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Walley P. — Statistical reasoning with imprecise probabilities
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Íàçâàíèå: Statistical reasoning with imprecise probabilitiesÀâòîð: Walley P. Àííîòàöèÿ: This text presents a theory of probabilistic reasoning, statistical inference and decision. The book is concerned with the problems of reasoning under conditions of uncertainty, partial information and ignorance. It is argued that, in order to give appropriate weight to both ignorance and uncertainty, imprecise probabilities need to be assessed. The imprecision can be modelled mathematically by upper and lower probabilities or (more generally) upper and lower previsions. The degree of imprecision can reflect both the amount of information on which probabilities are based and the extent of conflict between different types of information. The book develops mathematical methods for reasoning using imprecise probabilities. These include methods for assessing probabilities, modifying the assessments to achieve coherence, updating them to take account of new information, and combining them to calculate other probabilities, draw conclusions and make decisions. The methods are extended in the second half of the book to construct a general theory of conditional probability and statistical inference. The mathematical theory is based on simple and compelling principles of avoiding sure loss, coherence and natural extension. Careful attention is given to the philosophical foundations, interpretation and justification of the theory. It is compared with alternate theories of inference, including Bayesian theories (which require all probability assesments to be precise), Bayesian sensitivity analysis, the Neyman-Pearson theory of confidence intervals, the Dempster-Shafer theory of belief functions and the theory of fuzzy sets. The theory is applicable to a wide range of disciplines including statistics, decision theory, economics, psychology, philosophy of science, management science, operations research, engineering and artificial intelligence. (References are given to related work in these fields). In fact, the theory has important implications for any field in which the problems of uncertainty and limited information are taken seriously. This book should be of interest to researchers in statistics and those in related disciplines.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1991Êîëè÷åñòâî ñòðàíèö: 706Äîáàâëåíà â êàòàëîã: 10.12.2005Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
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action 24 160—61 235—41 Action as a rationality axiom 32 326—27 Action, admissible 161 Action, Bayes 162 165—66 237—40 Action, Behaviour Additivity 89—92 Action, countable 323—27 453—54 Action, de Finettib’s arguments against 326 Action, finite versus countable, examples 70 97—99 310—11 313 321—23 325—26 365 390 396 420—21 450 Action, identified with gamble 160—61 Action, maximal 161—66 Action, minimax 163 166 240 269—70 619 630 Action, optimal, see Action, Bayes reasonable but not Bayes 165—66 Action, relation to conglomerability 310 13 323—24 Action, satisfactory 239—41 (see also Decision) Action, sub- and super 30 84 600 Aczel, J. 523 534 540 Adams, J.B. 50 Admissibility 161 231 Admissibility, compared to coherence 345 509 527 Admissibility, relation to Bayesian optimality 163—64 253 Agassi, J. 500 Aggregation (see Combination of assessments) Agreement, interpersonal 111—13 361 Algorithms for checking coherence 597—98 Algorithms for computing extreme points 174 76 511 Algorithms for computing joint previsions 316 17 454—55 Algorithms for generalized Bayes rule 550 581 Algorithms or computing natural