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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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Daroczy, Z 523 540 Darwall, S.L. 484 485 500 DATA (see Observation) Davis, M. 507 Dawid, A.P. 361 488 500 527 565 566 568 587 590 591 592 Day, J.P. 514 Day, M.M. 504 505 608 de Finetti, B. viii ix 3 16 20 43 44 45 54 55 57 66 81 86 111 138 241 242 243 246 247 282 295 320 321 322 326 327 361 365 443 460 463 465 466 478 479 480 481 482 483 484 486 487 488 489 490 491 493 494 496 497 498 499 500 501 503 507 510 511 512 513 515 516 528 531 532 533 534 537 539 541 545 546 548 549 552 553 554 555 556 557 558 559 560 566 567 580 585 586 587 589 590 591 592 601 624 632 644 De Finettib’s elimination of aleatory probabilities 111—13 361 465—67 De Finettib’s fundamental theorem of probability 138 De Finettib’s representation theorem 463 De Morgan, A. 43 479 De Sousa, R. 510 537 Death risk 352 354—55 360 Decision 24—26 160—66 235—41 Decision rules, Bayes 162 164 Decision rules, minimax regret 508 Decision rules, P-minimax 163—64 240 269—70 619 630 Decision rules, statistical 164—65 240 Decision, Bayesian theories 43 164—65 237—38 Decision, experimental studies 48 116—17 Decision, frequentist theory 164—65 Decision, fuzzy analysis 263 Decision, role of probability models in 24—25 161 Decision, sensitivity analysis 24 44 238 256 Decision, statistical 164—65 Decision, strategies for 24—25 238—41 Decision, versus inference 21 109—10 572 Decision, with imprecise utilities 165—66 Decision, with precise utilities 160—65 Decisiveness 235—36 243 Definition, operational 20 102—03 243 247 624 DeGroot, M.H. 43 45 241 516 518 523 528 532 647 Dempster, A.P. 11 44 47 182 208 272 273 274 275 281 358 479 489 511 513 536 542 543 544 545 564 568 Dempster-Shafer theory (see Belief functions) Dempsterb’s rule of combination 275—81 Dempsterb’s rule of conditioning 278—81 Dennett, D.C. 18 481 482 491 529 Density function (see also Distribution) Density function, as a reference prior 233—34 583 Density function, behavioural meaning 230—31 Density function, conditional 331 Density function, conjugate (see Density function natural-conjugate) Density function, dependence on sampling model 229 231—32 374—76 583 Density function, for a chance 228—29 Density function, for a location parameter 229—34 370—72 376 420—21 Density function, for a scale parameter 372—73 (see also Density function improper) Density function, for location-scale parameters 232 373 383 Density function, generates incoherent inferences 231 369—77 Density function, Haldane 223 228 374—76 417—18 Density function, improper 228 229—30 369 Density function, joint 331—32 450 Density function, marginal 331 393 Density function, motivations for 234—35 Density function, natural-conjugate 205 Density function, noninformative 226—35 Density function, objections to 230—34 Density function, on finite set 227—28 267 Density function, on integers 310 Density function, on positive integers 97—98 143—44 311 321—22 326 365 450 Density function, on real line 99—100 136 144 420 improper) Density function, on unit interval 98—99 220 223 228 437—39 Density function, posterior 393 Density function, posterior prior (see Prevision prior) Density function, prior Studentb’s 373 382—83 Density function, regular 437—39 Density function, relation to proper previsions 233 376 399 417—18 420—21 Density function, sampling 369 393 436—37 Density function, uniform (see Density function improper uniform) Density function, uniform on bounded interval 372 384—85 Density function, uniform on surface of a sphere 328 Dependence (see also Independence) Dependence, degree of 589 Dependence, non-negative 473 DeRobertis, L. 498 517 535 536 576 Desirability 150—53 155 156—60 614 16 Desirability, almost 151—53 Desirability, axioms for, refer to Index of axioms B 287 294 Desirability, behavioural interpretation 60 151 614 Desirability, correspondence with conditional previsions 615—16 Desirability, correspondence with linear previsions 158 Desirability, correspondence with lower previsions 64 156 615 Desirability, correspondence with preference 153 Desirability, more