Inferences from multinomial data: Learning about a bag of marbles. (With discussion). (English) Zbl 0834.62004

Walley, Peter

J. R. Stat. Soc., Ser. B 58, No. 1, 3-57 (1996).

Summary: A new method is proposed for making inferences from multinomial data in cases where there is no prior information. A paradigm is the problem of predicting the colour of the next marble to be drawn from a bag whose contents are (initially) completely unknown. In such problems we may be unable to formulate a sample space because we do not know what outcomes are possible. This suggests an invariance principle: inferences based on observations should not depend on the sample space in which the observations and future events of interest are represented. Objective Bayesian methods do not satisfy this principle.
This paper describes a statistical model, called the imprecise Dirichlet model, for drawing coherent inferences from multinomial data. Inferences are expressed in terms of posterior upper and lower probabilities. The probabilities are initially vacuous, reflecting prior ignorance, but they become more precise as the number of observations increases. This model does satisfy the invariance principle.
Two sets of data are analysed in detail. In the first example one red marble is observed in six drawings from a bag. Inferences from the imprecise Dirichlet model are compared with objective Bayesian and frequentist inferences. The second example is an analysis of data from medical trials which compared two treatments for cardiorespiratory failure in newborn babies. There are two problems: to draw conclusions about which treatment is more effective and to decide when the randomized trials should be terminated. This example shows how the imprecise Dirichlet model can be used to analyse data in the form of a contingency table.

MSC:

62A01	Foundations and philosophical topics in statistics
62P10	Applications of statistics to biology and medical sciences; meta analysis

Keywords:

Bernoulli trials; Dirichlet distribution; imprecise probabilities; inductive logic; randomized clinical trials; new method; multinomial data; invariance principle; imprecise Dirichlet model; coherent inferences; posterior upper and lower probabilities; prior ignorance; frequentist inferences; cardiorespiratory failure; contingency table

Software:

PULCinella

References:

[1]	Adlassnig, K. -P.; Kolarz, G.: CADIAG-2: computer-assisted medical diagnosis using fuzzy subsets. Approximate reasoning in decision analysis, 219-247 (1982)
[2]	Andersen, S. K.; Olesen, K. G.; Jensen, F. V.; Jensen, F.: HUGIN–a shell for building Bayesian belief univeises for expert systems. Proceedings IJCAI-89, 1080-1085 (1989) · Zbl 0713.68051
[3]	Beyth-Marom, R.: How probable is probable? numerical translations of verbal probability expressions. J. forecasting 1, 257-269 (1982)
[4]	Biswas, G.; Anand, T. S.: Using the Dempster-Shafer scheme in a mixed-initiative expert system shell. Uncertainty in artificial intelligence 3, 223-239 (1989)
[5]	Biswas, G.; Oliff, M.; Abramczyk, R.: OASES (Operations analysis expert system): an application in fiberglass manufacturing. Int. J. Expert syst. Res. appl. 1, 193-216 (1988)
[6]	Buchanan, B. G.; Shortliffe, E. H.: Rule-based expert systems. (1984)
[7]	Buisson, J. -C.; Farreny, H.; Prade, H.: The development of a medical expert system and the treatment of imprecision in the framework of possibility theory. Inf. sci. 37, 211-226 (1985)
[8]	Buxton, R.: Modelling uncertainty in expert systems. Int. J. Man-Mach stud. 31, 415-476 (1989)
[9]	De Campos, L. M.; Lamata, M. T.; Moral, S.: The concept of conditional fuzzy measure. Int. J. Intell. syst. 5, 237-246 (1990) · Zbl 0694.68058
[10]	Cano, J. E.; Moral, S.; Verdegay, J. F.: Propagation of convex sets of probabilities in directed acyclic networks. Proceedings fourth IPMU conference, 289-292 (1992)
[11]	Cheeseman, P.: Probabilistic versus fuzzy reasoning. Uncertainty in artificial intelligence, 85-102 (1986) · Zbl 0607.68076
[12]	Cohen, L. J.: The probable and the provable. (1977)
[13]	Cohen, P. R.: Heuristic reasoning about uncertainty: an artificial intelligence approach. (1985)
[14]	Cowell, R. G.: BAIES–a probabilistic expert system shell with qualitative and quantitative learning. Bayesian statistics 4, 595-600 (1992)
[15]	Dempster, A. P.: Upper and lower probabilities induced by a multivalued mapping. Ann. math. Stat. 38, 325-339 (1967) · Zbl 0168.17501
[16]	Dubois, D.; Prade, H.: An introduction to possibilistic and fuzzy logics. Non-standard logics for automated reasoning, 287-326 (1988)
[17]	Dubois, D.; Prade, H.: Possibility theory. (1988) · Zbl 0645.68108
[18]	Dubois, D.; Prade, H.: Modelling uncertain and vague knowledge in possibility and evidence theories. Uncertainty in artificial intelligence 4, 303-318 (1990)
[19]	Dubois, D.; Prade, H.: Evidence, knowledge, and belief functions. Int. J. Approx. reasoning 6, 295-319 (1992) · Zbl 0749.68085
[20]	Webber, B. W.; Nilsson, N. J.: Readings in artificial intelligence. 192-199 (1981) · Zbl 0498.68054
[21]	Duda, R. O.; Reboh, R.: AI and decision making: the PROSPECTOR experience. Artificial intelligence applications for business, 111-147 (1984)
[22]	Dutta, A.: Reasoning with imprecise knowledge in expert systems. Inf. sci. 37, 3-24 (1985)
[23]	Fagin, R.; Halpern, J. Y.: A new approach to updating beliefs. Uncertainty in artificial intelligence 6, 347-374 (1991) · Zbl 0742.68067
[24]	Farreny, H.; Prade, H.; Wyss, E.: Approximate reasoning in a rule-based expert system using possibility theory: a case study. Information processing ’86, 407-413 (1986)
[25]	Fertig, K. W.; Breese, J. S.: Interval influence diagrams. Uncertainty in artificial intelligence 5, 149-161 (1990) · Zbl 0721.68061
[26]	Fieschi, M.; Joubert, M.; Fieschi, D.; Botti, G.; Roux, M.: A program for expert diagnosis and therapeutic decision. Med. inf. 8, 127-135 (1983)
[27]	Fine, T. L.: Theories of probability. (1973) · Zbl 0275.60006
[28]	De Finetti, B.: Theory of probability. 1 (1974) · Zbl 0328.60002
[29]	De Finetti, B.: Theory of probability. 2 (1975) · Zbl 0328.60003
[30]	Van Der Gaag, L. C.: Computing probability intervals under independency constraints. Uncertainty in artificial intelligence 6, 457-466 (1991)
