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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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Hahn — Banach theorem 139 506 611 Hahn — Banach to joint previsions 314—17 405 415 24 452—57 Hahn — Banach to linear previsions 136—39 418—21 427—28 433 Hahn — Banach to posterior previsions 424—34 640—41 Hahn — Banach to predictive previsions 314 421—24 Hahn — Banach via generalized Bayes rule 318 425—27 429 431—32 Hahn — Banach, as lower envelope of Bayesian priors 418 Hahn — Banach, basic properties 123 409—10 Hahn — Banach, behavioural meaning 122 408—09 Hahn — Banach, computation 128 130 136 174—76 316 Hahn — Banach, existence of 416 422 Hahn — Banach, from a linear space 125—27 Hahn — Banach, independent 453—57 Hahn — Banach, maximal coherent 126 Hahn — Banach, natural 30—31 121—32 408—15 Hahn — Banach, regular 582 616 639—41 Hahn — Banach, relation to coherence 74 122—23 408—11 Hahn — Banach, role in assessment and elicitation 35—36 171—72 Hahn — Banach, role in statistical reasoning 406 Hahn — Banach, satisfies likelihood principle 427 431—32 Hahn — Banach, to prior previsions 415—24 Hahn — Banach, to unconditional previsions (see also Extension to Extension to Hahn — Banach, typically non-vacuous 368 417 Hahn — Banach, uniqueness of 129 138 419—20 428 Hailperin, T. 489 503 Haldane density (see Density function Haldane) Haldane, J.B.S. 223 228 526 527 542 Hall, W.I. 228 518 525 Halmos, P.R. 502 506 588 Hamblin, C.L. 514 Hampton, J.M. 510 Harsanyi, J.C. 519 Hartigan, J.A. 230 488 498 517 527 528 535 536 568 576 Heath, D.C. 374 390 495 496 502 557 566 567 568 571 573 575 579 590 591 602 Hempel, C.G. 482 Hernandez, A. 517 518 536 Hersh, H.M. 539 Hewitt, E. 590 591 Hey, J.D. 519 Hill, B.M. 485 527 528 555 556 557 558 568 Hinkley, D.V. 386 483 489 500 525 526 527 528 567 569 570 572 573 583 592 Hisdal, E. 539 Hodges, J.L. 501 Hogarth, R.M. 44 48 480 482 488 501 510 532 535 536 623 Holmes, R.B. 505 608 610 611 612 613 Hora, R.B. 568 Horse racing, betting on 94—95 602—07 Huber, G.P. 510 Huber, P.J. 44 49 479 493 494 495 496 497 502 503 505 517 518 536 563 564 Hull, C.L. 481 Hunter, J.S. 565 Hunter, W.G. 565 Hyperparameter 260 Hyperplane separating 133 611 Hyperplane separating, simplex representation for 176 Hyperplane separating, strictly separating 611—12 Hyperplane separating, strongly separating 612 Hyperplane separating, supporting 146 Hyperplane theorems 611—12 Hyperprior 260 Hypotheses, statistical (see Models sampling) Hypothesis testing, examples of 233 583 Idealization in model building 39 117 Idealization in model building of precise measurement 329 436—40 Idealization in model building of precise probabilities 249—50 Identifiability 350 471 563 Ignorance 226—35 Ignorance, about a chance 228—29 Ignorance, about a finite space 227—28 Ignorance, about a location parameter 229—34 Ignorance, about a positive integer 322 Ignorance, Bayesian models for 223—24 226—35 Ignorance, beta model for 218—21 621 641 Ignorance, complete 92—93 226—27 270 445 458 462 Ignorance, consistent with precise prior 310 391—92 Ignorance, consistent with prior ignorance 308 391 696 Ignorance, implies posterior ignorance 367—69 428—29 vacuous) Ignorance, near 206 218 235 Ignorance, prior 367 417 Imprecision in probabilities 210 Imprecision in probabilities as measure of information 211—12 222 Imprecision in probabilities in football experiment 633—34 Imprecision in probabilities in second-order probabilities 260—61 Imprecision in probabilities, arguments against 235—53 Imprecision in probabilities, arguments for 3—8 212—17 Imprecision in probabilities, degree of 210 219 224—26 468 Imprecision in probabilities, due to conflict between information 213 222—26 Imprecision in probabilities, due to lack of information 212—13 217—22 224—26 Imprecision in probabilities, models for 50—51 159—60 253—81 Imprecision in probabilities, sources of 212—17 (see also Indeterminacy Precision Improper prior (see Density function improper) Inadmissibility (see Admissibility) Incoherence examples of 29 67 72 85—86 198 99 300 601 605 636 Incoherence examples of standard statistical models 370 372 373—76 382—83 Incoherence examples, models 104—05 173 212 215—17 Inconsistency (see Conflict) Indecision 226 235—41 Indecision, versus suspension of judgement 239 530 