fundamental than prevision 159—60 Desirability, real 160 614—16 Desirability, role in elicitation 152 169 171—72 Desirability, strict 155 Determinacy (see Indeterminacy) Determinism 354—55 358—59 Dewey, J. 484 Diaconis, P. 518 544 547 559 560 Dias, P. 540 Dickey, J.M. 44 499 528 536 537 568 Direct inference (see Principles of rationality direct Disagreement (see Conflict) Dispositions behavioural 18—20 61—63 Dispositions behavioural, beliefs 285—86 (see also Beliefs) Dispositions behavioural, physical (see Chance to update) Distribution beta 205 218 395 Distribution beta with unknown variance 355—57 373 382—83 Distribution beta, binomial 374—75 395 Distribution beta, degenerate 149 209 Distribution beta, double exponential 384 Distribution beta, geometric 375—76 Distribution beta, imprecise 398—99 399—400 401—03 405 423—24 524 620 Distribution beta, maximum entropy 266—72 Distribution beta, noninformative (see Density function noninformative) Distribution beta, Normal 370—72 Distribution beta, posterior (see Prevision posterior) Distribution function 130 Distribution function, upper and lower 130 203—05 Distribution, uniform, upper and lower 200—02 Domain of conditional previsions 289 291 293 Domain of lower prevision 61 Domain of statistical model 363 388 393 Domain of upper prevision 70 Dominance 78 132—35 Donnell, M.L. 50 262 538 672 Drawing pin (see Thumbtack) Dretske, F. 482 Dubins, L.E. 313 496 528 552 554 557 Dubois, D. 538 669 Duda, R.O. 49 Dunford, N. 608 Dunsmore, I.R. 523 Dutch book 29 (see also Loss) Economics, examples of difference between buying and selling prices 65 72 94—95 95—97 602—07 627—28 Edwards, A.W.F. 524 568 Edwards, W. 43 44 480 488 492 501 510 528 533 536 538 546 623 Eells, E. 485 Einhorn, H.J. 44 48 480 532 535 536 Elicitation 167—74 Elicitation of beliefs about football game 172 73 175—77 632—33 Elicitation, Bayesian procedures 173—74 Elicitation, general procedure 167—76 Elicitation, incomplete 104 106 173 215—17 Elicitation, operational procedures 173 622—31 Elicitation, using structural judgements 474—76 Elicitation, versus assessment 14—15 167 Judgement Measurement Ellis, R.L. 43 480 526 Ellsberg, D. 48 530 532 Elster, J. 484 485 532 Embedding principle (see Principles of rationality embedding) entropy 211 266 Entropy for continuous distributions 271 Entropy upper and lower 541 Envelopes of Bayesian models 397—400 402—03 405 418—19 429—30 432 Envelopes of Bayesian models of coherent linear collections 308 312—13 316 327 349 414—15 642—44 Envelopes of Bayesian models of countably additive measures 327 454 Envelopes of Bayesian models of linear previsions 89 132—36 Envelopes of Bayesian models, independent lower 446—48 454—57 Envelopes of Bayesian models, of lower previsions 78 348—49 Envelopes of Bayesian models, of noninformative priors 223—24 Envelopes of Bayesian models, of variances 618 (see also Sensitivity analysis) Envelopes of Bayesian models, relation to coherence 134—35 145 Envelopes of Bayesian models, relation to natural extension 136 414 Epstein, R.A. 602 Equivalence judgements 472 Equivalence of models for uncertainty 156 159—60 Estimation interval (see Confidence intervals Credible Events 54 81 Events of probability zero 306—07 328—34 436—40 Events, as zero-one valued gambles 81 (see also Gamble) Events, conditioning 284 289 Events, exchangeable 460—67 Events, independent 443—48 Events, irrelevant 444 Events, non-negatively dependent 473 Events, non-negatively relevant 473 Events, observable 118 465—66 Events, permutable 458—59 Events, probable 188—91 262 Events, standard 196 Events, sure 54 evidence (see Information) Exchangeability 361 460—67 Exchangeability as a direct judgement of desirability 472 Exchangeability for infinite sequence of events 463—66 Exchangeability, grounds for 461—62 466 Exchangeability, stronger than permutability 461—62 Expectation 128—29 493 694 Experiments auxiliary 196 Experiments auxiliary binary (see Bernoulli models exchangeable) Experiments auxiliary binary, independent 448 Experiments auxiliary binary, permutable 457 (see also Trials) Expert Systems 49—50 Expert systems for assessing probabilities 40 512 Explanation