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Introduction to Bayesian Approach: Methodology for Data and Expert Judgment Integration

This document presents a comprehensive overview of the Bayesian approach, a methodology that effectively combines observed data with expert judgment. It introduces the concept of treating all parameters as random variables, illustrated through discrete and continuous cases. The text covers essential terminology, including prior and posterior probabilities, and provides examples such as the proportion of defective concrete piles, demonstrating how to update probabilities using new data. Practical formulas and methods for estimating parameters based on prior and observed information are also discussed.

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Introduction to Bayesian Approach: Methodology for Data and Expert Judgment Integration

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  1. Bayesian Approach Jake Blanchard Fall 2010

  2. Introduction • This is a methodology for combining observed data with expert judgment • Treats all parameters are random variables

  3. Discrete Case • Suppose parameter i has k discrete values • Also, let pi represent the prior relative likelihoods (in a pmf) (based on old information) • If we get new data, we want to modify the pmf to take it into account (systematically)

  4. Terminology • pi=P(= i)=prior relative likelihoods (data available prior to experiment providing ) • =observed outcome • P(= i|)=posterior probability of = I (after incorporating ) • P´(= i)=prior probability • P´´ (= i)=posterior probability • Estimator of parameter  is given by

  5. Useful formulas

  6. Example • Variable is proportion of defective concrete piles • Engineer estimates that probabilities are:

  7. Prior PMF

  8. Find Posterior Probabilities • Engineer orders one additional pile and it is defective • Probabilities must be updated

  9. Posterior PMF

  10. What if next sample had been good? • Switch to p representing good (rather than defective)

  11. Find Posterior Probabilities • Engineer orders one additional pile and it is good • Probabilities must be updated

  12. Continuous Case • Prior pdf=f´()

  13. Example • Defective piles • Assume uniform distribution • Then, single inspection identifies defective pile

  14. Solution

  15. Sampling • Suppose we have a population with a prior standard deviation (´) and mean (´) • Assume we then sample to get sample mean (x)and standard deviation ()

  16. With Prior Information Weighted average of prior mean and sample mean

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