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Walley P. Ч Statistical reasoning with imprecise probabilities
Walley P. Ч Statistical reasoning with imprecise probabilities

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Ќазвание: Statistical reasoning with imprecise probabilities

јвтор: Walley P.

јннотаци€:

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: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1991

 оличество страниц: 706

ƒобавлена в каталог: 10.12.2005

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
action      24 160Ч61 235Ч41
Action as a rationality axiom      32 326Ч27
Action, admissible      161
Action, Bayes      162 165Ч66 237Ч40
Action, Behaviour Additivity      89Ч92
Action, countable      323Ч27 453Ч54
Action, de FinettibТs arguments against      326
Action, finite versus countable, examples      70 97Ч99 310Ч11 313 321Ч23 325Ч26 365 390 396 420Ч21 450
Action, identified with gamble      160Ч61
Action, maximal      161Ч66
Action, minimax      163 166 240 269Ч70 619 630
Action, optimal, see Action, Bayes reasonable but not Bayes      165Ч66
Action, relation to conglomerability      310 13 323Ч24
Action, satisfactory      239Ч41 (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      163Ч64 253
Agassi, J.      500
Aggregation      (see Combination of assessments)
Agreement, interpersonal      111Ч13 361
Algorithms for checking coherence      597Ч98
Algorithms for computing extreme points      174 76 511
Algorithms for computing joint previsions      316 17 454Ч55
Algorithms for generalized Bayes rule      550 581
Algorithms or computing natural extension      136 174Ч76
Allais, M.      429
Amaral Ч Turkman, M.A.      523
Ambiguity      216 261Ч64 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      32Ч42 167
Assessment as a sequential process      177Ч78
Assessment of class of conjugate priors      205Ч06 218Ч19 221 225
Assessment of intervals of measures      201Ч02
Assessment of neighbourhoods      202Ч03
Assessment of prior previsions      198 200Ч06 221 403 424 429
Assessment of sampling models      359Ч60
Assessment of upper and lower density functions      200Ч02
Assessment of upper and lower distribution functions      203Ч05
Assessment of upper and lower probabilities      197 99
Assessment of upper and lower probability ratios      199Ч200
Assessment of upper and lower quantiles      204Ч05
Assessment strategies      (see Strategies assessment)
Assessment, combination of      (see Combination of assessments in football)
Assessment, experiment      634Ч35
Assessment, guided by an expert system      40 512
Assessment, incomplete      173 212 215Ч16 256Ч57 285Ч86
Assessment, inconsistent      187Ч88 213Ч14 216Ч17
Assessment, reliable      285 288Ч89
Assessment, using conditional previsions      286 300
Assessment, versus elicitation      15 22Ч23 167
Assessment, well grounded in evidence      113Ч14 (see also Strategies assessment Judgement Elicitation Extension)
Assignment, probability      273
Aumann, R.J.      241 490 533
Avoiding sure loss      28Ч29 67Ч72 115Ч16 293Ч94 346
Avoiding sure loss in betting on horses      603Ч05
Avoiding sure loss in football experiment      635
Avoiding sure loss, comparison with coherence      67 74
Avoiding sure loss, justification for      68 295 343Ч45
Avoiding sure loss, objections to the emphasis on      114Ч16
Avoiding sure loss, relation to dominating linear previsions      134Ч35 (see also Loss)
Axiom of Choice      505 611
Axioms, index of      680
Axioms, role of      31Ч32
Ayer, A.J.      487 501
Banach limits      143Ч44
Barnard, G.A.      374 501 558 559 568 584 585
Barndorff Ч 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      285Ч86 335 40
Bayes rule as coherence relation (axiom C      12 297 300 303Ч04 335
Bayes rule as natural extension      318Ч19 411Ч12 424Ч27
Bayes rule for imprecise likelihood functions      337Ч38 430Ч34
Bayes rule for precise likelihood functions      392 400Ч03 424Ч29
Bayes rule, algorithms for solving      550 581
Bayes rule, conditions for applying      300Ч01 334 37
Bayes rule, examples of      298Ч300 308Ч13
Bayes rule, generalized (GBR)      185 297Ч301 400Ч03 424Ч34
Bayes rule, may yield imprecise updated probabilities      298Ч301 334 336 338Ч40 conditional)
Bayes rule, psychological model for      17Ч19 25
Bayes rule, relation to beliefs      17Ч21 22 109Ч10 Decision Interpretations behavioural)
Bayes rule, Updating Behaviour      17Ч21
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      331Ч33 393Ч96 438Ч40
Bayesian sensitivity analysis in statistical problems      393Ч97 427Ч9 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      19Ч20
Belief functions      183Ч84 272Ч81 358
Belief functions, behavioural interpretation of      280Ч81
Belief functions, generated through multivalued mapping      182Ч84 273Ч74
Belief functions, not sufficiently general      184 274Ч75
Beliefs      17Ч19
Beliefs as behavioural dispositions      18Ч20 481
Beliefs of experts      47 113 213Ч14 622Ч23 626
Beliefs, combination of      (see Combination of assessments)
Beliefs, conflicting      213Ч14
Beliefs, evolution of      177Ч78
Beliefs, group      5 105 113 186Ч88
Beliefs, indeterminate      209Ч10
Beliefs, mathematical models for      51 156 159Ч60
Beliefs, rational      26Ч29 32
Beliefs, role in statistical inference      109Ч11
