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Walley P. — Statistical reasoning with imprecise probabilities
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Название: Statistical reasoning with imprecise probabilitiesАвтор: Walley P. Аннотация: 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.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1991Количество страниц: 706Добавлена в каталог: 10.12.2005Операции: Положить на полку |
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Предметный указатель
Reasoning 26—27 30—31 35—42 Reasoning, inductive (see Reasoning probabilistic) Reasoning, Logic, inductive principles of (see Principles of rationality probabilistic) Reasoning, standards of 26—27 Reasoning, statistical 406 Reasoning, versus rationality 26 (see also Rationality Inference statistical Extension natural Strategies assessment) Reductionism (see Interpretations of probability and prevision reductionist) Refinement (of a possibility space) 56—57 180—81 337 Regazzini, E. 549 555 642 Regret 508 Reichenbach, H. 480 534 562 Relevance (see Information relevant) Relevant subset 381—86 Renyi, A. 549 553 Rescher, N. 514 515 Reward from action 160 Reward from action in elicitation 624 629—31 Rios, S. 47 506 507 528 536 Risk function 164 Robinson, G.K. 377 569 570 571 572 573 Robustness 5—6 Robustness of Bayesian inferences 6 256 Robustness of independence judgements 589 Robustness of inferences from conjugate priors 206 Robustness of likelihood function 431 Robustness of sampling models 356—57 Rockafellar, R.T. 511 Roles of probability-utility models 17 Roles of probability-utility models, constructive 21—26 30—31 35—42 Roles of probability-utility models, explanatory (descriptive) 17 19 116 17 Roles of probability-utility models, in decision 24—25 160—61 Roles of probability-utility models, in inference 21—22 26 30—31 Roles of probability-utility models, normative 27—35 41—42 Rorty, R. 500 Rosenblatt, J. 530 536 Rosenkrantz, R.D. 43 267 525 528 540 541 Ross, L. 483 488 Royden, H.L. 493 498 504 559 574 588 Rubin, H. 484 Russell, B. 44 499 532 543 562 Ryle, G. 19 481 565 Sachs, NJ. 480 536 Saddle point 613 619 Sage, A.P. 490 Sahlin, N. 44 49 507 530 535 538 Salmon, W.C. 480 562 Sample equivalent to prior information 221—22 Sample space 362 Sample space, continuous 328—33 435—40 Sampling models (see Models sampling) Satisfaction level 240 Satisficing 239—40 Savage, L.J. 16 43 44 45 241 242 246 247 478 483 484 485 489 490 491 492 496 500 515 516 521 528 529 531 533 534 536 537 546 547 548 557 590 591 623 631 Schefe, P. 539 Schervish, M.J. 323 555 556 659 Schlaifer, R. 516 518 521 523 533 545 547 Schoemaker, P.J.H. 480 536 Schum, D.A. 490 538 Schwartz, J.J. 608 Scores 630 Scoring rules 247—48 630—31 Scoring rules as operational definition of prevision 247 533—34 Scoring rules, absolute error 248 Scoring rules, deterministic 534 Scoring rules, logarithmic 269 631 Scoring rules, proper 631 Scoring rules, quadratic 247 270 631 Scott, D. 516 Seidenberg, A. 515 601 Seidenfeld, T. 323 555 568 569 659 667 Self-conjugacy 66 87 90 Semigroup of transformations 139 Semigroup of transformations, Abelian 139 142—45 Sen, A.K. 529 530 Sensitivity analysis 6—7 44 105—08 171 253—58 Sensitivity analysis in decision problems 238 Sensitivity analysis in statistical inference 257—58 397 400 Sensitivity analysis, Bergerb’s account 256—57 Sensitivity analysis, models for independence 446—48 454—57 Sensitivity analysis, models for permutation-invariance 460—61 Sensitivity analysis, practical differences from behavioural theory 107—08 477 Sensitivity analysis, similarities to behavioural theory 107—08 257—58 Separating hyperplane theorems 611—12 Separating hyperplane theorems, strength of 505—06 Sequences of previsions 79—81 326 Sets, compact (see Compactness convex) Sets, non-measurable 149—50 Sex 481 Shackle, G.L.S. 50 539 Shafer, G. ix 11 23 47 208 272 273 274 275 280 281 358 483 489 490 491 513 528 542 543 544 545 546 547 548 550 555 559 560 564 Shannon, C.E. 268 523 540 Shilov, G.E. 493 Shimony, A. 494 540 Shore, J.E. 540 Shortliffe, E.H. 49 50 490 536 Sierpinski, W. 506 Simon, H.A. 484 488 530 Skinner, B.F. 19 482 Skyrms, B. 537 560 Slovic, P. 480 488 501 510 536 659 Smets, P. 490 Smith, A.F.M. 538 591 647 Smith, C.A.B. ix 44 46 106 255 482 483 485 489 490 492 494 495 501 503 506 507 531 563 Smith, L.D. 481 Smokier, H.E. 488 Snell, E.J. 488 565 Solovay, R.M. 505 Space dual 609 Space dual, finite dimensional 609 612 Space dual, half 146 610 Space dual, linear 63 608 Space dual, locally convex 612 Space dual, parameter (see Parameter space possibility) Spetzler, C.S. 510 516 Spiegelhalter, D.J. 49 50 490 Sprott, D.A. 568 Stael von Holstein, C.S. 510 516 Stakes function 629 630—31 Stallings, W. 538 Standard deviation, dependence on sample size 620—21 (see also Variance) Standard deviation, upper and lower 617 State of affairs 54 State