extension 136 174—76 Allais, M. 429 Amaral — Turkman, M.A. 523 Ambiguity 216 261—64 266 Amenability 504 Anderson, B.D.O. 536 Anderson, J.R. 482 Andrews, S.E. 490 Anger, B. 502 Anscombe, F.J. 241 533 Approximation in model building 117 (see also Idealization) Arbitrage 72 Aristotle 484 485 Armstrong, D.M. 481 482 491 Armstrong, T.E. 516 artificial intelligence (see Expert systems) Asai, K. 538 Assessment 32—42 167 Assessment as a sequential process 177—78 Assessment of class of conjugate priors 205—06 218—19 221 225 Assessment of intervals of measures 201—02 Assessment of neighbourhoods 202—03 Assessment of prior previsions 198 200—06 221 403 424 429 Assessment of sampling models 359—60 Assessment of upper and lower density functions 200—02 Assessment of upper and lower distribution functions 203—05 Assessment of upper and lower probabilities 197 99 Assessment of upper and lower probability ratios 199—200 Assessment of upper and lower quantiles 204—05 Assessment strategies (see Strategies assessment) Assessment, combination of (see Combination of assessments in football) Assessment, experiment 634—35 Assessment, guided by an expert system 40 512 Assessment, incomplete 173 212 215—16 256—57 285—86 Assessment, inconsistent 187—88 213—14 216—17 Assessment, reliable 285 288—89 Assessment, using conditional previsions 286 300 Assessment, versus elicitation 15 22—23 167 Assessment, well grounded in evidence 113—14 (see also Strategies assessment Judgement Elicitation Extension) Assignment, probability 273 Aumann, R.J. 241 490 533 Avoiding sure loss 28—29 67—72 115—16 293—94 346 Avoiding sure loss in betting on horses 603—05 Avoiding sure loss in football experiment 635 Avoiding sure loss, comparison with coherence 67 74 Avoiding sure loss, justification for 68 295 343—45 Avoiding sure loss, objections to the emphasis on 114—16 Avoiding sure loss, relation to dominating linear previsions 134—35 (see also Loss) Axiom of Choice 505 611 Axioms, index of 680 Axioms, role of 31—32 Ayer, A.J. 487 501 Banach limits 143—44 Barnard, G.A. 374 501 558 559 568 584 585 Barndorff — Nielsen, O. 570 Barnett, V. 489 568 Baron, J. 484 Basu, D. 583 584 Bayes decision rule 164 Bayes rule as a lower envelope 298 429 432 Bayes rule as an updating strategy 285—86 335 40 Bayes rule as coherence relation (axiom C 12 297 300 303—04 335 Bayes rule as natural extension 318—19 411—12 424—27 Bayes rule for imprecise likelihood functions 337—38 430—34 Bayes rule for precise likelihood functions 392 400—03 424—29 Bayes rule, algorithms for solving 550 581 Bayes rule, conditions for applying 300—01 334 37 Bayes rule, examples of 298—300 308—13 Bayes rule, generalized (GBR) 185 297—301 400—03 424—34 Bayes rule, may yield imprecise updated probabilities 298—301 334 336 338—40 conditional) Bayes rule, psychological model for 17—19 25 Bayes rule, relation to beliefs 17—21 22 109—10 Decision Interpretations behavioural) Bayes rule, Updating Behaviour 17—21 Bayes, T. 43 220 228 377 493 521 Bayesian inference (see Inference statistical) Bayesian sensitivity analysis (see Sensitivity analysis) Bayesian sensitivity analysis as an updating strategy 336 560 Bayesian sensitivity analysis for density functions 331—33 393—96 438—40 Bayesian sensitivity analysis in statistical problems 393—97 427—9 432 Bayesian sensitivity analysis not valid for arbitrary conditioning variables 332 559 Bayesian sensitivity analysis, not sufficient for coherence 305 321 Beach, B.H. 501 Becker, S.W. 48 Behaviourism 19—20 Belief functions 183—84 272—81 358 Belief functions, behavioural interpretation of 280—81 Belief functions, generated through multivalued mapping 182—84 273—74 Belief functions, not sufficiently general 184 274—75 Beliefs 17—19 Beliefs