Confidence, likelihood, probability: statistical inference with confidence distributions
This lively book lays out a methodology of confidence distributions and puts them through their paces. Among other merits, they lead to optimal combinations of confidence from different sources of information, and they can make complex models amenable to objective and indeed prior-free analysis for...
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2016
|
| Series: | Cambridge series in statistical and probabilistic mathematics
41 |
| Subjects: | |
| Online Access: | DE-12 DE-92 Volltext Inhaltsverzeichnis |
| Summary: | This lively book lays out a methodology of confidence distributions and puts them through their paces. Among other merits, they lead to optimal combinations of confidence from different sources of information, and they can make complex models amenable to objective and indeed prior-free analysis for less subjectively inclined statisticians. The generous mixture of theory, illustrations, applications and exercises is suitable for statisticians at all levels of experience, as well as for data-oriented scientists. Some confidence distributions are less dispersed than their competitors. This concept leads to a theory of risk functions and comparisons for distributions of confidence. Neyman–Pearson type theorems leading to optimal confidence are developed and richly illustrated. Exact and optimal confidence distribution is the gold standard for inferred epistemic distributions. Confidence distributions and likelihood functions are intertwined, allowing prior distributions to be made part of the likelihood. Meta-analysis in likelihood terms is developed and taken beyond traditional methods, suiting it in particular to combining information across diverse data sources |
| Item Description: | Title from publisher's bibliographic system (viewed on 08 Mar 2016) |
| Physical Description: | 1 online resource (xx, 500 pages) |
| ISBN: | 9781139046671 |
| DOI: | 10.1017/CBO9781139046671 |
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Record in the Search Index
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Titel: Confidence, likelihood, probability
Autor: Schweder, Tore
Jahr: 2016
Contents Preface page xiii 1 Confidence, likelihood, probability: An invitation 1 1.1 Introduction 1 1.2 Probability 4 1.3 Inverse probability 6 1.4 Likelihood 7 1.5 Frequentism 8 1.6 Confidence and confidence curves 10 1.7 Fiducial probability and confidence 14 1.8 Why not go Bayesian? 16 1.9 Notes on the literature 19 2 Inference in parametric models 23 2.1 Introduction 23 2.2 Likelihood methods and first-order large-sample theory 24 2.3 Sufficiency and the likelihood principle 30 2.4 Focus parameters, pivots and profile likelihoods 32 2.5 Bayesian inference 40 2.6 Related themes and issues 42 2.7 Notes on the literature 48 Exercises 50 3 Confidence distributions 55 3.1 Introduction 55 3.2 Confidence distributions and statistical inference 56 3.3 Graphical focus summaries 65 3.4 General likelihood-based recipes 69 3.5 Confidence distributions for the linear regression model 72 3.6 Contingency tables 78 3.7 Testing hypotheses via confidence for alternatives 80 3.8 Confidence for discrete parameters 83 vii
Contents Preface page xiii 1 Confidence, likelihood, probability: An invitation 1 1.1 Introduction 1 1.2 Probability 4 1.3 Inverse probability 6 1.4 Likelihood 7 1.5 Frequenti sm 8 1.6 Confidence and confidence curves 10 1.7 Fiducial probability and confidence 14 1.8 Why not go Bayesian? 16 1.9 Notes on the literature 19 2 Inference in parametric models 23 2.1 Introduction 23 2.2 Likelihood methods and first-order large-sample theory 24 2.3 Sufficiency and the likelihood principle 30 2.4 Focus parameters, pivots and profile likelihoods 32 2.5 Bayesian inference 40 2.6 Related themes and issues 42 2.7 Notes on the literature 48 Exercises 50 3 Confidence distributions 55 3.1 Introduction 55 3.2 Confidence distributions and statistical inference 56 3.3 Graphical focus summaries 65 3.4 General likelihood-based recipes 69 3.5 Confidence distributions for the linear regression model 72 3.6 Contingency tables 78 3.7 Testing hypotheses via confidence for alternatives 80 3.8 Confidence for discrete parameters 83 vii