[31]	Gale, W. A.: Artificial intelligence and statistics. (1986) · Zbl 0618.90014
[32]	Giles, R.: Foundations for a theory of possibility. Fuzzy information and decision processes, 183-195 (1982) · Zbl 0514.94028
[33]	Giles, R.: Semantics for fuzzy reasoning. Int. J. Man-Mach stud. 17, 401-415 (1982) · Zbl 0515.03013
[34]	Gordon, J.; Shortliffe, E. H.: A method for managing evidential reasoning in a hierarchical hypothesis space. Artif. intell. 26, 323-357 (1985) · Zbl 1134.68530
[35]	Grosof, B. N.: An inequality paradigm for probabilistic knowledge. Uncertainty in artificial intelligence, 259-275 (1986) · Zbl 0608.68076
[36]	Halpern, J. Y.; Fagin, R.: Two views of belief: belief as generalized probability and belief as evidence. Artif. intell. 54, 275-317 (1992) · Zbl 0762.68055
[37]	Heckerman, D.; Horvitz, E.; Nathwani, B.: Toward normative expert systems I: The PATHFINDER project. Methods inf. Med. 31, 90-105 (1992)
[38]	Henrion, M.; Breese, J. S.; Horvitz, E. J.: Decision analysis and expert systems. AI magazine 12, 64-91 (1991)
[39]	Hsia, Y. -T.; Shenoy, P. P.: An evidential language for expert systems. Methodologies for intelligent systems 4, 9-16 (1989)
[40]	Huber, P. J.: Robust statistics. (1981) · Zbl 0536.62025
[41]	Jaffray, J. -Y.: Bayesian updating and belief functions. IEEE trans. Syst. man cybern. 22, 1144-1152 (1992) · Zbl 0769.62001
[42]	Jaynes, E. T.: Papers on probability, statistics and statistical physics. (1983) · Zbl 0501.01026
[43]	Jensen, F. V.; Andersen, S. K.; Kjaerulff, U.; Andreassen, S.: MUNIN–on the case for probabilities in medical expert systems. Lecture notes in medical informatics 33 (1987)
[44]	Kak, A. C.; Andress, K. M.; Lopez-Abadia, C.; Caroll, M. S.; Lewis, J. R.: Hierarchical evidence accumulation in the PSEIKI system. Uncertainty in artificial intelligence 5 (1990) · Zbl 0721.68098
[45]	Kent, S.: Words of estimated probability. Stud. intell. 8, 49-65 (1964)
[46]	Klir, G. J.; Folger, T. A.: Fuzzy sets, uncertainty, and information. (1988) · Zbl 0675.94025
[47]	Krause, P.; Clark, D.: Representing uncertain knowledge. (1993) · Zbl 0865.68116
[48]	Jr, H. E. Kyburg: Bayesian and non-Bayesian evidential updating. Artif. intell. 31, 271-293 (1987) · Zbl 0622.68069
[49]	Laskey, K. B.; Cohen, M. S.; Martin, A. W.: Representing and eliciting knowledge about uncertain evidence and its implications. IEEE trans. Syst. man cybern. 19, 536-557 (1989)
[50]	Lauritzen, S. L.; Spiegelhalter, D. J.: Local computations with probabilities on graphical structures and their application to expert systems (with discussion). J. roy. Stat. soc. Ser. B 50, 157-224 (1988) · Zbl 0684.68106
[51]	Lebailly, J.; Martin-Clouaire, R.; Prade, H.: Use of fuzzy logic in rule-based systems in petroleum geology. Approximate reasoning in intelligent systems, decision and control, 125-144 (1987)
[52]	Lemmer, J. F.: Generalised Bayesian updating of incompletely specified distributions. Large scale syst. 5, 51-68 (1933) · Zbl 0545.62026
[53]	Leung, K. S.; Wong, W. S. F.; Lam, W.: Applications of a novel fuzzy expert system shell. Expert syst. 6, 2-10 (1939)
[54]	Levi, I.: Potential surprise: its role in inference and decision making. Applications of inductive logic, 1-27 (1980)
[55]	Levi, I.: Consonance, dissonance and evidentiary mechanisms. Library of theoria 15, 27-43 (1983)
[56]	Lindley, D. V.: Making decisions. (1971)
[57]	Loui, R. P.: Interval-based decisions for reasoning systems. Uncertainty in artificial intelligence, 459-472 (1986)
[58]	Lowrance, J. D.: Automated argument construction. J. stat. Planning inference 20, 369-387 (1988) · Zbl 0655.68120
[59]	Moore, R. C.: Semantical considerations on nonmonotonic logic. Artif. intell. 25, 75-94 (1985) · Zbl 0569.68079
[60]	Nilsson, N. J.: Probabilistic logic. Artif. intell. 28, 71-87 (1986) · Zbl 0589.03007
[61]	Orponen, P.: Dempster’s rule is #P-complete. Artif. intell. 44, 245-253 (1990) · Zbl 0717.68048