as behavioural dispositions 18—20 481 Beliefs of experts 47 113 213—14 622—23 626 Beliefs, combination of (see Combination of assessments) Beliefs, conflicting 213—14 Beliefs, evolution of 177—78 Beliefs, group 5 105 113 186—88 Beliefs, indeterminate 209—10 Beliefs, mathematical models for 51 156 159—60 Beliefs, rational 26—29 32 Beliefs, role in statistical inference 109—11 Beliefs, second-order 258—61 262 264 510 Indeterminacy Uncertainty) Bellman, R.E. 538 539 Beran, R.J. 489 545 Berberian, S.K. 504 608 Berger, J.O. 43 44 255 256 257 374 478 479 487 488 492 497 500 501 507 508 510 517 518 520 525 526 527 528 529 530 532 535 536 538 540 569 570 573 583 584 Berliner, L.M. 497 518 536 Bernardo, J.M. 228 229 233 525 526 528 540 583 584 Bernoulli models 205 217 467 Bernoulli models, approximate 356 454—55 Bernoulli models, compared to exchangeability 112—13 463—66 Bernoulli models, robust 467—71 Bernoulli, J. 489 543 Beta distribution (see Distribution beta Beta distribution on confidence intervals 380—82 386 Beta distribution on football games 632—38 Beta distribution on horses 94—95 245 602—07 Beta distribution, objections to the emphasis on 114—16 Beta distribution, on parameter values 118—19 367—68 429 Beta distribution, pari-mutuel system 94—95 130—31 135 147 202—03 604—06 690 Betting procedure, biased 381—82 384 565—66 Betting rates 82 602—03 Betting rates of bookmakers 87—88 95 245 602 605 625 28 Betting rates, fair (two-sided) 89 243—44 251 624—25 Betting rates, lower 82 251 632 Betting rates, upper 82 602 632 Probability additive Probability lower Probability upper) Beyth — Marom, R. 514 539 Bhaskara Rao, K.P.S. 496 498 504 Bhaskara Rao, M. 496 498 504 Bias in elicitation 624—28 Bias of measuring instrument, model for 399—400 405 423—24 Biller, W.F. 538 Billingsley, P. 498 553 558 559 574 587 588 Birnbaum, A. 483 572 583 584 Blachman, N.M. 523 Black, M. 514 539 Blackburn, S. 563 Blackwell, D. 528 Block, N. 482 491 Blum, J.R. 530 536 Boes, D.C. 569 Boesky, I.F. 495 Bondar, J.V. 569 Bookmakers (see also Betting rates) 95 245 625—28 Boole, G. 43 44 487 489 494 503 526 586 Borel cr-field 98 Borel, E. 43 46 481 489 624 625 Bounds, upper and lower for chances 356—57 468 Bounds, upper and lower for chances for conditional probabilities 296 301 Bounds, upper and lower for chances for density functions 199—201 Bounds, upper and lower for chances for posterior previsions 432 Bounds, upper and lower for chances for probabilities 35 46—47 216 Envelopes) Box, G.E.P. 43 228 229 233 500 525 526 528 538 564 565 567 568 583 584 593 Braithwaite, R.B. 481 501 Breiman, L. 553 Bridgman, P.W. 482 Brown, L.D. 383 573 Brown, R.V. 499 500 534 536 537 Brownson, F.O. 48 Buchanan, B.G. 50 490 536 Budescu, D.V. 48 510 539 672 Buehler, R.J. 47 377 381 383 494 506 530 568 570 Burgess, J.P. 514 Calibration 39 Campello de Souza, F.M. viii 47 488 500 508 520 529 Cano, J.A. 517 518 536 Cantelli — Levy paradox 365 450 Caramazza, A. 539 Carnap, R. 16 23 34 43 113 238 480 Casella, G. 569 571 572 Certainty 54 209 Certainty factors (MYCIN) 50 Chamberlain, G. 536 Chance 351—59 465—66 Chance, imprecise 358—59 454 468 Chance, lower bound for 356—57 468 588 Chemoff, H. 491 Choice 160—66 236—41 Choice, arbitrary 19 23—41 210 Choice, between models for uncertainty 159 60 Choice, strategies for 239—40 Choice, versus preference 236—37 242 247—48 Action Preference) Choquet, G. 502 505 Chuang, D.T. 536 Classification of probable events (see Probability classificatory) Coarsening (of a possibility space) 181—82 Cohen, L.J. 39 50 488 490 501 539 Coherence 29—30 63—66 72—76 346—47 Coherence for statistical models 366 380 391 392 397 401 404 Coherence for win-and-place betting on horses 605—06 Coherence in football experiment 