Beliefs, second-order      258Ч61 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 454Ч55
Bernoulli models, compared to exchangeability      112Ч13 463Ч66
Bernoulli models, robust      467Ч71
Bernoulli, J.      489 543
Beta distribution      (see Distribution beta
Beta distribution on confidence intervals      380Ч82 386
Beta distribution on football games      632Ч38
Beta distribution on horses      94Ч95 245 602Ч07
Beta distribution, objections to the emphasis on      114Ч16
Beta distribution, on parameter values      118Ч19 367Ч68 429
Beta distribution, pari-mutuel system      94Ч95 130Ч31 135 147 202Ч03 604Ч06 690
Betting procedure, biased      381Ч82 384 565Ч66
Betting rates      82 602Ч03
Betting rates of bookmakers      87Ч88 95 245 602 605 625 28
Betting rates, fair (two-sided)      89 243Ч44 251 624Ч25
Betting rates, lower      82 251 632
Betting rates, upper      82 602 632 Probability additive Probability lower Probability upper)
Beyth Ч Marom, R.      514 539
Bhaskara Rao, K.P.S.      496 498 504
Bhaskara Rao, M.      496 498 504
Bias in elicitation      624Ч28
Bias of measuring instrument, model for      399Ч400 405 423Ч24
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 625Ч28
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      356Ч57 468
Bounds, upper and lower for chances for conditional probabilities      296 301
Bounds, upper and lower for chances for density functions      199Ч201
Bounds, upper and lower for chances for posterior previsions      432
Bounds, upper and lower for chances for probabilities      35 46Ч47 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 Ч 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      351Ч59 465Ч66
Chance, imprecise      358Ч59 454 468
Chance, lower bound for      356Ч57 468 588
Chemoff, H.      491
Choice      160Ч66 236Ч41
Choice, arbitrary      19 23Ч41 210
Choice, between models for uncertainty      159 60
Choice, strategies for      239Ч40
Choice, versus preference      236Ч37 242 247Ч48 Action Preference)
Choquet, G.      502 505
Chuang, D.T.      536
Classification of probable events      (see Probability classificatory)
Coarsening (of a possibility space)      181Ч82
Cohen, L.J.      39 50 488 490 501 539
Coherence      29Ч30 63Ч66 72Ч76 346Ч47
Coherence for statistical models      366 380 391 392 397 401 404
Coherence for win-and-place betting on horses      605Ч06
Coherence in football experiment      636
Coherence of Bayesian models      394 405
Coherence of conditional with unconditional previsions      293Ч96 301Ч05
Coherence of lower previsions      72Ч76 134
Coherence of lower probabilities      84Ч86 135 600
Coherence, algorithm for checking      597Ч98
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      342Ч49
Coherence, justification for      60Ч61 64 73Ч74 295Ч96 347
Coherence, relation to dominating linear previsions      134Ч35 145Ч46
Coherence, relation to natural extension      74 122 23 408Ч09
Coherence, separate      289Ч93
Coherence, strength of      74 116Ч18
Coherence, strict      32 495 581
Coherence, weak      346Ч47 391 566
Coin tossing, models for      119 274Ч75 298Ч99
Collective (von Mises)      351 564
Combination of assessments      30 39 170Ч72 178Ч80 186Ч88 275Ч81 338 452Ч57
Compactness      610
Comparability      (see Completeness)
Completeness      157 190 195Ч97 241Ч48 601
Computations      (see Algorithms)
Conclusions      21Ч22
Conclusions, indeterminate      2 5Ч6 209Ч10 226
Conclusions, robust      5Ч6 256 Inference statistical Prevision posterior)
Conditioning      185Ч86 282Ч340
Conditioning on continuous variables      330Ч33 436Ч40
Conditioning on events of probability zero      306Ч07 328Ч34 391Ч92 615Ч16
Conditioning, de FinettinТs theory      282 326Ч27
Conditioning, examples of      185Ч86 298Ч300 308Ч13 321
Conditioning, in frequentist inference      382 387Ч88
Conditioning, KolmogorovТs theory      282 306Ч08 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 378Ч80 388
Confidence coefficient, data-dependent      387
confidence intervals      377Ч88
Confidence intervals, axioms for coherence      380
Confidence intervals, examples of incoherence      379 382Ч85
Confidence intervals, interpretations of      378 382 385Ч87
Confidence intervals, need for conditioning      382 387Ч88
Confidence intervals, posterior confidence in      378Ч80 388
Confidence intervals, zero confidence in      379 385 569
Conflict      187Ч88 213Ч14 222Ч26
Conflict, between expert opinions      47 213Ч14 520
Conflict, between probability models      187Ч88 213Ч14 216Ч17
Conflict, degree of      223 224Ч26
Conflict, information      6 213Ч14 222Ч26
Conglomerability      317Ч27 614Ч16 31Ч 317
Conglomerability for uncountable spaces      324Ч26 3
Conglomerability, examples of failure      311 321Ч22
Conglomerability, full      317 614
Conglomerability, justification for      319Ч20
Conglomerability, relation to countable additivity      310 13 323Ч24
Conglomerability, relation to sensitivity analysis      327
Conglomerative principle      (see Principles of rationality conglomerative)
Conjugacy      64Ч65 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      3Ч4 219 220 223Ч25 470Ч71 620Ч21
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      145Ч46
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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