for actions, psychological 17—19 Explanation for actions, versus prediction 360 466 Extension of conditional and marginal previsions 314—17 413 Extension of conditional and marginal previsions rom a field 125 127—32 Extension of conditional and marginal previsions to conditional previsions 317—33 412 615—16 639—41 Extreme point 146 613 Extreme points of J((P) 146—48 159 Extreme points of J((P), computation 175 511 Extreme points of J((P), examples 147 Extreme points of J((P), existence 146 148 Extreme points of J((P), lower envelopes of 147 149 429 432 Family, location (see Parameter location) Farquhar, P.H. 492 Feagans, T.B. 538 Feddersen, A.P. 383 Feller, W. 526 541 591 Fellner, W. 44 47 48 520 521 535 536 Ferguson, T.S. 508 509 Field 90 Field, extension from 125 127—32 Filter 100—01 131 147—49 190 310 Fine, T.L. ix 44 45 46 47 49 478 479 480 487 489 491 496 501 507 514 515 516 517 525 526 528 531 532 533 539 540 541 542 543 547 562 563 564 565 586 587 588 589 591 592 600 Finite additivity (see Additivity Football) Finite additivity, examples of assessments 172 175 179 180 181 182 185 187 190 194 198 199 201 202 270 288 634 Finite additivity, World Cup experiment 632—38 Fischhoff, B. 480 510 662 Fishbum, P.C. 46 241 489 490 499 507 515 521 531 532 533 536 Fisher, R.A. 44 232 362 373 382 483 488 521 522 524 526 528 536 562 568 569 570 571 Fodor, J.A. 482 491 Folks, J.L. 536 Folks, L. 386 483 558 559 562 569 572 573 584 Forsyth, B. 672 Forsyth, R. 49 538 Fraser, D.A.S. 373 501 568 570 571 585 Freedman, D.A. 579 Freeling, A.N.S. 50 263 538 539 540 French, S. 516 684 Frequency, apparent convergence 464—65 Frequency, convergence 351 354 464 Frequency, divergence 80—81 358—59 469 frequency) Frequency, relative 219 350—52 357—58 Fubinib’s theorem 588 Function (see Belief function Density etc.) Functional bivariate 613 618 Functional bivariate, continuous 609 611—13 Functional bivariate, evaluation 609 Functional bivariate, linear 609 Functional bivariate, positive 89 609 Functional bivariate, variance 618 Fuzzy sets 50 261—66 Gain in football experiment 635 637—38 Gain in the stock market 72 Gain of bookmakers 95 627—28 Gain, avoiding 87 Gain, from betting on horses 603—05 Gain, sure 64 Gaines, B.R. 538 539 Gale, D. 511 612 Gale, W.A. 490 Gamble 58 Gamble, admissible 161 Gamble, Bayes 162 Gamble, called off 56 184—85 284 Gamble, contingent 284 Gamble, desirable (see Desirability indeterminate) Gamble, j-measurable 129 Gamble, marginal 68 289 Gamble, maximal 161 Gamble, measurable 291 Gamble, nondesirable 474 Gamble, P-minimax 163 Gamble, permuted 457 Gamble, simple 129 174 Gamble, super-relevant 381 Gamble, two-stage 291 294 303—04 Gamble, undesirable 474 gambling (see Betting) Gardenfors, P. 44 49 507 530 535 538 Gardner, H. 482 Gauss, C.F. 527 567 569 Geisser, S. 521 526 Generalized Bayes rule (GBR) (see Bayes rule generalized) Geometric representation of probability models 176—77 494 512 Geometry, Euclidean 249—50 Gerber, H.U. 498 Giere, R.N. 480 562 Giertz, M. 539 Giles, R. 47 499 500 503 506 510 519 539 Gillies, D.A. 480 562 Giron, F.J. 47 506 507 528 536 Goel, P.K. 523 Goldstein, M. 536 546 547 549 552 555 557 560 592 Good, I.J. ix 43 44 46 255 256 257 478 479 481 484 485 487 488 500 501 509 519 523 524 525 528 529 534 535 536 537 538 540 551 579 589 590 591 592 Granirer, E. 504 Graybill, F.A. 569 Greenleaf, F.P. 504 Griffiths, A.P. 481 Grize, Y. — L. 49 592 Gurevich, B.L. 493 Hacking, I. 478 479 480 481 482 489 491 493 526 543 545 547 559 562 563 568 569 592 Hagen, O. 492 Hahn — Banach 138—39 Hahn — Banach as lower envelope of Bayesian posteriors 429 432 Hahn — Banach as lower envelope of linear extensions 136 316 414 418 429 432 Hahn — Banach extensions 138—39 Hahn — Banach for imprecise sampling models 430—34 440—41 Hahn — Banach in betting on horses 606—07 Hahn — Banach of classificatory judgements 189 Hahn — Banach of comparative probability ordering 192 Hahn — Banach of desirable gambles 151 53 Hahn — Banach of independent marginals 452—57 Hahn — Banach of Lebesgue measure (see Lebesgue measure) Hahn — Banach of structural judgements 475
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