Contents viii 3.9 Notes on the literature 91 Exercises 92 4 Further developments for confidence distribution 100 4.1 Introduction 100 4.2 Bounded parameters and bounded confidence 100 4.3 Random and mixed effects models 107 4.4 The Neyman-Scott problem 111 4.5 Multimodality 115 4.6 Ratio of two normal means 117 4.7 Hazard rate models 122 4.8 Confidence inference for Markov chains 128 4.9 Time series and models with dependence 133 4.10 Bivariate distributions and the average confidence density 138 4.11 Deviance intervals versus minimum length intervals 140 4.12 Notes on the literature 142 Exercises 144 5 Invariance, sufficiency and optimality for confidence distributions 154 5.1 Confidence power 154 5.2 Invariance for confidence distributions 157 5.3 Loss and risk functions for confidence distributions 161 5.4 Sufficiency and risk for confidence distributions 165 5.5 Uniformly optimal confidence for exponential families 173 5.6 Optimality of component confidence distributions 177 5.7 Notes on the literature 179 Exercises 180 6 The fiducial argument 185 6.1 The initial argument 185 6.2 The controversy 188 6.3 Paradoxes 191 6.4 Fiducial distributions and Bayesian posteriors 193 6.5 Coherence by restricting the range: Invariance or irrelevance? 194 6.6 Generalised fiducial inference 197 6.7 Further remarks 200 6.8 Notes on the literature 201 Exercises 202 7 Improved approximations for confidence distributions 204 7.1 Introduction 204 7.2 From first-order to second-order approximations 205
Contents IX 7.3 Pivot tuning 208 7.4 Bartlett corrections for the deviance 210 7.5 Median-bias correction 214 7.6 The t-bootstrap and abc-bootstrap method 217 7.7 Saddlepoint approximations and the magic formula 219 7.8 Approximations to the gold standard in two test cases 222 7.9 Further remarks 227 7.10 Notes on the literature 228 Exercises 229 8 Exponential families and generalised linear models 233 8.1 The exponential family 233 8.2 Applications 235 8.3 A bivariate Poisson model 241 8.4 Generalised linear models 246 8.5 Gamma regression models 249 8.6 Flexible exponential and generalised linear models 252 8.7 Strauss, Ising, Potts, Gibbs 256 8.8 Generalised linear-linear models 260 8.9 Notes on the literature 264 Exercises 266 9 Confidence distributions in higher dimensions 274 9.1 Introduction 274 9.2 Normally distributed data 275 9.3 Confidence curves from deviance functions 278 9.4 Potential bias and the marginalisation paradox 279 9.5 Product confidence curves 280 9.6 Confidence bands for curves 284 9.7 Dependencies between confidence curves 291 9.8 Notes on the literature 292 Exercises 292 10 Likelihoods and confidence likelihoods 295 10.1 Introduction 295 10.2 The normal conversion 298 10.3 Exact conversion 301 10.4 Likelihoods from prior distributions 302 10.5 Likelihoods from confidence intervals 305 10.6 Discussion 311 10.7 Notes on the literature 312 Exercises 313
X Contents 11 Confidence in non- and semiparametric models 317 11.1 Introduction 317 11.2 Confidence distributions for distribution functions 318 11.3 Confidence distributions for quantiles 318 11.4 Wilcoxon for location 324 11.5 Empirical likelihood 325 11.6 Notes on the literature 332 Exercises 333 12 Predictions and confidence 336 12.1 Introduction 336 12.2 The next data point 337 12.3 Comparison with Bayesian prediction 343 12.4 Prediction in regression models 346 12.5 Time series and kriging 350 12.6 Spatial regression and prediction 353 12.7 Notes on the literature 356 Exercises 356 13 Meta-analysis and combination of information 360 13.1 Introduction 360 13.2 Aspects of scientific reporting 363 13.3 Confidence distributions in basic meta-analysis 364 13.4 Meta-analysis for an ensemble of parameter estimates 371 13.5 Binomial count data 374 13.6 Direct combination of confidence distributions 375 13.7 Combining confidence likelihoods 376 13.8 Notes on the literature 379 Exercises 380 14 Applications 383 14.1 Introduction 383 14.2 Golf putting 384 14.3 Bowheads 387 14.4 Sims and economic prewar development in the United States 389 14.5 Olympic unfairness 391 14.6 Norwegian income 396 14.7 Meta-analysis of two-by-two tables from clinical trials 401 14.8 Publish (and get cited) or perish 409 14.9 Notes on the literature 412 Exercises 413
X Contents 11 Confidence in non- and semiparametric models 317 11.1 Introduction 317 11.2 Confidence distributions for distribution functions 318 11.3 Confidence distributions for quantiles 318 11.4 Wilcoxon for location 324 11.5 Empirical likelihood 325 11.6 Notes on the literature 332 Exercises 333 12 Predictions and confidence 336 12.1 Introduction 336 12.2 The next data point 337 12.3 Comparison with Bayesian prediction 343 12.4 Prediction in regression models 346 12.5 Time series and kriging 350 12.6 Spatial regression and prediction 353 12.7 Notes on the literature 356 Exercises 356 13 Meta-analysis and combination of information 360 13.1 Introduction 360 13.2 Aspects of scientific reporting 363 13.3 Confidence distributions in basic meta-analysis 364 13.4 Meta-analysis for an ensemble of parameter estimates 371 13.5 Binomial count data 374 13.6 Direct combination of confidence distributions 375 13.7 Combining confidence likelihoods 376 13.8 Notes on the literature 379 Exercises 380 14 Applications 383 14.1 Introduction 383 14.2 Golf putting 384 14.3 Bowheads 387 14.4 Sims and economic prewar development in the United States 389 14.5 Olympic unfairness 391 14.6 Norwegian income 396 14.7 Meta-analysis of two-by-two tables from clinical trials 401 14.8 Publish (and get cited) or perish 409 14.9 Notes on the literature 412 Exercises 413
