Convex Sets of Probabilities Simulation Algorithms for the Propagation of Convex Sets of Probabilities on a Tree of Cliques , 1993 .

[38]  Andrew P. Sage,et al.  Uncertainty in Artificial Intelligence , 1987, IEEE Transactions on Systems, Man, and Cybernetics.

[39]  Gregory F. Cooper,et al.  The ALARM Monitoring System: A Case Study with two Probabilistic Inference Techniques for Belief Networks , 1989, AIME.

[40]  Prakash P. Shenoy,et al.  Axioms for probability and belief-function proagation , 1990, UAI.

[41]  Terrence L. Fine.  An Argument for Comparative Probability , 1977 .

[42]  Prakash P. Shenoy,et al.  A valuation-based language for expert systems , 1989, Int. J. Approx. Reason..

[43]  Craig Boutilier,et al.  Context-Specific Independence in Bayesian Networks , 1996, UAI.

[44]  Serafín Moral,et al.  Calculating Uncertainty Intervals from Conditional Convex Sets of Probabilities , 1992, UAI.

[45]  Fabio Gagliardi Cozman,et al.  Robustness Analysis of Bayesian Networks with Local Convex Sets of Distributions , 1997, UAI.

[46]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems , 1988 .

[47]  Zhaoyu Li,et al.  Efficient inference in Bayes networks as a combinatorial optimization problem , 1994, Int. J. Approx. Reason..

[48]  B. Ripley,et al.  Robust Statistics , 2018, Encyclopedia of Mathematical Geosciences.

[49]  J. Ross Quinlan,et al.  Induction of Decision Trees , 1986, Machine Learning.

[50]  L. M. Berliner,et al.  Robust Bayes and Empirical Bayes Analysis with $_\epsilon$-Contaminated Priors , 1986 .

[51]  Kurt Weichselberger.  The theory of interval-probability as a unifying concept for uncertainty , 2000, Int. J. Approx. Reason..

[52]  Serafín Moral,et al.  Propagación exacta y aproximada mediante árboles de probabilidad en redes causales , 1997 .

[53]  Fabio Gagliardi Cozman,et al.  Robustness Analysis of Bayesian Networks with Finitely Generated Convex Sets of Distributions , 1997 .

[54]  Andrés Cano,et al.  Convex Sets Of Probabilities Propagation By Simulated Annealing , 1994 .

[55]  Andrés Cano,et al.  A Genetic algorithm to approximate convex sets of probabilities , 1996 .

[56]  John S. Breese,et al.  Interval Influence Diagrams , 1989, UAI.

[57]  Serafín Moral,et al.  Removing partial inconsistency in valuation‐based systems , 1997 .

[58]  Didier Dubois,et al.  Constraint Propagation with Imprecise Conditional Probabilities , 1991, UAI.

[59]  J. Hartigan.  Theories of Probability , 1983 .

[60]  Philippe Smets.  Non-standard logics for automated reasoning , 1988 .

[61]  Serafín Moral,et al.  Penniless propagation in join trees , 2000 .

[62]  Luis M. de Campos,et al.  Probability Intervals: a Tool for uncertain Reasoning , 1994, Int. J. Uncertain. Fuzziness Knowl. Based Syst..

[63]  Serafín Moral,et al.  Combination of Upper and Lower Probabilities , 1991, UAI.

[64]  Luis M. de Campos,et al.  Removing partial inconsistency in valuation-based systems , 1997, Int. J. Intell. Syst..

[65]  Klemens Szaniawski,et al.  Formal methods in the methodology of empirical sciences : proceedings of the Conference for Formal Methods in the Methodology of Empirical Sciences, Warsaw, June 17-21, 1974 , 1976 .

[66]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[67]  Nevin Lianwen Zhang,et al.  Exploiting Causal Independence in Bayesian Network Inference , 1996, J. Artif. Intell. Res..

[68]  Judea Pearl,et al.  Probabilistic reasoning in intelligent systems - networks of plausible inference , 1991, Morgan Kaufmann series in representation and reasoning.