[62]	Paass, G.: Probabilistic logic. Non-standard logics for automated reasoning, 213-251 (1988)
[63]	Pearl, J.: Probabilistic reasoning in intelligent systems. (1988) · Zbl 0649.68104
[64]	Pearl, J.: Reasoning with belief functions: an analysis of compatibility. Int. J. Approx. reasoning 4, 363-389 (1990) · Zbl 0706.68086
[65]	Pearl, J.: Rejoinder to comments on ”reasoning with belief functions: an analysis of compatibility”. Int. J. Approx. reasoning 6, 425-443 (1992) · Zbl 0751.68063
[66]	Quinlan, J. R.: Inferno: a cautious approach to uncertain inference. Comput. J. 26, 255-269 (1983)
[67]	Reiter, R.: A logic for default reasoning. Artif. intell. 13, 81-132 (1980) · Zbl 0435.68069
[68]	Reiter, R.: Nonmonotonic reasoning. Ann. rev. Comput. sci. 2, 147-186 (1987)
[69]	Saffiotti, A.; Umkehrer, E.: PULCINELLA: a general tool for propagating uncertainty in valuation networks. Proceedings seventh international conference on uncertainty in AI, 323-331 (1991)
[70]	Shackle, G. L. S.: Decision, order, and time in human affairs. (1969)
[71]	Shafer, G.: A mathematical theory of evidence. (1976) · Zbl 0359.62002
[72]	Shafer, G.: Constructive probability. Synthese 48, 1-60 (1981) · Zbl 0522.60001
[73]	Shafer, G.: Belief functions and parametric models (with discussion). J. roy. Stat. soc. Ser. B 44, 322-352 (1982) · Zbl 0499.62007
[74]	Shafer, G.: Probability judgement in artificial intelligence and expert systems. Stat. sci. 2, 3-16 (1987) · Zbl 0955.68506
[75]	Shafer, G.: Perspectives on the theory and practice of belief functions. Int. J. Approx. reasoning 4, 323-362 (1990) · Zbl 0714.62001
[76]	Shafer, G.: Rejoinders to comments on ”perspectives on the theory and practice of belief functions”. Int. J. Approx. reasoning 6, 445-480 (1992) · Zbl 0749.68081
[77]	Shafer, G.; Logan, R.: Implementing Dempster’s rule for hierarchical evidence. Artif. intell. 33, 271-298 (1987) · Zbl 0633.68093
[78]	Shenoy, P.; Shafer, G.: Propagating belief functions with local computations. IEEE expert 1, No. 3, 43-52 (1986)
[79]	Sheridan, F. K. J.: A survey of techniques for inference under uncertainty. Artif. intell. Rev. 5, 89-119 (1991)
[80]	Shortliffe, E. H.: Computer-based medical consultations: MYCIN. (1976)
[81]	Shortliffe, E. H.; Buchanan, B. G.: A model of inexact reasoning in medicine. Math. biosci. 23, 351-379 (1975)
[82]	Smets, P.: Belief functions. Non-standard logics for automated reasoning, 253-286 (1988)
[83]	Smets, P.: The transferable belief model and other interpretations of Dempster-Shafer’s model. Uncertainty in artificial intelligence 6, 375-383 (1991)
[84]	Smets, P.: Resolving misunderstandings about belief functions. Int. J. Approx. reasoning 6, 321-344 (1992) · Zbl 0749.68087
[85]	Smets, P.; Kennes, R.: The transferable belief model. Artif. intell. 66, 191-234 (1994) · Zbl 0807.68087
[86]	Smets, P.; Mamdani, A.; Dubois, D.; Prade, H.: Non-standard logics for automated reasoning. (1988) · Zbl 0669.03002
[87]	Smith, C. A. B.: Consistency in statistical inference and decision (with discussion). J. roy. Stat. soc. Ser. B 23, 1-37 (1961) · Zbl 0124.09603
[88]	Smithson, M.: Ignorance and uncertainty. (1989)
[89]	Spiegelhalter, D. J.: A statistical view of uncertainty in expert systems. Artificial intelligence and statistics (1986)
[90]	Spiegelhalter, D. J.; Dawid, A. P.; Lauritzen, S. L.; Cowell, R. G.: Bayesian analysis in expert systems (with discussion). Stat. sci. 8, 219-283 (1993) · Zbl 0955.62523
[91]	Spiegelhalter, D. J.; Knill-Jones, R. P.: Statistical and knowledge-based approaches to clinical decision-support systems, with an application in gastroenterology (with discussion). J. roy. Stat. soc. Ser. A 147, 35-76 (1984) · Zbl 0559.62089
[92]	Strat, T. M.: Decision analysis using belief functions. Int. J. Approx. reasoning 4, 391-417 (1990) · Zbl 0706.68089
