Applied Bayesian Statistics: With R and OpenBUGS Examples: 98 (Springer Texts in Statistics)

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Introduction to Bayesian data analysis - part 1: What is Bayes?

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Chapter 1What Is Bayesian Statistics? Assess the relevant information already available. If yes, draw appropriate conclusions, make appropriate decisions, and take appropriate action. If no, proceed to step 3. Determine what additional information is needed and design a study or experi- ment to attempt to obtain it. Carry out the study designed in step 3. Use the data obtained in step 4 to update what was previously known. Return to step 2.

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Statistics is central to steps 2, 3, and 5. Bayesian statistics is particularly wellsuited to steps 2 and 5, because it provides a quantitative framework for representingcurrent knowledge and for rationally integrating new information. Destroying over 13, houses, 89churches, and dozens of public buildings, the Great Fire led to the rise of insuranceprotection as we understand it today. By the early eighteenth century, the idea of lifeinsurance as well as property insurance was taking hold in England.

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However, lackof adequate vital statistics and of probability theory led to the failure of many earlylife insurers. Enter Thomas Bayes. Born in London in , Bayes became an ordainedPresbyterian minister by profession and a mathematician and scientist by avocation.

He applied his mind to the questions urgently raised by the insurers and laid outhis resulting theory of probability in his Essay towards solving a problem in thedoctrine of chances.

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However, George Boole questioned themin his treatise on logic called Laws of Thought. Until about , the application of Bayesian methods to statistical analysisin real-world problems was very limited because the necessary mathematicalcomputations could be done analytically only for very simple models. Thesubsequent explosion of interest in Bayesian statistics has led not only to extensiveresearch in Bayesian methodology but also to the use of Bayesian methods toaddress pressing questions in diverse application areas such as astrophysics, weatherforecasting, health-care policy, and criminal justice.

Today, Bayesian statistics is widely used to guide learning and decision-makingin business and industry as well as in science. McGrayne b offers a lively introduc-tion to the history and current status of Bayesian statistics. The National Cancer Institute recommends that women aged 40 and above shouldhave mammograms every one to two years. A mammogram produces x-ray imagesof tissues and structure inside the breast and may help detect and identify canceroustumors, benign cysts, and other breast conditions.

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My friend does not have a family history of breast cancer, and beforebeing referred for the screening mammogram, she had given no thought whatsoeverto breast cancer as something that could conceivably happen to her. However, asthe date of the mammogram approached, she began to wonder about her chances ofbeing diagnosed with breast cancer. In other words, she was uncertainabout her status with respect to breast cancer and wanted to learn more about it. Wewill introduce the notion of using probabilities to quantify knowledge or uncertaintyand of using data to update such probabilities in a rational way.

Two different interpretations ofprobability are in common use. For example, Moore , page says: The probability of any outcome of a random phenomenon is the proportion of the times the outcome would occur in a very long series of repetitions. Saying thatthe probability of a fair coin coming up heads is 0. Although this interpretation is useful and the frequentist or classical approachto statistics is based on it , it has serious shortcomings.

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