Bayesian Approach

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