636 Coherence of Bayesian models 394 405 Coherence of conditional with unconditional previsions 293—96 301—05 Coherence of lower previsions 72—76 134 Coherence of lower probabilities 84—86 135 600 Coherence, algorithm for checking 597—98 Coherence, axioms, refer to Index of axioms 680 Coherence, compared to avoiding sure loss 67 74 347 Coherence, countable 32 495 556 Coherence, general concepts 342—49 Coherence, justification for 60—61 64 73—74 295—96 347 Coherence, relation to dominating linear previsions 134—35 145—46 Coherence, relation to natural extension 74 122 23 408—09 Coherence, separate 289—93 Coherence, strength of 74 116—18 Coherence, strict 32 495 581 Coherence, weak 346—47 391 566 Coin tossing, models for 119 274—75 298—99 Collective (von Mises) 351 564 Combination of assessments 30 39 170—72 178—80 186—88 275—81 338 452—57 Compactness 610 Comparability (see Completeness) Completeness 157 190 195—97 241—48 601 Computations (see Algorithms) Conclusions 21—22 Conclusions, indeterminate 2 5—6 209—10 226 Conclusions, robust 5—6 256 Inference statistical Prevision posterior) Conditioning 185—86 282—340 Conditioning on continuous variables 330—33 436—40 Conditioning on events of probability zero 306—07 328—34 391—92 615—16 Conditioning, de Finettin’s theory 282 326—27 Conditioning, examples of 185—86 298—300 308—13 321 Conditioning, in frequentist inference 382 387—88 Conditioning, Kolmogorov’s theory 282 306—08 327 conditional Conglomerability Bayes generalized Updating) Cone, convex 76 Confidence coefficient 378 Confidence coefficient as a posterior lower probability 386 Confidence coefficient as a posterior probability 22 378—80 388 Confidence coefficient, data-dependent 387 confidence intervals 377—88 Confidence intervals, axioms for coherence 380 Confidence intervals, examples of incoherence 379 382—85 Confidence intervals, interpretations of 378 382 385—87 Confidence intervals, need for conditioning 382 387—88 Confidence intervals, posterior confidence in 378—80 388 Confidence intervals, zero confidence in 379 385 569 Conflict 187—88 213—14 222—26 Conflict, between expert opinions 47 213—14 520 Conflict, between probability models 187—88 213—14 216—17 Conflict, degree of 223 224—26 Conflict, information 6 213—14 222—26 Conglomerability 317—27 614—16 31— 317 Conglomerability for uncountable spaces 324—26 3 Conglomerability, examples of failure 311 321—22 Conglomerability, full 317 614 Conglomerability, justification for 319—20 Conglomerability, relation to countable additivity 310 13 323—24 Conglomerability, relation to sensitivity analysis 327 Conglomerative principle (see Principles of rationality conglomerative) Conjugacy 64—65 70 Conjugate prior (see Density function natural-conjugate) conjunction rule 186 Consensus (see Agreement interpersonal Combination Consistency level 386 Constant odds-ratio (COR) model (see Odds ratio constant) Construction of probability models (see Assessment Strategies assessment Model statistical) Convergence pointwise 79 Convergence pointwise of relative frequencies (see Frequency relative) Convergence pointwise of relative frequencies of upper and lower probabilities 3—4 219 220 223—25 470—71 620—21 Convex combination 79 93 Convex hull 611 613 Convexity 611 Convexity of set of coherent lower previsions 79 Convexity of set of dominating linear previsions 145—46 Convexity of set of gambles 162 Cornfield, J. 569 571 Countable additivity (see Additivity countable) Cox, D.R. 386 483 487 489 500 525 526 527 528 565 567 569 570 572 573 583 591 Cox, R.T. 44 534 Credible intervals 383 387 Curley, S.P. 48 Currency, probability 25 59 Dalai, S.R. 518 Daniell integral 493 Daniell, P.J. 493
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