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Wiley (2004) Introduction to Bayesian Statistics

'Wiley (2004) Introduction to Bayesian Statistics'
Preface How This Text Was Developed This text grew out of the course notes for an Introduction to Bayesian Statistics course that I have been teaching at the University of Waikato for the past few years. My goal in developing this course was to introduce Bayesian methods at the earliest possible stage, and cover a similar range of topics as a traditional introductory statistics course. There is currently an upsurge in using Bayesian methods in applied statistical analysis, yet the Introduction to Statistics course most students take is almost always taught from a frequentist perspective. In my view, this is not right. Students with a reasonable mathematics background should be exposed to Bayesian methodsfromthebeginning, becausethatisthedirectionappliedstatisticsismoving. Mathematical Background Required Bayesian statistics uses the rules of probability to make inferences, so students must have good algebraic skills for recognizing and manipulating formulas. A general knowledge of calculus would be an advantage in reading this book. In particular, the student should understand that the area under a curve is found by integration, and that the location of a maximum or a minimum of a continuous differentiable function is found by setting the derivative function equal to zero and solving. The book is self-contained with a calculus appendix students can refer to. However, the actual calculus used is minimal. xiii xivPREFACE Features of the Text InthistextIhaveintroducedBayesianmethodsusingastepbystepdevelopmentfrom conditional probability. In Chapter 4, the universe of an experiment is set up with two dimensions, the horizontal dimension is observable, and the vertical dimension is unobservable. Unconditional probabilities are found for each point in the universe using the multiplication rule and the prior probabilities of the unobservable events. Conditionalprobabilityistheprobabilityonthatpartoftheuniversethatoccurred,the reduced universe. It is found by dividing the unconditional probability by their sum over all the possible unobservable events. Because of way the universe is organized, this summing is down the column in the reduced universe. The division scales them up so the conditional probabilities sum to one. This result known as Bayes’ theorem is the key to this course. In Chapter 6 this pattern is repeated with the Bayesian universe. The horizontal dimension is the sample space, the set of all possible values of the observable random variable. The vertical dimension is the parameter space, the set of all possible values of the unobservable parameter. The reduced universe is the vertical slice that we observed.The conditional probabilities given what we observed are the unconditional probabilities found by using the multiplication rule (prior × likelihood) divided by their sum over all possible parameter values. Again, this sum is taken down the column. The division rescales the probabilities so they sum to one.This gives Bayes’ theorem for
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