Indecision, ways of resolving 238—41 Independence 443—57 Independence as a structural judgement 473 Independence of events 443—48 Independence of experiments 448—57 Independence, behavioural meaning 443 448 Independence, conditional 451 Independence, epistemic 443—44 448 Independence, grounds for 444—45 Independence, linear 596 Independence, logical 445 Independence, physical 444 Independence, sensitivity analysis definitions 446—48 454—57 Independence, standard definitions 445—46 448—51 Indeterminacy 209—10 Indeterminacy in conditional previsions 306—07 328—29 Indeterminacy in likelihood function 436 Indeterminacy in posterior previsions 395—96 436— 40 Indeterminacy, experimental evidence for 48 116—17 254 532 633 Indeterminacy, judgements of 474 476 Indeterminacy, may produce indecision 161—62 236 Indeterminacy, physical 215 357—58 468 Indeterminacy, sources of 212—15 Indeterminacy, versus incompleteness 104—06 212 Indicator function 81 Indifference, principle of (see Principle of indifference) Induction (see Logic inductive) Inference (see Reasoning) Inference, arguments for 434—37 568 Inference, Bayesian 5 42—43 393—97 Inference, behavioural theories 21—23 109 10 Inference, examples of incoherence 369—76 379 382—85 Inference, fiducial 373 382 Inference, frequentist 377—88 (see also Confidence intervals) Inference, from imprecise priors 217—26 397—403 424—34 Inference, from improper priors 369—77 417 18 420—21 Inference, likelihood 377 Inference, Neyman-Pearson (see Confidence intervals) Inference, not consistent with prior ignorance 368—69 417—18 420—21 Inference, objections to 109—119 Inference, objective 43 226—35 266—72 369—77 Inference, objective methods 367 373—74 377 Inference, pivotal 374 Inference, requires non-vacuous prior beliefs 110 367—69 387 391 417 429 Inference, rom noninformative priors 226—35 369—77 383 Inference, standard 393—97 Inference, statistical 21—22 406 424 Inference, structural 373 Inference, subjective 43 Inference, versus decision 21 109 Infimum 58 information 33 Information, amount of 222 Information, assessment of (see Assessment Strategies assessment Information, can increase indeterminacy 223—26 298—300 433 551 Information, combination of (see Combination of assessments conflicting) Information, comparison of measures 211—12 268 523—24 Information, incomplete (see Information partial Information, modelled as an event 300—01 336 39 Information, needed to determine conditional previsions 329—34 437—40 Information, new 336—38 Information, partial 212—13 Information, prior 110 114 221—22 Information, reflected in degree of imprecision 3—4 211 220 222 Information, relation to sample size 213 222 Information, relevant 33 444 Information, Shannonb’s theory of 268 523—24 Information, statistical 434—35 Information, symmetric 33 227 458 461—62 Ignorance) Information, versus old 338 Initial conditions 355 358—59 Instability in elicitation 216—17 Instability in elicitation of relative frequencies 359 469 Instability in elicitation physical 215 358—59 Insurance premiums 95 245 532 Integral, constructed from measure 128—29 Integral, Daniell 493 Integral, Lebesgue 132 493 Intentional systems 17—18 Interpretations allows irrational judgements 32 110 253 Interpretations and conglomerability 312—13 327 Interpretations as version of sensitivity analysis 107 Interpretations in decision making 238 Interpretations in inductive logic 34—35 44—45 48—49 Interpretations objective, see aleatory or logical observational 14—16 Interpretations of elicited probabilities 173 477 Interpretations of probability and prevision 13—17 101—108 Interpretations of probability and prevision, aleatory (see also Frequency or propensity) Interpretations of probability and prevision, behavioural 14—24 61—63 109—10 Interpretations of probability and prevision, constructive 14—16 Interpretations of probability and prevision, contingent 284—85 Interpretations of probability and prevision, descriptive 361 Interpretations of probability and prevision, direct 105 357 Interpretations of probability and prevision, dispositional 15 16 18—20 Interpretations of probability and prevision, empirical 14—17 102 Interpretations of probability and prevision, epistemological, see logical evidential 14 22—24 Interpretations of probability and prevision, exhaustive 62 104—05 251 476 Interpretations of probability