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 p i 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 p i =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: Defective FractionProbability.2.30.4.40.6.15.8.10 1..05

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) “Good” FractionProbability 0.05.2.10.4.15.6.40.8.30 1..00

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