[93]	Sundberg, C.; Wagner, C.: Generalized finite differences and Bayesian conditioning of Choquet capacities. (1991) · Zbl 0757.60001
[94]	Tessem, B.: Interval probability propagation. Int. J. Approx. reasoning 7, 95-120 (1992) · Zbl 0769.68113
[95]	Voorbraak, F.: On the justification of Dempster’s rule of combination. Artif. intell. 48, 171-197 (1991) · Zbl 1117.68496
[96]	Walley, P.: Coherent lower (and upper) probabilities. Statistics research report (1981)
[97]	Walley, P.: The elicitation and aggregation of beliefs. Statistics research report (1982)
[98]	Walley, P.: Belief function representations of statistical evidence. Ann. stat. 15, 1439-1465 (1987) · Zbl 0645.62003
[99]	Walley, P.: Statistical reasoning with imprecise probabilities. (1991) · Zbl 0732.62004
[100]	Walley, P.: Measures of uncertainty in expert systems. Statistics research report (1991) · Zbl 0850.62716
[101]	Walley, P.: Inferences from multinomial data: learning about a bag of marbles (with discussion). J. roy. Stat. soc. Ser. B 58, 3-57 (1996) · Zbl 0834.62004
[102]	Walley, P.: Statistical inferences based on a second-order possibility distribution (with discussion). J. am. Stat. assoc. 91 (1996) · Zbl 0905.62026
[103]	Willey, P.; De Souza, F. M. Campello: Uncertainty and indeterminacy in assessing the economic viability of energy options: a case study of solar heating systems in Brazil. Energy syst. Policy 14, 281-304 (1990)
[104]	Walley, P.; Fine, T. L.: Varieties of modal (classificatory) and comparative probability. Synthese 41, 321-374 (1979) · Zbl 0442.60005
[105]	Walley, P.; Fine, T. L.: Towards a frequentist theory of upper and lower probability. Ann. stat. 10, 741-761 (1982) · Zbl 0488.62004
[106]	Wallsten, T. S.; Budescu, D. V.; Rapoport, A.; Zwick, R.; Forsyth, B.: Measuring the vague meanings of probability terms. J. exper. Psych. general 115, 348-365 (1986)
[107]	Wasserman, L. A.: Comments on Shafer’s ”perspectives on the theory and practice of belief functions”. Int. J. Approx. reasoning 6, 367-375 (1992) · Zbl 0749.68088
[108]	Wasserman, L. A.; Kadane, J.: Bayes theorem for Choquet capacities. Ann. stat. 18, 1328-1339 (1990) · Zbl 0736.62026
[109]	Watson, S. R.; Weiss, J. J.; Donnell, M. L.: Fuzzy decision analysis. IEEE trans. Syst. man cybern. 9, 1-9 (1979)
[110]	Williams, P. M.: Indeterminate probabilities. Formal methods in the methodology of empirical sciences, 229-246 (1976) · Zbl 0395.60003
[111]	Wilson, N.: A Monte-Carlo algorithm for Dempster-Shafer belief. Proceedings seventh international conference on uncertainty in AI, 414-417 (1991)
[112]	Wilson, N.: The combination of belief: when and how fast?. Int. J. Approx. reasoning 6, 377-388 (1992) · Zbl 0749.68089
[113]	Wilson, N.; Moral, S.: A logical view of probability. Proceedings eleventh European conference on artificial intelligence, 386-390 (1994)
[114]	Xu, H.: An efficient implementation of belief function propagation. Proceedings seventh international conference on uncertainty in AI, 425-432 (1991)
[115]	Yager, R. R.: An introduction to applications of possibility theory (with discussion). Human syst. Manage. 3, 246-269 (1983)
[116]	Zadeh, L. A.: Fuzzy sets. Inf. control 8, 338-353 (1965) · Zbl 0139.24606
[117]	Zadeh, L. A.: The concept of a linguistic variable and its application to approximate reasoning (Part 3). Inf. sci. 9, 43-80 (1976) · Zbl 0404.68075
[118]	Zadeh, L. A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy sets syst. 1, 3-28 (1978) · Zbl 0377.04002
[119]	Zadeh, L. A.: A theory of approximate reasoning. Machine intelligence 9, 149-194 (1979)
[120]	Zadeh, L. A.: The role of fuzzy logic in the management of uncertainty in expert systems. Fuzzy sets syst. 11, 199-227 (1983) · Zbl 0553.68049
[121]	Zadeh, L. A.: Is probability theory sufficient for dealing with uncertainty in AI: a negative view. Uncertainty in artificial intelligence, 103-116 (1986) · Zbl 0607.68075