Introduction to statistical decision: utility theory and causal analysis

305 Pages·2020·1.898 MB·English

by  Bacci S., Chiandotto B

Introduction to
Statistical Decision
Theory
Utility Theory and Causal
Analysis

Introduction to
Statistical Decision
Theory
Utility Theory and Causal
Analysis
Silvia Bacci
Bruno Chiandotto
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To my parents, Mauro and Giovanna.
And to my decisions without uncertainty:
Massimo, Lorenzo, Serena.
S. Bacci
In memory of my (cid:12)rst teacher
of probability and statistics: Giuseppe Pompilj.
B. Chiandotto

Contents
Authors xi
Preface xiii
1 Statistics and decisions 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Decision theory . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Value theory and utility theory . . . . . . . . . . . . . . . . 4
1.4 Decisions and informational background . . . . . . . . . . . 6
1.5 Statistical inference and decision theory . . . . . . . . . . . 10
1.6 The decision-making approach to statistics . . . . . . . . . 13
2 Probability and statistical inference 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Random experiments, events, and probability . . . . . . . . 18
2.3 Bayes’ rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Univariate random variables . . . . . . . . . . . . . . . . . . 25
2.5 Multivariate random variables . . . . . . . . . . . . . . . . . 36
2.6 The exponential family . . . . . . . . . . . . . . . . . . . . . 46
2.7 Descriptive statistics and statistical inference . . . . . . . . 48
2.8 Sample distributions . . . . . . . . . . . . . . . . . . . . . . 51
2.9 Classical statistical inference . . . . . . . . . . . . . . . . . 56
2.9.1 Optimal point estimators . . . . . . . . . . . . . . . . 58
2.9.2 Point estimation methods . . . . . . . . . . . . . . . 62
2.9.3 Con(cid:12)dence intervals. . . . . . . . . . . . . . . . . . . 69
2.9.4 Hypothesis testing . . . . . . . . . . . . . . . . . . . 71
2.10 Bayesian statistical inference . . . . . . . . . . . . . . . . . 78
2.10.1 Conjugate prior distributions . . . . . . . . . . . . . 84
2.10.2 Uninformative prior distributions . . . . . . . . . . . 88
2.10.3 Bayesian point and interval estimation . . . . . . . . 91
2.10.4 Bayesian hypothesis testing . . . . . . . . . . . . . . 92
2.11 Multiple linear regression model . . . . . . . . . . . . . . . 94
2.11.1 The statistical model . . . . . . . . . . . . . . . . . . 95
2.11.2 Least squares estimator and maximum likelihood
estimator. . . . . . . . . . . . . . . . . . . . . . . . . 96
vii
viii Contents
2.11.3 Hypothesis testing . . . . . . . . . . . . . . . . . . . 98
2.11.4 Bayesian regression . . . . . . . . . . . . . . . . . . . 100
2.12 Structural equation model . . . . . . . . . . . . . . . . . . . 101
3 Utility theory 109
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.2 Binary relations and preferences . . . . . . . . . . . . . . . 110
3.3 Decisions under certainty: Value theory . . . . . . . . . . . 111
3.4 Decisions under risk: Utility theory . . . . . . . . . . . . . . 118
3.4.1 von Neumann and Morgenstern’s theory . . . . . . . 120
3.4.2 Savage’s theory . . . . . . . . . . . . . . . . . . . . . 127
3.5 Empirical failures of rational behavioral axioms . . . . . . . 131
3.5.1 Violation of transitivity. . . . . . . . . . . . . . . . . 131
3.5.2 Certainty e(cid:11)ect . . . . . . . . . . . . . . . . . . . . . 133
3.5.3 Pseudo-certainty e(cid:11)ect and isolation e(cid:11)ect . . . . . . 134
3.5.4 Framing e(cid:11)ect . . . . . . . . . . . . . . . . . . . . . . 135
3.5.5 Extreme probability e(cid:11)ect . . . . . . . . . . . . . . . 136
3.5.6 Aversion to uncertainty. . . . . . . . . . . . . . . . . 137
3.6 Alternative utility theories . . . . . . . . . . . . . . . . . . . 138
3.6.1 Rank-dependent utility theory . . . . . . . . . . . . . 142
3.6.2 Prospect theory and cumulative prospect theory . . . 143
4 Utility function elicitation 147
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.2 Attitude towards risk . . . . . . . . . . . . . . . . . . . . . 148
4.3 A measure of risk aversion . . . . . . . . . . . . . . . . . . . 155
4.4 Classical elicitation paradigm . . . . . . . . . . . . . . . . . 156
4.4.1 Standard gamble methods . . . . . . . . . . . . . . . 158
4.4.2 Paired gamble methods . . . . . . . . . . . . . . . . . 162
4.4.3 Other classical elicitation methods . . . . . . . . . . 166
4.5 Multi-step approaches . . . . . . . . . . . . . . . . . . . . . 167
4.6 Partial preference information paradigm . . . . . . . . . . . 169
4.7 Combining multiple preferences . . . . . . . . . . . . . . . . 172
4.8 Case study: Utility elicitation for banking
foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5 Classical and Bayesian statistical decision theory 179
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.2 Structure of the decision-making process . . . . . . . . . . . 180
5.3 Decisions under uncertainty (classical decision
theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.4 Decisions with sample information (classical
statistical decision theory) . . . . . . . . . . . . . . . . . . . 188
5.5 Decisions with sample and prior information
(Bayesian statistical decisional theory) . . . . . . . . . . . . 195
Contents ix
5.6 Perfect information and sample information . . . . . . . . . 204
5.7 Case study: Seeding hurricanes . . . . . . . . . . . . . . . . 214
6 Statistics, causality, and decisions 225
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.2 Causality and statistical inference . . . . . . . . . . . . . . 226
6.3 Causal inference . . . . . . . . . . . . . . . . . . . . . . . . 230
6.3.1 Statistical causality . . . . . . . . . . . . . . . . . . . 231
6.3.2 Modern causal inference . . . . . . . . . . . . . . . . 232
6.3.3 Structural equation approach to causal inference. . . 236
6.4 Causal decision theory . . . . . . . . . . . . . . . . . . . . . 244
6.5 Case study: Subscription fees of the RAI -
Radiotelevisione Italiana . . . . . . . . . . . . . . . . . . . . 249
6.6 Case study: Customer satisfaction for the RAI -
Radiotelevisione Italiana . . . . . . . . . . . . . . . . . . . . 256
References 267
Index 283