and prevision, finite 350—51 357—58 Interpretations of probability and prevision, frequency 16 49 Interpretations of probability and prevision, incomplete 104—05 285 Interpretations of probability and prevision, instrumentalist 360—61 698 Interpretations of probability and prevision, limiting 49 80—81 351 358 378 Interpretations of probability and prevision, minimal 20—21 61—63 Interpretations of probability and prevision, needed in inference 15 100 Interpretations of probability and prevision, objections to 109—19 Interpretations of probability and prevision, of sampling models 111 14 349 59 465—66 Interpretations of probability and prevision, relation to aleatory 17 353—54 357 Interpretations of probability and prevision, relation to epistemic 17 102 353 54 357 Interpretations of probability and prevision, versus evidential 22—24 Interpretations of probability and prevision, versus sensitivity analysis 105—08 476—77 Interpretations of structural judgements 108 446—48 456—57 460 477 Interpretations operational 15 20 102—03 Interpretations personalist 14 43 Interpretations propensity 16—17 351—55 358 Interpretations rationalistic 14—15 102 Interpretations reductionist 112—13 361 465—67 Interpretations sensitivity analysis 7 105—08 253—58 Interpretations, based on dogma of ideal precision 7 105 254 Interpretations, intersubjective 361 Interpretations, logical 14—16 43 102 267 Interpretations, subjective 15—16 103 Interpretations, updated 284—86 Interpretations, versus behavioural interpretation 107—08 477 Interval estimation (see Confidence intervals Credible Interval of measures 201—02 Invariance 139—45 231 Invariance, permutation 457 Invariance, shift 97—98 143—44 Invariance, translation 98—100 140 144—45 229—30 Irrelevance 444 (see also Independence) Isaacs, H.H. 536 Jain, R. 538 Jameson, G.J.O. 506 608 Janis, I.L. 484 Jaynes, E.T. 11 23 43 228 266 267 271 486 525 526 527 528 540 541 542 590 591 592 Jech, TJ. 506 Jeffrey, R.C. 43 339 481 483 485 490 491 492 525 533 543 544 560 Jeffreyb’s rule 339 Jeffreys, H. 16 23 43 45 113 223 228 229 232 238 241 244 245 362 373 478 486 500 520 525 526 527 528 532 567 571 Jenkins, G.M. 558 646 Jepson, C. 664 Johnson, R.W. 540 Johnson, W.E. 589 Judgement 167—71 (see also Assessment Elicitation) Judgement, ambiguous versus imprecise 263 Judgement, arbitrary 173—74 210 237 241 Judgement, classificatory 188—91 Judgement, comparative 191—97 Judgement, effect of new 177—80 Judgement, equivalence 472 Judgement, of desirability 169—70 474 Judgement, structural 472—77 Judgement, suspending 226 239 Judgement, types of 169—71 472—74 Kadane, J.B. 323 496 536 555 557 558 559 667 Kahneman, D. 488 501 536 Kanal, L.N. 490 Kaplan, M. 515 516 588 Karmarkar, N. 511 Kekes, J. 484 Kelley, J.L. 506 608 609 610 Kemeny, J. 494 Kempthorne, O. 386 483 558 559 562 569 572 573 584 585 Kent, S. 514 539 Keynes, J.M. ix 23 34 44 45 113 238 244 486 487 489 497 515 516 522 523 526 532 585 Keynesb’s theory of probability 44—45 Kickert, W.J.M. 538 539 540 Kiefer, J. 387 570 572 Kingman, J.F.C. 590 Kmietowicz, Z.W. 48 507 536 Kneale, W. 514 Knight, F.H. 209 489 519 Knill — Jones, R.P. 50 490 Kolmogorov, A.N. 32 282 306 307 327 489 549 553 558 561 Kolmogorov’s condition 306 Kolmogorov’s theory of conditional probability 306—08 Kolmogorov’s theory of conditional probability, compared to de Finettib’s theory 282 327 Koopman, B.O. 45 515 557 Kraft, C. 515 601 Krantz, D.H. 515 664 Krein — Milman theorem 613 Kudo, H. 517 518 536 Kuhn, T.S. 478 Kullback, S. 523 Kumar, A. 49 592 Kunda, Z. 664 Kunreuther, H. 48 501 532 Kyburg, H.E. 34 44 48 238 479 480 484 486 487 488 501 529 531 536 562 563 590 591 592 Lad, F.R. 482 488 499 Lane, D.A. 555 556 566 568 573 579 Laplace, P.S. de 43 220 228 377 521 522 525 527 569 Laplaceb’s rule of succession 220 Lauritzen, S.L. 49 490 Laws of large numbers 352 Laws of large numbers for imprecise probabilities 359 564 Laws, deterministic versus statistical 352 355 Learner, E.E. 44 47 492 493 518 521 525 536 565 623 Lebesgue lower prevision 98—99 132 Lebesgue lower prevision is fully conglomerable 325 Lebesgue measure (on real line) 229—30 (see also Density function improper uniform) Lebesgue measure (on unit interval) 98—99 Lebesgue measure (on unit interval), additive extensions 132 136—37 144—45 325—27
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