Introduction to Bayesian Statistics

CONTENTS......Page 6
Preface......Page 11
INTRODUCTION TO STATISTICAL SCIENCE......Page 17
1.1 THE SCIENTIFIC METHOD: A PROCESS FOR LEARNI......Page 19
1.2 THE ROLE OF STATISTICS IN THE SCIENTIFIC METHOD......Page 20
1.3 MAIN APPROACHES TO STATISTICS......Page 21
1.4 PURPOSE AND ORGANIZATION OF THIS TEXT......Page 24
SCIENTIFIC DATA GATHERING......Page 28
2.1 SAMPLING FROM A REAL POPULATION......Page 29
2.2 OBSERVATIONAL STUDIES AND DESIGNED EXPERIMENTS......Page 32
3.1 GRAPHICALLY DISPLAYING A SINGLE VARIABLE......Page 44
3.2 GRAPHICALLY COMPARING TWO SAMPLES......Page 52
3.3 MEASURES OF LOCATION......Page 54
3.4 MEASURES OF SPREAD......Page 57
3.5 DISPLAYING RELATIONSHIPS BETWEEN TWO OR MORE VARIABLES......Page 59
3.6 MEASURES OF ASSOCIATION FOR TWO OR MORE VARIABLES......Page 61
LOGIC, PROBABILITY, AND UNCERTAINTY......Page 70
4.1 DEDUCTIVE LOGIC AND PLAUSIBLE REASONING......Page 71
4.2 PROBABILITY......Page 73
4.3 AXIOMS OF PROBABILITY......Page 74
4.4 JOINT PROBABILITY AND INDEPENDENT EVENTS......Page 75
4.5 CONDITIONAL PROBABILITY......Page 77
4.6 BAYES’ THEOREM......Page 78
4.7 ASSIGNING PROBABILITIES......Page 83
4.8 ODDS RATIOS AND BAYES FACTOR......Page 84
DISCRETE RANDOM VARIABLES......Page 90
5.1 DISCRETE RANDOM VARIABLES......Page 91
5.2 PROBABILITY DISTRIBUTION OF A DISCRETE RANDOM VARIABLE......Page 93
5.3 BINOMIAL DISTRIBUTION......Page 96
5.4 HYPERGEOMETRIC DISTRIBUTION......Page 98
5.5 JOINT RANDOM VARIABLES......Page 99
5.6 CONDITIONAL PROBABILITY FOR JOINT RANDOM VARIABLES......Page 103
BAYESIAN INFERENCE FOR DISCRETE RANDOM VARIABLES......Page 110
6.1 TWO EQUIVALENT WAYS OF USING BAYES’ THEOREM......Page 115
6.2 BAYES’ THEOREM FOR BINOMIAL WITH DISCRETE PRIOR......Page 117
6.3 IMPORTANT CONSEQUENCES OF BAYES’ THEOREM......Page 120
CONTINUOUS RANDOM VARIABLES......Page 125
7.1 PROBABILITY DENSITY FUNCTION......Page 127
7.2 SOME CONTINUOUS DISTRIBUTIONS......Page 130
7.3 JOINT CONTINUOUS RANDOM VARIABLES......Page 136
7.4 JOINT CONTINUOUS AND DISCRETE RANDOM VARIABLES......Page 137
BAYESIAN INFERENCE FOR BINOMIAL PROPORTION......Page 142
8.1 USING A UNIFORM PRIOR......Page 143
8.2 USING A BETA PRIOR......Page 144
8.3 CHOOSING YOUR PRIOR......Page 146
8.4 SUMMARIZING THE POSTERIOR DISTRIBUTION......Page 149
8.5 ESTIMATING THE PROPORTION......Page 152
8.6 BAYESIAN CREDIBLE INTERVAL......Page 153
9.1 FREQUENTIST INTERPRETATION OF PROBABILITY AND PARAMETERS......Page 160
9.2 POINT ESTIMATION......Page 162
9.3 COMPARING ESTIMATORS FOR PROPORTION......Page 164
9.4 INTERVAL ESTIMATION......Page 166
9.5 HYPOTHESIS TESTING......Page 168
9.6 TESTING A ONE-SIDED HYPOTHESIS......Page 170
9.7 TESTING A TWO-SIDED HYPOTHESIS......Page 172
10.1 BAYES’ THEOREM FOR NORMAL MEAN WITH A DISCRETE PRIOR......Page 181
10.2 BAYES’ THEOREM FOR NORMAL MEAN WITH A CONTINUOUS PRIOR......Page 187
10.3 CHOOSING YOUR NORMAL PRIOR......Page 191
10.4 BAYESIAN CREDIBLE INTERVAL FOR NORMAL MEAN......Page 193
10.5 PREDICTIVE DENSITY FOR NEXT OBSERVATION......Page 196
11.1 COMPARING FREQUENTIST AND BAYESIAN POINT ESTIMATORS......Page 205
11.2 COMPARING CONFIDENCE AND CREDIBLE INTERVALS FOR MEAN......Page 208
11.3 TESTING A ONE-SIDED HYPOTHESIS ABOUT A NORMAL MEAN......Page 210
11.4 TESTING A TWO-SIDED HYPOTHESIS ABOUT A NORMAL MEAN......Page 214
BAYESIAN INFERENCE FOR DIFFERENCE BETWEEN MEANS......Page 221
12.2 CASE 1: EQUAL VARIANCES......Page 222
12.3 CASE 2: UNEQUAL VARIANCES......Page 227
12.4 BAYESIAN INFERENCE FOR DIFFERENCE BETWEEN TWO PROPORTIONS USING NORMAL APPROXIMATION......Page 230
12.5 NORMAL RANDOM SAMPLES FROM PAIRED EXPERIMENTS......Page 232
BAYESIAN INFERENCE FOR SIMPLE LINEAR REGRESSION......Page 247
13.1 LEAST SQUARES REGRESSION......Page 248
13.2 EXPONENTIAL GROWTH MODEL......Page 252
13.3 SIMPLE LINEAR REGRESSION ASSUMPTIONS......Page 253
13.4 BAYES’ THEOREM FOR THE REGRESSION MODEL......Page 256
13.5 PREDICTIVE DISTRIBUTION FOR FUTURE OBSERVATION......Page 260
ROBUST BAYESIAN METHODS......Page 273
14.1 EFFECT OF MISSPECIFIED PRIOR......Page 274
14.2 BAYES’ THEOREM WITH MIXTURE PRIORS......Page 275
INTRODUCTION TO CALCULUS......Page 287
USE OF STATISTICAL TABLES......Page 306
USING THE INCLUDED MINITAB MACROS......Page 318
USING THE INCLUDED R FUNCTIONS......Page 327
ANSWERS TO SELECTED EXERCISES......Page 338
References......Page 357
Index......Page 359