of affairs, observable versus theoretical 54—55 118 State of affairs, personal interest in 623 624 State of affairs, possible (see Possibility Statistics theories Stein, C. 527 Stephanou, H.E. 490 Stock, M. 49 Stoer, J. 608 613 Stone, M. 527 528 556 566 567 568 Strassen, V. 49 497 502 563 Strategies assessment 8 35—42 286 Strategies assessment as justification for assessments 41—42 Strategies assessment in statistical problems 406 Strategies assessment, conflicting 214 Strategies assessment, criteria for selection 38—40 Strategies assessment, inference 21—22 406 424 Strategies assessment, types of 37—38 (see also Assessment Extension Strategies assessment, updating 285—86 335—40 406 424 Studentb’s t-distribution (see Distribution) Studentb’s t-Subjectivity (see Objectivity) Subspace, linear 608 Sudderth, W.D. 374 390 495 496 502 527 557 566 567 568 571 573 575 579 590 591 602 Suppes, P. 44 46 480 485 489 493 499 514 515 516 521 529 533 534 563 564 600 660 Support of gamble 344 363 Supremum 58 Symmetry, in information 227 458 461—62 Szolovits, P. 490 Tanaka, H. 538 Taxation, capital gains 95—96 Taxation, models for 96 604 Taylor, C. 481 Taylor, S.J. 559 588 Teller, P. 545 550 Thole, U. 539 Thomas, H. 510 Thorp, J. 47 Thumbtack (drawing pin) for imprecise chances 433—34 454— 55 467—68 Thumbtack (drawing pin) for prior-data conflict 225—26 Thumbtack (drawing pin) for uncertain perception 339 Thumbtack (drawing pin) objectivity of 112—13 465—66 Thumbtack (drawing pin) or near-ignorance 4 218—20 Thumbtack (drawing pin) robust Bernoulli model 467—71 Thumbtack (drawing pin) standard Bernoulli model 217 467 470 Thumbtack (drawing pin), probability models 36—37 Tiao, G.C. 43 228 229 233 500 525 526 528 538 564 567 568 583 584 Tolman, E.C. 481 topology 609—10 Topology, supremum-norm 609 Topology, weak 145—46 609—10 Tractability of models 105 116—17 216 252 Transitivity of preferences 28—29 154 247 Translation-invariance (see Invariance Trials) Tribus, M. 267 487 534 540 542 544 Tuomela, R. 481 562 563 Tversky, A. 485 488 499 500 501 534 536 537 659 660 Ulam, S. 498 557 Ultrafilter 101 147—49 Ultrafilter theorem 148 149 Unanimity rule 188 214 Uncertainty 209 Uncertainty, degree of 211—12 266 617 Uniform distribution (see Distribution uniform Density improper uniform) Updating 285—86 334—40 Updating, a general strategy 337—38 Updating, coherent 317—19 321—22 Updating, difficulties in 334—37 Urn, drawings from 462—63 Utility 17—19 24—26 58—61 Utility function, identified with gamble 25 160—61 Utility, construction of scale 59—60 Utility, expected 18 24 162 Utility, imprecise 165—66 490 Utility, precise (linear) 25—26 59—61 64 115 160 Utility, quadratic 58 231 619 Utility, unbounded 58 527 Vacuous prevision (see Prevision vacuous) Vagueness 245—46 261 519 Indeterminacy) Valentine, E.R. 483 Values 17—19 (see also Utility) Variable, continuous 330—33 436—40 Variable, random (see Gamble) Variance determinate (see Determinacy indeterminate) Variance determinate, choice between 159—60 Variance determinate, equivalence of 156 Variance determinate, literature on 42—50 Variance determinate, need for posterior measures 22 110 252 388 Variance determinate, types of 113—14 (see also Beliefs Indeterminacy Prevision Probability) Variance of a gamble precise 617 619—20 Variance of a gamble precise, upper and lower 617—21 Variance of a sample 373 Vasely, W.E. 532 Venn, J. 43 479 480 526 Verification procedure 54 118 Verification procedure, approximate 119 (see also Definition operational) Vickers, J.M. 485 491 Villegas, C. 228 515 516 525 532 568 von Mises, R. 44 351 480 528 536 564 von Neumann, J. 490 492 613 von Winterfeldt, D. 43 480 488 492 501 510 536 538 546 623 Wallace, D.L. 487 500 536 565 569 Walley, P. vii viii 44 46 47 49 478 479 488 489 495 496 497 500 502 503 508 510 512 514 515 516 517 518 519 520 528 529 535 536 537 539 542 543 545 551 559 563 564 565 568 576 582 587 589 590 591 592 Wallsten, T.S. 510 539 648 Waterman, D.A. 49 Watson, J.B. 19 482 Watson, S.R. 50 262 263 264 536 538 539 Weatherford, R. 479 Weaver, W. 523 540 Weinberg, S. 648 Weiss, J.J. 50 262 538 672 Weisweiller, R. 495 Well-groundedness (of probability assessments) 35 113—14 White, A.R. 481 490 514 Whittle, P. 493 497 Wilkinson, G.N. 568 Williams, P.M. ix 13 46 106 414 489 491 493 495 499 502 503 506 519 540 542 543 544 548 549 558 561 577 614 642 643 Williamsb’ theorem 643 Winkler, R.L. 501 510 511 521 533 536 623 632 638 Winsten, C.B. 558 646 Witness, unreliable 274 276 Witzgall, C. 608 613 Wolfenson, M 47 507 517 600 Wolpert, R.L. 500 536 569 570 573 583 584 Wong, Y. — C. 506 Yates, F. 569 571 Yates, J.F. 48 Ylvisaker, D. 518 Zabell, S. 547 559 560 Zadeh, L.A. 50 261 265 519 538 539 Zanotti, M. 489 Zellner, A. 228 229 525 528 538 584 Zidek, J.V. 527 568 Zimmerman, H.J. 539 Zukowski, L.G. 48 Zwick, R. 672 Zysno, P. 539
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