Contents xi 15 Finale: Summary, and a look into the future 418 15.1 A brief summary of the book 418 15.2 Theories of epistemic probability and evidential reasoning 423 15.3 Why the world need not be Bayesian after all 428 15.4 Unresolved issues 430 15.5 Finale 435 Overview of examples and data 437 Appendix: Large-sample theory with applications 447 A. 1 Convergence in probability 447 A. 2 Convergence in distribution 448 A. 3 Central limit theorems and the delta method 449 A.4 Minimisers of random convex functions 452 A. 5 Likelihood inference outside model conditions 454 A.6 Robust parametric inference 458 A. 7 Model selection 462 A. 8 Notes on the literature 464 Exercises 464 References 471 Name index 489 Subject index 495 |
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| spelling | Schweder, Tore 1943-2024 Verfasser (DE-588)171527011 aut Confidence, likelihood, probability statistical inference with confidence distributions Tore Schweder, University of Oslo, Nils Lid Hjort, University of Oslo Cambridge Cambridge University Press 2016 1 online resource (xx, 500 pages) txt rdacontent c rdamedia cr rdacarrier Cambridge series in statistical and probabilistic mathematics 41 Title from publisher's bibliographic system (viewed on 08 Mar 2016) This lively book lays out a methodology of confidence distributions and puts them through their paces. Among other merits, they lead to optimal combinations of confidence from different sources of information, and they can make complex models amenable to objective and indeed prior-free analysis for less subjectively inclined statisticians. The generous mixture of theory, illustrations, applications and exercises is suitable for statisticians at all levels of experience, as well as for data-oriented scientists. Some confidence distributions are less dispersed than their competitors. This concept leads to a theory of risk functions and comparisons for distributions of confidence. Neyman–Pearson type theorems leading to optimal confidence are developed and richly illustrated. Exact and optimal confidence distribution is the gold standard for inferred epistemic distributions. Confidence distributions and likelihood functions are intertwined, allowing prior distributions to be made part of the likelihood. Meta-analysis in likelihood terms is developed and taken beyond traditional methods, suiting it in particular to combining information across diverse data sources Observed confidence levels (Statistics) Mathematical statistics Probabilities Konfidenzintervall (DE-588)4644801-9 gnd rswk-swf Statistische Schlussweise (DE-588)4182963-3 gnd rswk-swf Statistische Schlussweise (DE-588)4182963-3 s Konfidenzintervall (DE-588)4644801-9 s 1\p DE-604 Hjort, Nils Lid 1953- Sonstige (DE-588)137124562 oth Erscheint auch als Druckausgabe 978-0-521-86160-1 https://doi.org/10.1017/CBO9781139046671 Verlag URL des Erstveröffentlichers Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029349327&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Schweder, Tore 1943-2024 Confidence, likelihood, probability statistical inference with confidence distributions Observed confidence levels (Statistics) Mathematical statistics Probabilities Konfidenzintervall (DE-588)4644801-9 gnd Statistische Schlussweise (DE-588)4182963-3 gnd |
| subject_GND | (DE-588)4644801-9 (DE-588)4182963-3 |
| title | Confidence, likelihood, probability statistical inference with confidence distributions |
| title_auth | Confidence, likelihood, probability statistical inference with confidence distributions |
| title_exact_search | Confidence, likelihood, probability statistical inference with confidence distributions |
| title_full | Confidence, likelihood, probability statistical inference with confidence distributions Tore Schweder, University of Oslo, Nils Lid Hjort, University of Oslo |
| title_fullStr | Confidence, likelihood, probability statistical inference with confidence distributions Tore Schweder, University of Oslo, Nils Lid Hjort, University of Oslo |
| title_full_unstemmed | Confidence, likelihood, probability statistical inference with confidence distributions Tore Schweder, University of Oslo, Nils Lid Hjort, University of Oslo |
| title_short | Confidence, likelihood, probability |
| title_sort | confidence likelihood probability statistical inference with confidence distributions |
| title_sub | statistical inference with confidence distributions |
| topic | Observed confidence levels (Statistics) Mathematical statistics Probabilities Konfidenzintervall (DE-588)4644801-9 gnd Statistische Schlussweise (DE-588)4182963-3 gnd |
| topic_facet | Observed confidence levels (Statistics) Mathematical statistics Probabilities Konfidenzintervall Statistische Schlussweise |
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